Integrand size = 17, antiderivative size = 264 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=-\frac {17-\left (1+\frac {1}{x}\right )^2}{2 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {\left (59-17 \left (1+\frac {1}{x}\right )^2\right ) \left (1+\frac {1}{x}\right )}{10 \left (5-2 \left (1+\frac {1}{x}\right )^2+\left (1+\frac {1}{x}\right )^4\right )}+\frac {7}{4} \arctan \left (\frac {1}{2} \left (-1+\left (1+\frac {1}{x}\right )^2\right )\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \sqrt {5}\right )} \arctan \left (\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (5959+2665 \sqrt {5}\right )} \arctan \left (\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )}+\frac {2}{x}}{\sqrt {2 \left (-1+\sqrt {5}\right )}}\right )-\frac {1}{20} \sqrt {\frac {1}{10} \left (-5959+2665 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )}{\sqrt {5}+\left (1+\frac {1}{x}\right )^2}\right ) \] Output:
-1/2*(17-(1+1/x)^2)/(5-2*(1+1/x)^2+(1+1/x)^4)+(59-17*(1+1/x)^2)*(1+1/x)/(5 0-20*(1+1/x)^2+10*(1+1/x)^4)+7/4*arctan(-1/2+1/2*(1+1/x)^2)-1/200*(59590+2 6650*5^(1/2))^(1/2)*arctan((2-(2+2*5^(1/2))^(1/2)+2/x)/(-2+2*5^(1/2))^(1/2 ))-1/200*(59590+26650*5^(1/2))^(1/2)*arctan((2+(2+2*5^(1/2))^(1/2)+2/x)/(- 2+2*5^(1/2))^(1/2))-1/200*(-59590+26650*5^(1/2))^(1/2)*arctanh((2+2*5^(1/2 ))^(1/2)*(1+1/x)/(5^(1/2)+(1+1/x)^2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\frac {1}{40} \left (\frac {38+84 x-32 x^2+72 x^3}{1+4 x+4 x^2+4 x^4}+\text {RootSum}\left [1+4 \text {$\#$1}+4 \text {$\#$1}^2+4 \text {$\#$1}^4\&,\frac {27 \log (x-\text {$\#$1})-16 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2}{1+2 \text {$\#$1}+4 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]
Output:
((38 + 84*x - 32*x^2 + 72*x^3)/(1 + 4*x + 4*x^2 + 4*x^4) + RootSum[1 + 4*# 1 + 4*#1^2 + 4*#1^4 & , (27*Log[x - #1] - 16*Log[x - #1]*#1 + 18*Log[x - # 1]*#1^2)/(1 + 2*#1 + 4*#1^3) & ])/40
Time = 0.81 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.33, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2504, 27, 2202, 2194, 27, 2191, 27, 1083, 217, 2206, 27, 1483, 1142, 25, 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.
| \(\displaystyle \int \frac {1}{\left (4 x^4+4 x^2+4 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -16 \int \frac {1}{16 \left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2 x^6}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2 x^6}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle -\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )-\int \frac {\left (-6 \left (1+\frac {1}{x}\right )^4-20 \left (1+\frac {1}{x}\right )^2-6\right ) \left (1+\frac {1}{x}\right )}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle -\frac {1}{2} \int -\frac {2 \left (3 \left (1+\frac {1}{x}\right )^4+10 \left (1+\frac {1}{x}\right )^2+3\right )}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )^2-\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {3 \left (1+\frac {1}{x}\right )^4+10 \left (1+\frac {1}{x}\right )^2+3}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )^2-\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{16} \int \frac {56}{\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}d\left (1+\frac {1}{x}\right )^2-\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{2} \int \frac {1}{\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}d\left (1+\frac {1}{x}\right )^2-\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -7 \int \frac {1}{-\left (1+\frac {1}{x}\right )^4-16}d\left (2 \left (1+\frac {1}{x}\right )^2-2\right )-\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\int \frac {\left (1+\frac {1}{x}\right )^6+15 \left (1+\frac {1}{x}\right )^4+15 \left (1+\frac {1}{x}\right )^2+1}{\left (\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5\right )^2}d\left (1+\frac {1}{x}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {1}{160} \int \frac {16 \left (27 \left (1+\frac {1}{x}\right )^2+61\right )}{\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}d\left (1+\frac {1}{x}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{10} \int \frac {27 \left (1+\frac {1}{x}\right )^2+61}{\left (1+\frac {1}{x}\right )^4-2 \left (1+\frac {1}{x}\right )^2+5}d\left (1+\frac {1}{x}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\int \frac {61 \sqrt {2 \left (1+\sqrt {5}\right )}-\left (61-27 \sqrt {5}\right ) \left (1+\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\int \frac {\left (61-27 \sqrt {5}\right ) \left (1+\frac {1}{x}\right )+61 \sqrt {2 \left (1+\sqrt {5}\right )}}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )-\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int -\frac {\sqrt {2 \left (1+\sqrt {5}\right )}-2 \left (1+\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )+\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )+\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {\sqrt {2 \left (1+\sqrt {5}\right )}-2 \left (1+\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )+\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {\sqrt {2 \left (1+\sqrt {5}\right )}-2 \left (1+\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )-2 \sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{2 \left (1-\sqrt {5}\right )-\left (2 \left (1+\frac {1}{x}\right )-\sqrt {2 \left (1+\sqrt {5}\right )}\right )^2}d\left (2 \left (1+\frac {1}{x}\right )-\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )-2 \sqrt {2 \left (5959+2665 \sqrt {5}\right )} \int \frac {1}{2 \left (1-\sqrt {5}\right )-\left (2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}\right )^2}d\left (2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{10} \left (-\frac {\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {\sqrt {2 \left (1+\sqrt {5}\right )}-2 \left (1+\frac {1}{x}\right )}{\left (1+\frac {1}{x}\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )+2 \sqrt {\frac {5959+2665 \sqrt {5}}{\sqrt {5}-1}} \arctan \left (\frac {2 \left (\frac {1}{x}+1\right )-\sqrt {2 \left (1+\sqrt {5}\right )}}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {\frac {1}{2} \left (61-27 \sqrt {5}\right ) \int \frac {2 \left (1+\frac {1}{x}\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\left (1+\frac {1}{x}\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (1+\frac {1}{x}\right )+\sqrt {5}}d\left (1+\frac {1}{x}\right )+2 \sqrt {\frac {5959+2665 \sqrt {5}}{\sqrt {5}-1}} \arctan \left (\frac {2 \left (\frac {1}{x}+1\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )+\frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7}{4} \arctan \left (\frac {1}{4} \left (2 \left (\frac {1}{x}+1\right )^2-2\right )\right )+\frac {1}{10} \left (-\frac {2 \sqrt {\frac {5959+2665 \sqrt {5}}{\sqrt {5}-1}} \arctan \left (\frac {2 \left (\frac {1}{x}+1\right )-\sqrt {2 \left (1+\sqrt {5}\right )}}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )-\frac {1}{2} \left (61-27 \sqrt {5}\right ) \log \left (\left (\frac {1}{x}+1\right )^2-\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}-\frac {2 \sqrt {\frac {5959+2665 \sqrt {5}}{\sqrt {5}-1}} \arctan \left (\frac {2 \left (\frac {1}{x}+1\right )+\sqrt {2 \left (1+\sqrt {5}\right )}}{\sqrt {2 \left (\sqrt {5}-1\right )}}\right )+\frac {1}{2} \left (61-27 \sqrt {5}\right ) \log \left (\left (\frac {1}{x}+1\right )^2+\sqrt {2 \left (1+\sqrt {5}\right )} \left (\frac {1}{x}+1\right )+\sqrt {5}\right )}{2 \sqrt {10 \left (1+\sqrt {5}\right )}}\right )-\frac {16-\frac {1}{x}}{2 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}+\frac {\left (59-17 \left (\frac {1}{x}+1\right )^2\right ) \left (\frac {1}{x}+1\right )}{10 \left (\left (\frac {1}{x}+1\right )^4-2 \left (\frac {1}{x}+1\right )^2+5\right )}\) |
Input:
Int[(1 + 4*x + 4*x^2 + 4*x^4)^(-2),x]
Output:
-1/2*(16 - x^(-1))/(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4) + ((59 - 17*(1 + x^(-1))^2)*(1 + x^(-1)))/(10*(5 - 2*(1 + x^(-1))^2 + (1 + x^(-1))^4)) + (7*ArcTan[(-2 + 2*(1 + x^(-1))^2)/4])/4 + (-1/2*(2*Sqrt[(5959 + 2665*Sqrt[ 5])/(-1 + Sqrt[5])]*ArcTan[(-Sqrt[2*(1 + Sqrt[5])] + 2*(1 + x^(-1)))/Sqrt[ 2*(-1 + Sqrt[5])]] - ((61 - 27*Sqrt[5])*Log[Sqrt[5] - Sqrt[2*(1 + Sqrt[5]) ]*(1 + x^(-1)) + (1 + x^(-1))^2])/2)/Sqrt[10*(1 + Sqrt[5])] - (2*Sqrt[(595 9 + 2665*Sqrt[5])/(-1 + Sqrt[5])]*ArcTan[(Sqrt[2*(1 + Sqrt[5])] + 2*(1 + x ^(-1)))/Sqrt[2*(-1 + Sqrt[5])]] + ((61 - 27*Sqrt[5])*Log[Sqrt[5] + Sqrt[2* (1 + Sqrt[5])]*(1 + x^(-1)) + (1 + x^(-1))^2])/2)/(2*Sqrt[10*(1 + Sqrt[5]) ]))/10
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 = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.30
| method | result | size |
| default | \(\frac {\frac {9}{20} x^{3}-\frac {1}{5} x^{2}+\frac {21}{40} x +\frac {19}{80}}{x^{4}+x^{2}+x +\frac {1}{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\left (18 \textit {\_R}^{2}-16 \textit {\_R} +27\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{40}\) | \(79\) |
| risch | \(\frac {\frac {9}{20} x^{3}-\frac {1}{5} x^{2}+\frac {21}{40} x +\frac {19}{80}}{x^{4}+x^{2}+x +\frac {1}{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )}{\sum }\frac {\left (18 \textit {\_R}^{2}-16 \textit {\_R} +27\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+2 \textit {\_R} +1}\right )}{40}\) | \(79\) |
Input:
int(1/(4*x^4+4*x^2+4*x+1)^2,x,method=_RETURNVERBOSE)
Output:
(9/20*x^3-1/5*x^2+21/40*x+19/80)/(x^4+x^2+x+1/4)+1/40*sum((18*_R^2-16*_R+2 7)/(4*_R^3+2*_R+1)*ln(x-_R),_R=RootOf(4*_Z^4+4*_Z^2+4*_Z+1))
Time = 0.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\frac {1368 \, x^{3} - 19 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} \log \left (76 \, x^{2} + 2 \, {\left (\sqrt {5} {\left (61 \, x - 49\right )} + 135 \, x - 110\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} + 19 \, \sqrt {5} + 19\right ) + 19 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} \log \left (76 \, x^{2} - 2 \, {\left (\sqrt {5} {\left (61 \, x - 49\right )} + 135 \, x - 110\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} + 19 \, \sqrt {5} + 19\right ) - 608 \, x^{2} + {\left (5320 \, x^{4} + 5320 \, x^{2} + {\left (23836 \, x^{4} + 23836 \, x^{2} + 2665 \, \sqrt {5} {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )} + 23836 \, x + 5959\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} + 5320 \, x + 1330\right )} \arctan \left (\frac {1}{2} \, \sqrt {5} {\left (2 \, x + 1\right )} + \frac {1}{19} \, {\left (\sqrt {5} {\left (159 \, x + 49\right )} + 355 \, x + 110\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} + x + \frac {1}{2}\right ) - {\left (5320 \, x^{4} + 5320 \, x^{2} - {\left (23836 \, x^{4} + 23836 \, x^{2} + 2665 \, \sqrt {5} {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )} + 23836 \, x + 5959\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} + 5320 \, x + 1330\right )} \arctan \left (-\frac {1}{2} \, \sqrt {5} {\left (2 \, x + 1\right )} + \frac {1}{19} \, {\left (\sqrt {5} {\left (159 \, x + 49\right )} + 355 \, x + 110\right )} \sqrt {\frac {533}{2} \, \sqrt {5} - \frac {5959}{10}} - x - \frac {1}{2}\right ) + 1596 \, x + 722}{760 \, {\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}} \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^2,x, algorithm="fricas")
Output:
1/760*(1368*x^3 - 19*(4*x^4 + 4*x^2 + 4*x + 1)*sqrt(533/2*sqrt(5) - 5959/1 0)*log(76*x^2 + 2*(sqrt(5)*(61*x - 49) + 135*x - 110)*sqrt(533/2*sqrt(5) - 5959/10) + 19*sqrt(5) + 19) + 19*(4*x^4 + 4*x^2 + 4*x + 1)*sqrt(533/2*sqr t(5) - 5959/10)*log(76*x^2 - 2*(sqrt(5)*(61*x - 49) + 135*x - 110)*sqrt(53 3/2*sqrt(5) - 5959/10) + 19*sqrt(5) + 19) - 608*x^2 + (5320*x^4 + 5320*x^2 + (23836*x^4 + 23836*x^2 + 2665*sqrt(5)*(4*x^4 + 4*x^2 + 4*x + 1) + 23836 *x + 5959)*sqrt(533/2*sqrt(5) - 5959/10) + 5320*x + 1330)*arctan(1/2*sqrt( 5)*(2*x + 1) + 1/19*(sqrt(5)*(159*x + 49) + 355*x + 110)*sqrt(533/2*sqrt(5 ) - 5959/10) + x + 1/2) - (5320*x^4 + 5320*x^2 - (23836*x^4 + 23836*x^2 + 2665*sqrt(5)*(4*x^4 + 4*x^2 + 4*x + 1) + 23836*x + 5959)*sqrt(533/2*sqrt(5 ) - 5959/10) + 5320*x + 1330)*arctan(-1/2*sqrt(5)*(2*x + 1) + 1/19*(sqrt(5 )*(159*x + 49) + 355*x + 110)*sqrt(533/2*sqrt(5) - 5959/10) - x - 1/2) + 1 596*x + 722)/(4*x^4 + 4*x^2 + 4*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 3834 vs. \(2 (211) = 422\).
Time = 2.02 (sec) , antiderivative size = 3834, normalized size of antiderivative = 14.52 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(4*x**4+4*x**2+4*x+1)**2,x)
Output:
(36*x**3 - 16*x**2 + 42*x + 19)/(80*x**4 + 80*x**2 + 80*x + 20) - sqrt(-59 59/16000 + 533*sqrt(5)/3200)*log(x**2 + x*(-1601676*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt( 5) + 36004639)/13543383425 - 1067784*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/10 16389 + 3131659367*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))/13543383425 + 29168 9395/1083470674 + 470215*sqrt(5)/2032778 + 94043*sqrt(-665*sqrt(10)*sqrt(- 5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/541735337) - 40634464149 111451*sqrt(5)*sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt (5) + 36004639)/27530691871904650 - 2885835544225227917282997/146738587677 251784500 - 83803227754187*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))/100111606806 926 - 50208805356*sqrt(2)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sq rt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/550613837438093 - 53 8485754891933*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(-665*sqrt(10)*sqrt( -5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639)/14673858767725178450 - 925321955096901411*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))/2934771753545035690 0 + 484304611938766076267*sqrt(5)/55061383743809300 + 22013036087014785403 *sqrt(-665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 36004639 )/6669935803511444750) + sqrt(-5959/16000 + 533*sqrt(5)/3200)*log(x**2 + x *(-94043*sqrt(665*sqrt(10)*sqrt(-5959 + 2665*sqrt(5)) + 221195*sqrt(5) + 3 6004639)/541735337 - 1601676*sqrt(10)*sqrt(-5959 + 2665*sqrt(5))*sqrt(6...
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\int { \frac {1}{{\left (4 \, x^{4} + 4 \, x^{2} + 4 \, x + 1\right )}^{2}} \,d x } \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^2,x, algorithm="maxima")
Output:
1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1) + 1/10*integr ate((18*x^2 - 16*x + 27)/(4*x^4 + 4*x^2 + 4*x + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (201) = 402\).
Time = 0.17 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(1/(4*x^4+4*x^2+4*x+1)^2,x, algorithm="giac")
Output:
-1/100*(19*sqrt(26650*sqrt(5) - 59590)/(2665*sqrt(5) - 5959) + 175)*(arcta n(19/2980) + arctan(-1/17646838*x*(2999*sqrt(5)*sqrt(17761522*sqrt(5) - 18 213038) + 17646838*sqrt(5) + 8959*sqrt(17761522*sqrt(5) - 18213038) + 1764 6838) - 745/8823419*sqrt(5)*sqrt(17761522*sqrt(5) - 18213038) - 1/2*sqrt(5 ) - 1509/17646838*sqrt(17761522*sqrt(5) - 18213038) - 1/2)) + 1/100*(19*sq rt(26650*sqrt(5) - 59590)/(2665*sqrt(5) - 5959) - 175)*(arctan(19/2980) + arctan(1/17646838*x*(2999*sqrt(5)*sqrt(17761522*sqrt(5) - 18213038) - 1764 6838*sqrt(5) + 8959*sqrt(17761522*sqrt(5) - 18213038) - 17646838) + 745/88 23419*sqrt(5)*sqrt(17761522*sqrt(5) - 18213038) - 1/2*sqrt(5) + 1509/17646 838*sqrt(17761522*sqrt(5) - 18213038) - 1/2)) - 1/400*sqrt(26650*sqrt(5) - 59590)*log(577600*(105858671120*sqrt(5)*x - 108549706480*x + 8880761*sqrt (5)*sqrt(17761522*sqrt(5) - 18213038) + 337468918*sqrt(5) - 9106519*sqrt(1 7761522*sqrt(5) - 18213038) - 346047722)^2 + 2310400*(337468918*sqrt(5)*x - 346047722*x - 26464667780*sqrt(5) + 8823419*sqrt(17761522*sqrt(5) - 1821 3038) + 27137426620)^2) + 1/400*sqrt(26650*sqrt(5) - 59590)*log(577600*(10 5858671120*sqrt(5)*x - 108549706480*x - 8880761*sqrt(5)*sqrt(17761522*sqrt (5) - 18213038) + 337468918*sqrt(5) + 9106519*sqrt(17761522*sqrt(5) - 1821 3038) - 346047722)^2 + 2310400*(337468918*sqrt(5)*x - 346047722*x - 264646 67780*sqrt(5) - 8823419*sqrt(17761522*sqrt(5) - 18213038) + 27137426620)^2 ) + 1/20*(36*x^3 - 16*x^2 + 42*x + 19)/(4*x^4 + 4*x^2 + 4*x + 1)
Time = 21.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {169\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}{100}+\frac {11\,x}{1600}+\frac {\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\,x\,131}{100}-\frac {{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2\,x\,72}{5}-{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3\,x\,36+\frac {59\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^2}{20}-16\,{\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )}^3+\frac {27}{1600}\right )\,\mathrm {root}\left (z^4+\frac {3021\,z^2}{1000}-\frac {133\,z}{8000}+\frac {29}{64000},z,k\right )\right )+\frac {\frac {9\,x^3}{20}-\frac {x^2}{5}+\frac {21\,x}{40}+\frac {19}{80}}{x^4+x^2+x+\frac {1}{4}} \] Input:
int(1/(4*x + 4*x^2 + 4*x^4 + 1)^2,x)
Output:
symsum(log((11*x)/1600 - (169*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k))/100 + (131*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29 /64000, z, k)*x)/100 - (72*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/ 64000, z, k)^2*x)/5 - 36*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64 000, z, k)^3*x + (59*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k)^2)/20 - 16*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k)^3 + 27/1600)*root(z^4 + (3021*z^2)/1000 - (133*z)/8000 + 29/64000, z, k), k, 1, 4) + ((21*x)/40 - x^2/5 + (9*x^3)/20 + 19/80)/(x + x^2 + x^4 + 1 /4)
\[ \int \frac {1}{\left (1+4 x+4 x^2+4 x^4\right )^2} \, dx=\int \frac {1}{16 x^{8}+32 x^{6}+32 x^{5}+24 x^{4}+32 x^{3}+24 x^{2}+8 x +1}d x \] Input:
int(1/(4*x^4+4*x^2+4*x+1)^2,x)
Output:
int(1/(16*x**8 + 32*x**6 + 32*x**5 + 24*x**4 + 32*x**3 + 24*x**2 + 8*x + 1 ),x)