Irrationality Measures to some constants of the form int( 1/(a+c*x^2),x=0..1) with 1<=a,c <=, 40 By Shalosh B. Ekhad Consider the constant 1 / | 1 C = | -------- dx | 2 / c x + a 0 with a,c, integers. We are interested in the diophantine approximations induced by the sequence 1 / n n | x (1 - x) E(n) = | ----------------- dx | 2 (n + 1) / (c x + a) 0 that can be expressed as E(n)=A(n)+B(n)*C for sequences of rational numbers A(n) and B(n) Using the Almkvist-Zeilberger algorithm it follows that E(n), and hence A(n)\ and B(n) satisfy the linear recurrence equation with polynomial coefficients (n + 1) F(n) - 2 a (2 n + 3) F(n + 1) - 4 a c (n + 2) F(n + 2) = 0 but of course with different initial conditions. It follows from the Poincarre lemma that A(n),B(n)=OMEGA(C1(a,c)^n), E(n)=OMEGA( C2(a,c)^n ) where C1(a,c) is the larger root and C2(a,c) the smaller root of the indicia\ l equation of the constant-coefficient recurrence approximating the above recurrence. That indicial polynomial, in N, happens to be 2 -4 N a c - 4 N a + 1 and in Maple format -4*N^2*a*c-4*N*a+1 Setting it equal to 0, and solving the quadratic equation, we get that the l\ arger root, C1(a,c) is 2 1/2 -a + (a + a c) ------------------ 2 a c and the smaller root, C2(a,c) is 2 1/2 a + (a + a c) - ----------------- 2 a c and in Maple format, they are respectively, 1/2*(-a+(a^2+a*c)^(1/2))/a/c, -1/2*(a+(a^2+a*c)^(1/2))/a/c Suppose that we discover that there exists a constant K (that is either an i\ nteger or a square-root of an integer) such that A1(n)=lcm(1..n)*K^n*A(n) and B1(n)=lcm(1..n)*K^n*B(n) are all integers. (if K is a square-root of an integer only do it for even n .) Let E1(n)=lcm(1..n)*K^n*E(n) , then E1(n)=A1(n)+B1(n)*c where now A1(n) and B1(n) are INTEGERS. Since lcm(1..n)=OMEGA(e^n), we have A1(n),B1(n)=OMEGA(K^n*e*C1(a,c))^n ), E(n)=OMEGA( (K^n*e*C2(a,c))^n ) Hence E(n)=OMEGA(1/max(A(n),B(n))^delta ), where delta is -log(K*e*C2(a,c))/log(K*e*C1(a,c)) | 2 1/2 | | -a + (a + a c) | ln(1/2 | ------------------ |) + ln(K) + 1 | a c | that equals , - ------------------------------------------ | 2 1/2 | | a + (a + a c) | ln(1/2 | ----------------- |) + ln(K) + 1 | a c | and in Maple format it is -(ln(1/2*abs((-a+(a^2+a*c)^(1/2))/a/c))+ln(K)+1)/(ln(1/2*abs((a+(a^2+a*c)^(1/2))/a/c))+ln(K)+1) So it is good news whenever this is positive. By searching for quadratic polynomials P with positive coefficients <=, 40, we found the following list of , 42, lucky cases. For each of the lists below The first entry is the polynomial in x, P(x), a certain irreducible quadrat\ ic. the second entry is the constant whose irrationality we are claiming , namel\ y the integral of 1/P(x) from x=0 to x=1. The third entry is the magic constant K, mentioned above such that multiplyi\ ng by K^n*lcm(1...n) makes everything integers. the fourth is the exact value of delta, the fifth, and last, entry is the implied irrationality measure, i.e. 1+1/delta, in floating-point. | 1/2 | | 12 | 1/2 1/2 ln(1/2 | -1 + ----- |) + ln(2 3 ) + 1 2 Pi 3 1/2 | 3 | [x + 3, -------, 2 3 , - ---------------------------------------, 18 | 1/2 | | 12 | 1/2 ln(1/2 | 1 + ----- |) + ln(2 3 ) + 1 | 3 | 8.3099863401554735212] 1/2 2 1/2 5 1/2 [x + 5, 1/5 5 arctan(----), 2 10 , 5 | 1/2 | | 30 | 1/2 ln(1/2 | -1 + ----- |) + ln(2 10 ) + 1 | 5 | - ----------------------------------------, 15.607845206093606137] | 1/2 | | 30 | 1/2 ln(1/2 | 1 + ----- |) + ln(2 10 ) + 1 | 5 | 1/2 2 1/2 7 1/2 [x + 7, 1/7 7 arctan(----), 2 7 , 7 | 1/2 | | 56 | 1/2 ln(1/2 | -1 + ----- |) + ln(2 7 ) + 1 | 7 | - ---------------------------------------, 4.8569705938289177865] | 1/2 | | 56 | 1/2 ln(1/2 | 1 + ----- |) + ln(2 7 ) + 1 | 7 | 1/2 2 1/2 2 1/2 [x + 8, 1/4 2 arctan(----), 8 2 , 4 | 1/2 | | 72 | 1/2 ln(1/2 | -1 + ----- |) + ln(8 2 ) + 1 | 8 | - ---------------------------------------, 50.653659172177507342] | 1/2 | | 72 | 1/2 ln(1/2 | 1 + ----- |) + ln(8 2 ) + 1 | 8 | 1/2 2 1/2 2 10 1/2 [x + 10, 1/20 10 arctan(-------), 4 10 , 9 | 1/2 | | 110 | 1/2 ln(1/2 | -1 + ------ |) + ln(4 10 ) + 1 | 10 | - -----------------------------------------, 21.305691483920764832] | 1/2 | | 110 | 1/2 ln(1/2 | 1 + ------ |) + ln(4 10 ) + 1 | 10 | 1/2 2 1/2 11 1/2 [x + 11, 1/22 11 arctan(-----), 2 11 , 5 | 1/2 | | 132 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 11 ) + 1 | 11 | - -----------------------------------------, 4.1879821595890442019] | 1/2 | | 132 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 11 ) + 1 | 11 | 1/2 2 1/2 3 1/2 [x + 12, 1/6 3 arctan(----), 8 3 , 6 | 1/2 | | 156 | 1/2 ln(1/2 | -1 + ------ |) + ln(8 3 ) + 1 | 12 | - ----------------------------------------, 14.892104575824395580] | 1/2 | | 156 | 1/2 ln(1/2 | 1 + ------ |) + ln(8 3 ) + 1 | 12 | 1/2 2 1/2 13 1/2 [x + 13, 1/26 13 arctan(-----), 2 26 , 6 | 1/2 | | 182 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 26 ) + 1 | 13 | - -----------------------------------------, 6.1577154229894323982] | 1/2 | | 182 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 26 ) + 1 | 13 | 1/2 2 1/2 14 1/2 [x + 14, 1/14 14 arctan(-----), 4 14 , 14 | 1/2 | | 210 | 1/2 ln(1/2 | -1 + ------ |) + ln(4 14 ) + 1 | 14 | - -----------------------------------------, 12.050569329465964540] | 1/2 | | 210 | 1/2 ln(1/2 | 1 + ------ |) + ln(4 14 ) + 1 | 14 | 1/2 2 1/2 15 1/2 [x + 15, 1/30 15 arctan(-----), 2 15 , 7 | 1/2 | | 240 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 15 ) + 1 | 15 | - -----------------------------------------, 3.8806942681404861578] | 1/2 | | 240 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 15 ) + 1 | 15 | | 1/2 | | 272 | ln(1/2 | -1 + ------ |) + ln(16) + 1 2 | 16 | [x + 16, 1/4 arctan(1/4), 16, - ------------------------------------, | 1/2 | | 272 | ln(1/2 | 1 + ------ |) + ln(16) + 1 | 16 | 10.432735096002347775] 1/2 2 1/2 1/2 51 Pi 1/2 [3 x + 17, -1/204 51 arctan(7 51 ) + --------, 6 17 , 204 | 1/2 | | 340 | 1/2 ln(1/2 | - 1/3 + ------ |) + ln(6 17 ) + 1 | 51 | - --------------------------------------------, 60.912947227427311284] | 1/2 | | 340 | 1/2 ln(1/2 | 1/3 + ------ |) + ln(6 17 ) + 1 | 51 | 1/2 2 1/2 2 1/2 [x + 18, 1/6 2 arctan(----), 12 2 , 6 | 1/2 | | 342 | 1/2 ln(1/2 | -1 + ------ |) + ln(12 2 ) + 1 | 18 | - -----------------------------------------, 9.3808596317363403341] | 1/2 | | 342 | 1/2 ln(1/2 | 1 + ------ |) + ln(12 2 ) + 1 | 18 | 1/2 2 1/2 19 1/2 [x + 19, 1/38 19 arctan(-----), 2 19 , 9 | 1/2 | | 380 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 19 ) + 1 | 19 | - -----------------------------------------, 3.6974006367249319472] | 1/2 | | 380 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 19 ) + 1 | 19 | 1/2 2 1/2 5 1/2 [x + 20, 1/10 5 arctan(----), 8 5 , 10 | 1/2 | | 420 | 1/2 ln(1/2 | -1 + ------ |) + ln(8 5 ) + 1 | 20 | - ----------------------------------------, 8.6379958426045502068] | 1/2 | | 420 | 1/2 ln(1/2 | 1 + ------ |) + ln(8 5 ) + 1 | 20 | 1/2 2 1/2 21 1/2 [x + 21, 1/21 21 arctan(-----), 2 42 , 21 | 1/2 | | 462 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 42 ) + 1 | 21 | - -----------------------------------------, 5.0585439651656041409] | 1/2 | | 462 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 42 ) + 1 | 21 | 1/2 2 1/2 22 1/2 [x + 22, 1/22 22 arctan(-----), 4 22 , 22 | 1/2 | | 506 | 1/2 ln(1/2 | -1 + ------ |) + ln(4 22 ) + 1 | 22 | - -----------------------------------------, 8.0828439211443457466] | 1/2 | | 506 | 1/2 ln(1/2 | 1 + ------ |) + ln(4 22 ) + 1 | 22 | 1/2 2 1/2 23 1/2 [x + 23, 1/23 23 arctan(-----), 2 23 , 23 | 1/2 | | 552 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 23 ) + 1 | 23 | - -----------------------------------------, 3.5728330278283835129] | 1/2 | | 552 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 23 ) + 1 | 23 | 1/2 2 1/2 6 1/2 [x + 24, 1/12 6 arctan(----), 8 6 , 12 | 1/2 | | 600 | 1/2 ln(1/2 | -1 + ------ |) + ln(8 6 ) + 1 | 24 | - ----------------------------------------, 7.6505623343655853958] | 1/2 | | 600 | 1/2 ln(1/2 | 1 + ------ |) + ln(8 6 ) + 1 | 24 | 1/2 2 1/2 3 [3 x + 25, 1/15 3 arctan(----), 30, 5 | 1/2 | | 700 | ln(1/2 | - 1/3 + ------ |) + ln(30) + 1 | 75 | - ---------------------------------------, 15.315550900340627733] | 1/2 | | 700 | ln(1/2 | 1/3 + ------ |) + ln(30) + 1 | 75 | 1/2 2 1/2 26 1/2 [x + 26, 1/26 26 arctan(-----), 4 26 , 26 | 1/2 | | 702 | 1/2 ln(1/2 | -1 + ------ |) + ln(4 26 ) + 1 | 26 | - -----------------------------------------, 7.3032836456607766307] | 1/2 | | 702 | 1/2 ln(1/2 | 1 + ------ |) + ln(4 26 ) + 1 | 26 | 1/2 2 1/2 3 1/2 [x + 27, 1/9 3 arctan(----), 6 3 , 9 | 1/2 | | 756 | 1/2 ln(1/2 | -1 + ------ |) + ln(6 3 ) + 1 | 27 | - ----------------------------------------, 3.4812632908955237133] | 1/2 | | 756 | 1/2 ln(1/2 | 1 + ------ |) + ln(6 3 ) + 1 | 27 | 1/2 2 1/2 7 1/2 [x + 28, 1/14 7 arctan(----), 8 7 , 14 | 1/2 | | 812 | 1/2 ln(1/2 | -1 + ------ |) + ln(8 7 ) + 1 | 28 | - ----------------------------------------, 7.0173748648955734301] | 1/2 | | 812 | 1/2 ln(1/2 | 1 + ------ |) + ln(8 7 ) + 1 | 28 | 1/2 2 1/2 29 1/2 [x + 29, 1/29 29 arctan(-----), 2 58 , 29 | 1/2 | | 870 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 58 ) + 1 | 29 | - -----------------------------------------, 4.5927237087173453573] | 1/2 | | 870 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 58 ) + 1 | 29 | 1/2 2 1/2 58 1/2 [2 x + 29, 1/58 58 arctan(-----), 8 29 , 29 | 1/2 | | 899 | 1/2 ln(1/2 | - 1/2 + ------ |) + ln(8 29 ) + 1 | 58 | - --------------------------------------------, 559.86750501368578137] | 1/2 | | 899 | 1/2 ln(1/2 | 1/2 + ------ |) + ln(8 29 ) + 1 | 58 | 1/2 2 1/2 87 1/2 [3 x + 29, 1/87 87 arctan(-----), 6 29 , 29 | 1/2 | | 928 | 1/2 ln(1/2 | - 1/3 + ------ |) + ln(6 29 ) + 1 | 87 | - --------------------------------------------, 12.223216944561103845] | 1/2 | | 928 | 1/2 ln(1/2 | 1/3 + ------ |) + ln(6 29 ) + 1 | 87 | 1/2 2 1/2 30 1/2 [x + 30, 1/30 30 arctan(-----), 4 30 , 30 | 1/2 | | 930 | 1/2 ln(1/2 | -1 + ------ |) + ln(4 30 ) + 1 | 30 | - -----------------------------------------, 6.7773028482094062475] | 1/2 | | 930 | 1/2 ln(1/2 | 1 + ------ |) + ln(4 30 ) + 1 | 30 | 1/2 2 1/2 31 1/2 [x + 31, 1/31 31 arctan(-----), 2 31 , 31 | 1/2 | | 992 | 1/2 ln(1/2 | -1 + ------ |) + ln(2 31 ) + 1 | 31 | - -----------------------------------------, 3.4103283713008726854] | 1/2 | | 992 | 1/2 ln(1/2 | 1 + ------ |) + ln(2 31 ) + 1 | 31 | 1/2 2 1/2 62 1/2 [2 x + 31, 1/62 62 arctan(-----), 8 31 , 31 | 1/2 | | 1023 | 1/2 ln(1/2 | - 1/2 + ------- |) + ln(8 31 ) + 1 | 62 | - ---------------------------------------------, 105.01969749473874703] | 1/2 | | 1023 | 1/2 ln(1/2 | 1/2 + ------- |) + ln(8 31 ) + 1 | 62 | 1/2 2 1/2 2 1/2 [x + 32, 1/8 2 arctan(----), 16 2 , 8 | 1/2 | | 1056 | 1/2 ln(1/2 | -1 + ------- |) + ln(16 2 ) + 1 | 32 | - ------------------------------------------, 6.5724250597283506285] | 1/2 | | 1056 | 1/2 ln(1/2 | 1 + ------- |) + ln(16 2 ) + 1 | 32 | 1/2 2 1/2 66 1/2 [2 x + 33, 1/66 66 arctan(-----), 8 33 , 33 | 1/2 | | 1155 | 1/2 ln(1/2 | - 1/2 + ------- |) + ln(8 33 ) + 1 | 66 | - ---------------------------------------------, 60.340314492883854179] | 1/2 | | 1155 | 1/2 ln(1/2 | 1/2 + ------- |) + ln(8 33 ) + 1 | 66 | 1/2 2 1/2 34 1/2 [x + 34, 1/34 34 arctan(-----), 4 34 , 34 | 1/2 | | 1190 | 1/2 ln(1/2 | -1 + ------- |) + ln(4 34 ) + 1 | 34 | - ------------------------------------------, 6.3951968273628847107] | 1/2 | | 1190 | 1/2 ln(1/2 | 1 + ------- |) + ln(4 34 ) + 1 | 34 | 1/2 2 1/2 35 1/2 [x + 35, 1/35 35 arctan(-----), 2 35 , 35 | 1/2 | | 1260 | 1/2 ln(1/2 | -1 + ------- |) + ln(2 35 ) + 1 | 35 | - ------------------------------------------, 3.3532818533703900760] | 1/2 | | 1260 | 1/2 ln(1/2 | 1 + ------- |) + ln(2 35 ) + 1 | 35 | 1/2 2 1/2 70 1/2 [2 x + 35, 1/70 70 arctan(-----), 8 35 , 35 | 1/2 | | 1295 | 1/2 ln(1/2 | - 1/2 + ------- |) + ln(8 35 ) + 1 | 70 | - ---------------------------------------------, 43.409003654012769933] | 1/2 | | 1295 | 1/2 ln(1/2 | 1/2 + ------- |) + ln(8 35 ) + 1 | 70 | | 1/2 | | 1332 | ln(1/2 | -1 + ------- |) + ln(24) + 1 2 | 36 | [x + 36, 1/6 arctan(1/6), 24, - -------------------------------------, | 1/2 | | 1332 | ln(1/2 | 1 + ------- |) + ln(24) + 1 | 36 | 6.2401149858787923537] 1/2 2 1/2 37 1/2 [x + 37, 1/37 37 arctan(-----), 2 74 , 37 | 1/2 | | 1406 | 1/2 ln(1/2 | -1 + ------- |) + ln(2 74 ) + 1 | 37 | - ------------------------------------------, 4.3242396092923923197] | 1/2 | | 1406 | 1/2 ln(1/2 | 1 + ------- |) + ln(2 74 ) + 1 | 37 | 1/2 2 1/2 74 1/2 [2 x + 37, 1/74 74 arctan(-----), 8 37 , 37 | 1/2 | | 1443 | 1/2 ln(1/2 | - 1/2 + ------- |) + ln(8 37 ) + 1 | 74 | - ---------------------------------------------, 34.490155336877812389] | 1/2 | | 1443 | 1/2 ln(1/2 | 1/2 + ------- |) + ln(8 37 ) + 1 | 74 | 1/2 2 1/2 111 1/2 [3 x + 37, 1/111 111 arctan(------), 6 37 , 37 | 1/2 | | 1480 | 1/2 ln(1/2 | - 1/3 + ------- |) + ln(6 37 ) + 1 | 111 | - ---------------------------------------------, 9.3833764155978820300] | 1/2 | | 1480 | 1/2 ln(1/2 | 1/3 + ------- |) + ln(6 37 ) + 1 | 111 | 1/2 2 1/2 38 1/2 [x + 38, 1/38 38 arctan(-----), 4 38 , 38 | 1/2 | | 1482 | 1/2 ln(1/2 | -1 + ------- |) + ln(4 38 ) + 1 | 38 | - ------------------------------------------, 6.1030672323035062852] | 1/2 | | 1482 | 1/2 ln(1/2 | 1 + ------- |) + ln(4 38 ) + 1 | 38 | 1/2 2 1/2 39 1/2 [x + 39, 1/39 39 arctan(-----), 2 39 , 39 | 1/2 | | 1560 | 1/2 ln(1/2 | -1 + ------- |) + ln(2 39 ) + 1 | 39 | - ------------------------------------------, 3.3060990040496475222] | 1/2 | | 1560 | 1/2 ln(1/2 | 1 + ------- |) + ln(2 39 ) + 1 | 39 | 1/2 2 1/2 78 1/2 [2 x + 39, 1/78 78 arctan(-----), 8 39 , 39 | 1/2 | | 1599 | 1/2 ln(1/2 | - 1/2 + ------- |) + ln(8 39 ) + 1 | 78 | - ---------------------------------------------, 28.977863588964496869] | 1/2 | | 1599 | 1/2 ln(1/2 | 1/2 + ------- |) + ln(8 39 ) + 1 | 78 | 1/2 2 1/2 10 1/2 [x + 40, 1/20 10 arctan(-----), 8 10 , 20 | 1/2 | | 1640 | 1/2 ln(1/2 | -1 + ------- |) + ln(8 10 ) + 1 | 40 | - ------------------------------------------, 5.9809160998957014118] | 1/2 | | 1640 | 1/2 ln(1/2 | 1 + ------- |) + ln(8 10 ) + 1 | 40 | and in Maple format [x^2+3, 1/18*Pi*3^(1/2), 2*3^(1/2), -(ln(1/2*abs(-1+1/3*12^(1/2)))+ln(2*3^(1/2) )+1)/(ln(1/2*abs(1+1/3*12^(1/2)))+ln(2*3^(1/2))+1), 8.3099863401554735212] [x^2+5, 1/5*5^(1/2)*arctan(1/5*5^(1/2)), 2*10^(1/2), -(ln(1/2*abs(-1+1/5*30^(1/ 2)))+ln(2*10^(1/2))+1)/(ln(1/2*abs(1+1/5*30^(1/2)))+ln(2*10^(1/2))+1), 15.60784\ 5206093606137] [x^2+7, 1/7*7^(1/2)*arctan(1/7*7^(1/2)), 2*7^(1/2), -(ln(1/2*abs(-1+1/7*56^(1/2 )))+ln(2*7^(1/2))+1)/(ln(1/2*abs(1+1/7*56^(1/2)))+ln(2*7^(1/2))+1), 4.856970593\ 8289177865] [x^2+8, 1/4*2^(1/2)*arctan(1/4*2^(1/2)), 8*2^(1/2), -(ln(1/2*abs(-1+1/8*72^(1/2 )))+ln(8*2^(1/2))+1)/(ln(1/2*abs(1+1/8*72^(1/2)))+ln(8*2^(1/2))+1), 50.65365917\ 2177507342] [x^2+10, 1/20*10^(1/2)*arctan(2/9*10^(1/2)), 4*10^(1/2), -(ln(1/2*abs(-1+1/10* 110^(1/2)))+ln(4*10^(1/2))+1)/(ln(1/2*abs(1+1/10*110^(1/2)))+ln(4*10^(1/2))+1), 21.305691483920764832] [x^2+11, 1/22*11^(1/2)*arctan(1/5*11^(1/2)), 2*11^(1/2), -(ln(1/2*abs(-1+1/11* 132^(1/2)))+ln(2*11^(1/2))+1)/(ln(1/2*abs(1+1/11*132^(1/2)))+ln(2*11^(1/2))+1), 4.1879821595890442019] [x^2+12, 1/6*3^(1/2)*arctan(1/6*3^(1/2)), 8*3^(1/2), -(ln(1/2*abs(-1+1/12*156^( 1/2)))+ln(8*3^(1/2))+1)/(ln(1/2*abs(1+1/12*156^(1/2)))+ln(8*3^(1/2))+1), 14.892\ 104575824395580] [x^2+13, 1/26*13^(1/2)*arctan(1/6*13^(1/2)), 2*26^(1/2), -(ln(1/2*abs(-1+1/13* 182^(1/2)))+ln(2*26^(1/2))+1)/(ln(1/2*abs(1+1/13*182^(1/2)))+ln(2*26^(1/2))+1), 6.1577154229894323982] [x^2+14, 1/14*14^(1/2)*arctan(1/14*14^(1/2)), 4*14^(1/2), -(ln(1/2*abs(-1+1/14* 210^(1/2)))+ln(4*14^(1/2))+1)/(ln(1/2*abs(1+1/14*210^(1/2)))+ln(4*14^(1/2))+1), 12.050569329465964540] [x^2+15, 1/30*15^(1/2)*arctan(1/7*15^(1/2)), 2*15^(1/2), -(ln(1/2*abs(-1+1/15* 240^(1/2)))+ln(2*15^(1/2))+1)/(ln(1/2*abs(1+1/15*240^(1/2)))+ln(2*15^(1/2))+1), 3.8806942681404861578] [x^2+16, 1/4*arctan(1/4), 16, -(ln(1/2*abs(-1+1/16*272^(1/2)))+ln(16)+1)/(ln(1/ 2*abs(1+1/16*272^(1/2)))+ln(16)+1), 10.432735096002347775] [3*x^2+17, -1/204*51^(1/2)*arctan(7*51^(1/2))+1/204*51^(1/2)*Pi, 6*17^(1/2), -( ln(1/2*abs(-1/3+1/51*340^(1/2)))+ln(6*17^(1/2))+1)/(ln(1/2*abs(1/3+1/51*340^(1/ 2)))+ln(6*17^(1/2))+1), 60.912947227427311284] [x^2+18, 1/6*2^(1/2)*arctan(1/6*2^(1/2)), 12*2^(1/2), -(ln(1/2*abs(-1+1/18*342^ (1/2)))+ln(12*2^(1/2))+1)/(ln(1/2*abs(1+1/18*342^(1/2)))+ln(12*2^(1/2))+1), 9.3\ 808596317363403341] [x^2+19, 1/38*19^(1/2)*arctan(1/9*19^(1/2)), 2*19^(1/2), -(ln(1/2*abs(-1+1/19* 380^(1/2)))+ln(2*19^(1/2))+1)/(ln(1/2*abs(1+1/19*380^(1/2)))+ln(2*19^(1/2))+1), 3.6974006367249319472] [x^2+20, 1/10*5^(1/2)*arctan(1/10*5^(1/2)), 8*5^(1/2), -(ln(1/2*abs(-1+1/20*420 ^(1/2)))+ln(8*5^(1/2))+1)/(ln(1/2*abs(1+1/20*420^(1/2)))+ln(8*5^(1/2))+1), 8.63\ 79958426045502068] [x^2+21, 1/21*21^(1/2)*arctan(1/21*21^(1/2)), 2*42^(1/2), -(ln(1/2*abs(-1+1/21* 462^(1/2)))+ln(2*42^(1/2))+1)/(ln(1/2*abs(1+1/21*462^(1/2)))+ln(2*42^(1/2))+1), 5.0585439651656041409] [x^2+22, 1/22*22^(1/2)*arctan(1/22*22^(1/2)), 4*22^(1/2), -(ln(1/2*abs(-1+1/22* 506^(1/2)))+ln(4*22^(1/2))+1)/(ln(1/2*abs(1+1/22*506^(1/2)))+ln(4*22^(1/2))+1), 8.0828439211443457466] [x^2+23, 1/23*23^(1/2)*arctan(1/23*23^(1/2)), 2*23^(1/2), -(ln(1/2*abs(-1+1/23* 552^(1/2)))+ln(2*23^(1/2))+1)/(ln(1/2*abs(1+1/23*552^(1/2)))+ln(2*23^(1/2))+1), 3.5728330278283835129] [x^2+24, 1/12*6^(1/2)*arctan(1/12*6^(1/2)), 8*6^(1/2), -(ln(1/2*abs(-1+1/24*600 ^(1/2)))+ln(8*6^(1/2))+1)/(ln(1/2*abs(1+1/24*600^(1/2)))+ln(8*6^(1/2))+1), 7.65\ 05623343655853958] [3*x^2+25, 1/15*3^(1/2)*arctan(1/5*3^(1/2)), 30, -(ln(1/2*abs(-1/3+1/75*700^(1/ 2)))+ln(30)+1)/(ln(1/2*abs(1/3+1/75*700^(1/2)))+ln(30)+1), 15.31555090034062773\ 3] [x^2+26, 1/26*26^(1/2)*arctan(1/26*26^(1/2)), 4*26^(1/2), -(ln(1/2*abs(-1+1/26* 702^(1/2)))+ln(4*26^(1/2))+1)/(ln(1/2*abs(1+1/26*702^(1/2)))+ln(4*26^(1/2))+1), 7.3032836456607766307] [x^2+27, 1/9*3^(1/2)*arctan(1/9*3^(1/2)), 6*3^(1/2), -(ln(1/2*abs(-1+1/27*756^( 1/2)))+ln(6*3^(1/2))+1)/(ln(1/2*abs(1+1/27*756^(1/2)))+ln(6*3^(1/2))+1), 3.4812\ 632908955237133] [x^2+28, 1/14*7^(1/2)*arctan(1/14*7^(1/2)), 8*7^(1/2), -(ln(1/2*abs(-1+1/28*812 ^(1/2)))+ln(8*7^(1/2))+1)/(ln(1/2*abs(1+1/28*812^(1/2)))+ln(8*7^(1/2))+1), 7.01\ 73748648955734301] [x^2+29, 1/29*29^(1/2)*arctan(1/29*29^(1/2)), 2*58^(1/2), -(ln(1/2*abs(-1+1/29* 870^(1/2)))+ln(2*58^(1/2))+1)/(ln(1/2*abs(1+1/29*870^(1/2)))+ln(2*58^(1/2))+1), 4.5927237087173453573] [2*x^2+29, 1/58*58^(1/2)*arctan(1/29*58^(1/2)), 8*29^(1/2), -(ln(1/2*abs(-1/2+1 /58*899^(1/2)))+ln(8*29^(1/2))+1)/(ln(1/2*abs(1/2+1/58*899^(1/2)))+ln(8*29^(1/2 ))+1), 559.86750501368578137] [3*x^2+29, 1/87*87^(1/2)*arctan(1/29*87^(1/2)), 6*29^(1/2), -(ln(1/2*abs(-1/3+1 /87*928^(1/2)))+ln(6*29^(1/2))+1)/(ln(1/2*abs(1/3+1/87*928^(1/2)))+ln(6*29^(1/2 ))+1), 12.223216944561103845] [x^2+30, 1/30*30^(1/2)*arctan(1/30*30^(1/2)), 4*30^(1/2), -(ln(1/2*abs(-1+1/30* 930^(1/2)))+ln(4*30^(1/2))+1)/(ln(1/2*abs(1+1/30*930^(1/2)))+ln(4*30^(1/2))+1), 6.7773028482094062475] [x^2+31, 1/31*31^(1/2)*arctan(1/31*31^(1/2)), 2*31^(1/2), -(ln(1/2*abs(-1+1/31* 992^(1/2)))+ln(2*31^(1/2))+1)/(ln(1/2*abs(1+1/31*992^(1/2)))+ln(2*31^(1/2))+1), 3.4103283713008726854] [2*x^2+31, 1/62*62^(1/2)*arctan(1/31*62^(1/2)), 8*31^(1/2), -(ln(1/2*abs(-1/2+1 /62*1023^(1/2)))+ln(8*31^(1/2))+1)/(ln(1/2*abs(1/2+1/62*1023^(1/2)))+ln(8*31^(1 /2))+1), 105.01969749473874703] [x^2+32, 1/8*2^(1/2)*arctan(1/8*2^(1/2)), 16*2^(1/2), -(ln(1/2*abs(-1+1/32*1056 ^(1/2)))+ln(16*2^(1/2))+1)/(ln(1/2*abs(1+1/32*1056^(1/2)))+ln(16*2^(1/2))+1), 6\ .5724250597283506285] [2*x^2+33, 1/66*66^(1/2)*arctan(1/33*66^(1/2)), 8*33^(1/2), -(ln(1/2*abs(-1/2+1 /66*1155^(1/2)))+ln(8*33^(1/2))+1)/(ln(1/2*abs(1/2+1/66*1155^(1/2)))+ln(8*33^(1 /2))+1), 60.340314492883854179] [x^2+34, 1/34*34^(1/2)*arctan(1/34*34^(1/2)), 4*34^(1/2), -(ln(1/2*abs(-1+1/34* 1190^(1/2)))+ln(4*34^(1/2))+1)/(ln(1/2*abs(1+1/34*1190^(1/2)))+ln(4*34^(1/2))+1 ), 6.3951968273628847107] [x^2+35, 1/35*35^(1/2)*arctan(1/35*35^(1/2)), 2*35^(1/2), -(ln(1/2*abs(-1+1/35* 1260^(1/2)))+ln(2*35^(1/2))+1)/(ln(1/2*abs(1+1/35*1260^(1/2)))+ln(2*35^(1/2))+1 ), 3.3532818533703900760] [2*x^2+35, 1/70*70^(1/2)*arctan(1/35*70^(1/2)), 8*35^(1/2), -(ln(1/2*abs(-1/2+1 /70*1295^(1/2)))+ln(8*35^(1/2))+1)/(ln(1/2*abs(1/2+1/70*1295^(1/2)))+ln(8*35^(1 /2))+1), 43.409003654012769933] [x^2+36, 1/6*arctan(1/6), 24, -(ln(1/2*abs(-1+1/36*1332^(1/2)))+ln(24)+1)/(ln(1 /2*abs(1+1/36*1332^(1/2)))+ln(24)+1), 6.2401149858787923537] [x^2+37, 1/37*37^(1/2)*arctan(1/37*37^(1/2)), 2*74^(1/2), -(ln(1/2*abs(-1+1/37* 1406^(1/2)))+ln(2*74^(1/2))+1)/(ln(1/2*abs(1+1/37*1406^(1/2)))+ln(2*74^(1/2))+1 ), 4.3242396092923923197] [2*x^2+37, 1/74*74^(1/2)*arctan(1/37*74^(1/2)), 8*37^(1/2), -(ln(1/2*abs(-1/2+1 /74*1443^(1/2)))+ln(8*37^(1/2))+1)/(ln(1/2*abs(1/2+1/74*1443^(1/2)))+ln(8*37^(1 /2))+1), 34.490155336877812389] [3*x^2+37, 1/111*111^(1/2)*arctan(1/37*111^(1/2)), 6*37^(1/2), -(ln(1/2*abs(-1/ 3+1/111*1480^(1/2)))+ln(6*37^(1/2))+1)/(ln(1/2*abs(1/3+1/111*1480^(1/2)))+ln(6* 37^(1/2))+1), 9.3833764155978820300] [x^2+38, 1/38*38^(1/2)*arctan(1/38*38^(1/2)), 4*38^(1/2), -(ln(1/2*abs(-1+1/38* 1482^(1/2)))+ln(4*38^(1/2))+1)/(ln(1/2*abs(1+1/38*1482^(1/2)))+ln(4*38^(1/2))+1 ), 6.1030672323035062852] [x^2+39, 1/39*39^(1/2)*arctan(1/39*39^(1/2)), 2*39^(1/2), -(ln(1/2*abs(-1+1/39* 1560^(1/2)))+ln(2*39^(1/2))+1)/(ln(1/2*abs(1+1/39*1560^(1/2)))+ln(2*39^(1/2))+1 ), 3.3060990040496475222] [2*x^2+39, 1/78*78^(1/2)*arctan(1/39*78^(1/2)), 8*39^(1/2), -(ln(1/2*abs(-1/2+1 /78*1599^(1/2)))+ln(8*39^(1/2))+1)/(ln(1/2*abs(1/2+1/78*1599^(1/2)))+ln(8*39^(1 /2))+1), 28.977863588964496869] [x^2+40, 1/20*10^(1/2)*arctan(1/20*10^(1/2)), 8*10^(1/2), -(ln(1/2*abs(-1+1/40* 1640^(1/2)))+ln(8*10^(1/2))+1)/(ln(1/2*abs(1+1/40*1640^(1/2)))+ln(8*10^(1/2))+1 ), 5.9809160998957014118] -------------------------- This took, 15847.050, second.