Integrand size = 17, antiderivative size = 300 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=-\frac {207+29 \left (1+\frac {4}{x}\right )^2}{336 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}+\frac {5 \left (5157+199 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right )}{87696 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right )}-\frac {17 \arctan \left (\frac {3-\left (1+\frac {4}{x}\right )^2}{6 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \arctan \left (\frac {2+\sqrt {6 \left (1+\sqrt {29}\right )}+\frac {8}{x}}{\sqrt {6 \left (-1+\sqrt {29}\right )}}\right )}{87696}-\frac {\sqrt {\frac {180983329+45923327 \sqrt {29}}{1218}} \arctan \left (\frac {8+\left (2-\sqrt {6 \left (1+\sqrt {29}\right )}\right ) x}{\sqrt {6 \left (-1+\sqrt {29}\right )} x}\right )}{87696}+\frac {\sqrt {\frac {-180983329+45923327 \sqrt {29}}{1218}} \text {arctanh}\left (\frac {\sqrt {6 \left (1+\sqrt {29}\right )} \left (1+\frac {4}{x}\right )}{3 \sqrt {29}+\left (1+\frac {4}{x}\right )^2}\right )}{87696} \] Output:
-1/336*(207+29*(1+4/x)^2)/(261-6*(1+4/x)^2+(1+4/x)^4)+5*(5157+199*(1+4/x)^ 2)*(1+4/x)/(22888656-526176*(1+4/x)^2+87696*(1+4/x)^4)-17/7056*arctan(1/42 *(3-(1+4/x)^2)*7^(1/2))*7^(1/2)-1/106813728*(220437694722+55934612286*29^( 1/2))^(1/2)*arctan((2+(6+6*29^(1/2))^(1/2)+8/x)/(-6+6*29^(1/2))^(1/2))-1/1 06813728*(220437694722+55934612286*29^(1/2))^(1/2)*arctan((8+(2-(6+6*29^(1 /2))^(1/2))*x)/(-6+6*29^(1/2))^(1/2)/x)+1/106813728*(-220437694722+5593461 2286*29^(1/2))^(1/2)*arctanh((6+6*29^(1/2))^(1/2)*(1+4/x)/(3*29^(1/2)+(1+4 /x)^2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\frac {544+1539 x-1146 x^2+784 x^3}{43848 \left (8+8 x-x^3+8 x^4\right )}+\frac {\text {RootSum}\left [8+8 \text {$\#$1}-\text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {2243 \log (x-\text {$\#$1})-1097 \log (x-\text {$\#$1}) \text {$\#$1}+392 \log (x-\text {$\#$1}) \text {$\#$1}^2}{8-3 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ]}{21924} \] Input:
Integrate[(8 + 8*x - x^3 + 8*x^4)^(-2),x]
Output:
(544 + 1539*x - 1146*x^2 + 784*x^3)/(43848*(8 + 8*x - x^3 + 8*x^4)) + Root Sum[8 + 8*#1 - #1^3 + 8*#1^4 & , (2243*Log[x - #1] - 1097*Log[x - #1]*#1 + 392*Log[x - #1]*#1^2)/(8 - 3*#1^2 + 32*#1^3) & ]/21924
Time = 1.38 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.44, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.059, Rules used = {2504, 27, 2202, 2194, 27, 2191, 27, 1083, 217, 2206, 27, 1483, 27, 1142, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (8 x^4-x^3+8 x+8\right )^2} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1024 \int \frac {\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^6}{4096 \left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{4} \int \frac {\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^6}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \frac {1}{4} \left (-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )-\int \frac {\left (-6144 \left (\frac {1}{4}+\frac {1}{x}\right )^4-1280 \left (\frac {1}{4}+\frac {1}{x}\right )^2-24\right ) \left (\frac {1}{4}+\frac {1}{x}\right )}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int -\frac {8 \left (768 \left (\frac {1}{4}+\frac {1}{x}\right )^4+160 \left (\frac {1}{4}+\frac {1}{x}\right )^2+3\right )}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )^2-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (4 \int \frac {768 \left (\frac {1}{4}+\frac {1}{x}\right )^4+160 \left (\frac {1}{4}+\frac {1}{x}\right )^2+3}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )^2-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{4} \left (4 \left (\frac {\int \frac {417792}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )^2}{258048}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (4 \left (\frac {34}{21} \int \frac {1}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )^2-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{4} \left (4 \left (-\frac {68}{21} \int \frac {1}{-\left (\frac {1}{4}+\frac {1}{x}\right )^4-258048}d\left (512 \left (\frac {1}{4}+\frac {1}{x}\right )^2-96\right )-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )-\int \frac {4096 \left (\frac {1}{4}+\frac {1}{x}\right )^6+3840 \left (\frac {1}{4}+\frac {1}{x}\right )^4+240 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^2}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{4} \left (-\frac {\int \frac {49152 \left (35888 \left (\frac {1}{4}+\frac {1}{x}\right )^2+12903\right )}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )}{134701056}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \int \frac {35888 \left (\frac {1}{4}+\frac {1}{x}\right )^2+12903}{256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}d\left (\frac {1}{4}+\frac {1}{x}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\int \frac {24 \left (4301 \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}-2 \left (4301-2243 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{24 \sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\int \frac {24 \left (2 \left (4301-2243 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )+4301 \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{24 \sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\int \frac {4301 \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}-2 \left (4301-2243 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\int \frac {2 \left (4301-2243 \sqrt {29}\right ) \left (\frac {1}{4}+\frac {1}{x}\right )+4301 \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\frac {1}{16} \left (4301-2243 \sqrt {29}\right ) \int -\frac {4 \left (\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{16} \left (4301-2243 \sqrt {29}\right ) \int \frac {4 \left (8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{2} \sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{96 \left (1-\sqrt {29}\right )-\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )^2}d\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )-\sqrt {\frac {3}{2} \left (1+\sqrt {29}\right )} \left (4301+2243 \sqrt {29}\right ) \int \frac {1}{96 \left (1-\sqrt {29}\right )-\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )^2}d\left (32 \left (\frac {1}{4}+\frac {1}{x}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{4} \left (-\frac {2 \left (\frac {\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {\sqrt {6 \left (1+\sqrt {29}\right )}-8 \left (\frac {1}{4}+\frac {1}{x}\right )}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{8} \sqrt {\frac {1+\sqrt {29}}{\sqrt {29}-1}} \left (4301+2243 \sqrt {29}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{4} \left (4301-2243 \sqrt {29}\right ) \int \frac {8 \left (\frac {1}{4}+\frac {1}{x}\right )+\sqrt {6 \left (1+\sqrt {29}\right )}}{16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{4}+\frac {1}{x}\right )+3 \sqrt {29}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {1}{8} \sqrt {\frac {1+\sqrt {29}}{\sqrt {29}-1}} \left (4301+2243 \sqrt {29}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{4} \left (4 \left (\frac {17 \arctan \left (\frac {512 \left (\frac {1}{x}+\frac {1}{4}\right )^2-96}{192 \sqrt {7}}\right )}{1008 \sqrt {7}}-\frac {464 \left (\frac {1}{x}+\frac {1}{4}\right )^2+207}{336 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )-\frac {2 \left (\frac {\frac {1}{8} \sqrt {\frac {1+\sqrt {29}}{\sqrt {29}-1}} \left (4301+2243 \sqrt {29}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )-4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )-\frac {1}{16} \left (4301-2243 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2-4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}+\frac {\frac {1}{8} \sqrt {\frac {1+\sqrt {29}}{\sqrt {29}-1}} \left (4301+2243 \sqrt {29}\right ) \arctan \left (\frac {32 \left (\frac {1}{x}+\frac {1}{4}\right )+4 \sqrt {6 \left (1+\sqrt {29}\right )}}{4 \sqrt {6 \left (\sqrt {29}-1\right )}}\right )+\frac {1}{16} \left (4301-2243 \sqrt {29}\right ) \log \left (16 \left (\frac {1}{x}+\frac {1}{4}\right )^2+4 \sqrt {6 \left (1+\sqrt {29}\right )} \left (\frac {1}{x}+\frac {1}{4}\right )+3 \sqrt {29}\right )}{\sqrt {174 \left (1+\sqrt {29}\right )}}\right )}{5481}+\frac {5 \left (3184 \left (\frac {1}{x}+\frac {1}{4}\right )^2+5157\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{5481 \left (256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261\right )}\right )\) |
Input:
Int[(8 + 8*x - x^3 + 8*x^4)^(-2),x]
Output:
((5*(5157 + 3184*(1/4 + x^(-1))^2)*(1/4 + x^(-1)))/(5481*(261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4)) + 4*(-1/336*(207 + 464*(1/4 + x^(-1))^2 )/(261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4) + (17*ArcTan[(-96 + 5 12*(1/4 + x^(-1))^2)/(192*Sqrt[7])])/(1008*Sqrt[7])) - (2*(((Sqrt[(1 + Sqr t[29])/(-1 + Sqrt[29])]*(4301 + 2243*Sqrt[29])*ArcTan[(-4*Sqrt[6*(1 + Sqrt [29])] + 32*(1/4 + x^(-1)))/(4*Sqrt[6*(-1 + Sqrt[29])])])/8 - ((4301 - 224 3*Sqrt[29])*Log[3*Sqrt[29] - 4*Sqrt[6*(1 + Sqrt[29])]*(1/4 + x^(-1)) + 16* (1/4 + x^(-1))^2])/16)/Sqrt[174*(1 + Sqrt[29])] + ((Sqrt[(1 + Sqrt[29])/(- 1 + Sqrt[29])]*(4301 + 2243*Sqrt[29])*ArcTan[(4*Sqrt[6*(1 + Sqrt[29])] + 3 2*(1/4 + x^(-1)))/(4*Sqrt[6*(-1 + Sqrt[29])])])/8 + ((4301 - 2243*Sqrt[29] )*Log[3*Sqrt[29] + 4*Sqrt[6*(1 + Sqrt[29])]*(1/4 + x^(-1)) + 16*(1/4 + x^( -1))^2])/16)/Sqrt[174*(1 + Sqrt[29])]))/5481)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.28
| method | result | size |
| default | \(\frac {\frac {7}{3132} x^{3}-\frac {191}{58464} x^{2}+\frac {57}{12992} x +\frac {17}{10962}}{x^{4}-\frac {1}{8} x^{3}+x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\left (392 \textit {\_R}^{2}-1097 \textit {\_R} +2243\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\right )}{21924}\) | \(83\) |
| risch | \(\frac {\frac {7}{3132} x^{3}-\frac {191}{58464} x^{2}+\frac {57}{12992} x +\frac {17}{10962}}{x^{4}-\frac {1}{8} x^{3}+x +1}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-\textit {\_Z}^{3}+8 \textit {\_Z} +8\right )}{\sum }\frac {\left (392 \textit {\_R}^{2}-1097 \textit {\_R} +2243\right ) \ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-3 \textit {\_R}^{2}+8}\right )}{21924}\) | \(83\) |
Input:
int(1/(8*x^4-x^3+8*x+8)^2,x,method=_RETURNVERBOSE)
Output:
(7/3132*x^3-191/58464*x^2+57/12992*x+17/10962)/(x^4-1/8*x^3+x+1)+1/21924*s um((392*_R^2-1097*_R+2243)/(32*_R^3-3*_R^2+8)*ln(x-_R),_R=RootOf(8*_Z^4-_Z ^3+8*_Z+8))
Time = 0.13 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="fricas")
Output:
1/1227744*(21952*x^3 - 2*(8*x^4 - x^3 + 8*x + 8)*sqrt(1479/4550065*(459233 27*sqrt(29) + 180983329)*sqrt(1583563/42*sqrt(29) - 180983329/1218) + 1108 4941/6*sqrt(29) + 3931186441/174)*arctan(-1/4750443236862691984257460*(159 252275*sqrt(29)*(7172449575781*x - 4063334408258) - 174*(sqrt(29)*(2385676 8674269633*x - 20350906238895184) + 117938631918963916*x - 102889113598186 588)*sqrt(1583563/42*sqrt(29) - 180983329/1218) + 5194358934701729637845*x - 4202037046136740466110)*sqrt(1479/4550065*(45923327*sqrt(29) + 18098332 9)*sqrt(1583563/42*sqrt(29) - 180983329/1218) + 11084941/6*sqrt(29) + 3931 186441/174)) + 2*(8*x^4 - x^3 + 8*x + 8)*sqrt(-1479/4550065*(45923327*sqrt (29) + 180983329)*sqrt(1583563/42*sqrt(29) - 180983329/1218) + 11084941/6* sqrt(29) + 3931186441/174)*arctan(1/4750443236862691984257460*(159252275*s qrt(29)*(7172449575781*x - 4063334408258) + 174*(sqrt(29)*(238567686742696 33*x - 20350906238895184) + 117938631918963916*x - 102889113598186588)*sqr t(1583563/42*sqrt(29) - 180983329/1218) + 5194358934701729637845*x - 42020 37046136740466110)*sqrt(-1479/4550065*(45923327*sqrt(29) + 180983329)*sqrt (1583563/42*sqrt(29) - 180983329/1218) + 11084941/6*sqrt(29) + 3931186441/ 174)) + 7*(8*x^4 - x^3 + 8*x + 8)*sqrt(1583563/42*sqrt(29) - 180983329/121 8)*log(1019214560*x^2 + 3*(3*sqrt(29)*(41665*x - 23116) + 1460875*x - 1897 76)*sqrt(1583563/42*sqrt(29) - 180983329/1218) - 63700910*x + 191102730*sq rt(29) + 63700910) - 7*(8*x^4 - x^3 + 8*x + 8)*sqrt(1583563/42*sqrt(29)...
Leaf count of result is larger than twice the leaf count of optimal. 3834 vs. \(2 (228) = 456\).
Time = 1.78 (sec) , antiderivative size = 3834, normalized size of antiderivative = 12.78 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(8*x**4-x**3+8*x+8)**2,x)
Output:
(784*x**3 - 1146*x**2 + 1539*x + 544)/(350784*x**4 - 43848*x**3 + 350784*x + 350784) - sqrt(-180983329/37468546762752 + 1583563*sqrt(29)/12920188538 88)*log(x**2 + x*(-62716756730859*sqrt(1218)*sqrt(-180983329 + 45923327*sq rt(29))*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)*sqrt(-18098 3329 + 45923327*sqrt(29)) + 40699873480352667)/227008323264998681573683424 - 267658292345340*sqrt(214095423017213*sqrt(29) + 47106822945*sqrt(1218)* sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/8435208206933660 878927 - 2157374520970352866823*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt (29))/113504161632499340786841712 + 3881045239007430*sqrt(29)/532672726436 1229 + 435853770857118353330297/33740832827734643515708 + 20905585576953*s qrt(42)*sqrt(-180983329 + 45923327*sqrt(29))/85227636229779664) - 29428140 74101429415084030510182204250067556953*sqrt(214095423017213*sqrt(29) + 471 06822945*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 406998734803526 67)/888496186751485201253966401139075287452416534006272 - 1425762563285631 4835831142972765102609010539559351093/277655058359839125391864500355961027 32888016687696 - 75184631502818837388875900060881355871*sqrt(1218)*sqrt(-1 80983329 + 45923327*sqrt(29))*sqrt(214095423017213*sqrt(29) + 47106822945* sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29)) + 40699873480352667)/30637 799543154662112205737970312940946635052896768 - 96331418179614125974885876 61065704878094062299*sqrt(1218)*sqrt(-180983329 + 45923327*sqrt(29))/30...
\[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - x^{3} + 8 \, x + 8\right )}^{2}} \,d x } \] Input:
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="maxima")
Output:
1/43848*(784*x^3 - 1146*x^2 + 1539*x + 544)/(8*x^4 - x^3 + 8*x + 8) + 1/21 924*integrate((392*x^2 - 1097*x + 2243)/(8*x^4 - x^3 + 8*x + 8), x)
\[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - x^{3} + 8 \, x + 8\right )}^{2}} \,d x } \] Input:
integrate(1/(8*x^4-x^3+8*x+8)^2,x, algorithm="giac")
Output:
integrate((8*x^4 - x^3 + 8*x + 8)^(-2), x)
Time = 0.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (\frac {2615257\,\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}{72918171648}+\frac {4225\,x}{40375589184}-\frac {\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )\,x\,34885379}{72918171648}-\frac {{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^2\,x\,191555}{475136}-\frac {{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^3\,x\,9261}{256}-\frac {11205\,{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^2}{59392}-\frac {24759\,{\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )}^3}{512}+\frac {10901}{107668237824}\right )\,\mathrm {root}\left (z^4+\frac {6630191\,z^2}{167270298048}+\frac {77351105\,z}{674433841729536}+\frac {1114096}{13723971258377709},z,k\right )\right )+\frac {\frac {7\,x^3}{3132}-\frac {191\,x^2}{58464}+\frac {57\,x}{12992}+\frac {17}{10962}}{x^4-\frac {x^3}{8}+x+1} \] Input:
int(1/(8*x - x^3 + 8*x^4 + 8)^2,x)
Output:
symsum(log((2615257*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/6 74433841729536 + 1114096/13723971258377709, z, k))/72918171648 + (4225*x)/ 40375589184 - (34885379*root(z^4 + (6630191*z^2)/167270298048 + (77351105* z)/674433841729536 + 1114096/13723971258377709, z, k)*x)/72918171648 - (19 1555*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z, k)^2*x)/475136 - (9261*root(z^4 + (6630191 *z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377 709, z, k)^3*x)/256 - (11205*root(z^4 + (6630191*z^2)/167270298048 + (7735 1105*z)/674433841729536 + 1114096/13723971258377709, z, k)^2)/59392 - (247 59*root(z^4 + (6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/13723971258377709, z, k)^3)/512 + 10901/107668237824)*root(z^4 + ( 6630191*z^2)/167270298048 + (77351105*z)/674433841729536 + 1114096/1372397 1258377709, z, k), k, 1, 4) + ((57*x)/12992 - (191*x^2)/58464 + (7*x^3)/31 32 + 17/10962)/(x - x^3/8 + x^4 + 1)
\[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^2} \, dx=\int \frac {1}{64 x^{8}-16 x^{7}+x^{6}+128 x^{5}+112 x^{4}-16 x^{3}+64 x^{2}+128 x +64}d x \] Input:
int(1/(8*x^4-x^3+8*x+8)^2,x)
Output:
int(1/(64*x**8 - 16*x**7 + x**6 + 128*x**5 + 112*x**4 - 16*x**3 + 64*x**2 + 128*x + 64),x)