Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {x}{32 \left (1-x^2\right )}+\frac {x \left (99-17 x^2\right )}{128 \left (1-6 x^2+x^4\right )}+\frac {5 \text {arctanh}(x)}{32}+\frac {1}{512} \left (-4+3 \sqrt {2}\right ) \text {arctanh}\left (\left (-1+\sqrt {2}\right ) x\right )+\frac {1}{512} \left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\left (1+\sqrt {2}\right ) x\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(91)=182\).
Time = 0.10 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.25, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2082, 213, 652, 632, 212, 646, 31} \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {17 \text {arctanh}\left (\frac {1-x}{\sqrt {2}}\right )}{256 \sqrt {2}}+\frac {5 \text {arctanh}(x)}{32}+\frac {17 \text {arctanh}\left (\frac {x+1}{\sqrt {2}}\right )}{256 \sqrt {2}}-\frac {41-17 x}{256 \left (-x^2+2 x+1\right )}+\frac {17 x+41}{256 \left (-x^2-2 x+1\right )}+\frac {1}{64 (1-x)}-\frac {1}{64 (x+1)}+\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (-x-\sqrt {2}+1\right )+\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (-x+\sqrt {2}+1\right )-\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )-\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right ) \]
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Rule 31
Rule 212
Rule 213
Rule 632
Rule 646
Rule 652
Rule 2082
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{64 (-1+x)^2}+\frac {1}{64 (1+x)^2}-\frac {5}{32 \left (-1+x^2\right )}+\frac {29-12 x}{64 \left (-1-2 x+x^2\right )^2}+\frac {6+x}{128 \left (-1-2 x+x^2\right )}+\frac {29+12 x}{64 \left (-1+2 x+x^2\right )^2}+\frac {6-x}{128 \left (-1+2 x+x^2\right )}\right ) \, dx \\ & = \frac {1}{64 (1-x)}-\frac {1}{64 (1+x)}+\frac {1}{128} \int \frac {6+x}{-1-2 x+x^2} \, dx+\frac {1}{128} \int \frac {6-x}{-1+2 x+x^2} \, dx+\frac {1}{64} \int \frac {29-12 x}{\left (-1-2 x+x^2\right )^2} \, dx+\frac {1}{64} \int \frac {29+12 x}{\left (-1+2 x+x^2\right )^2} \, dx-\frac {5}{32} \int \frac {1}{-1+x^2} \, dx \\ & = \frac {1}{64 (1-x)}-\frac {1}{64 (1+x)}+\frac {41+17 x}{256 \left (1-2 x-x^2\right )}-\frac {41-17 x}{256 \left (1+2 x-x^2\right )}+\frac {5}{32} \tanh ^{-1}(x)-\frac {17}{256} \int \frac {1}{-1-2 x+x^2} \, dx-\frac {17}{256} \int \frac {1}{-1+2 x+x^2} \, dx+\frac {1}{512} \left (2-7 \sqrt {2}\right ) \int \frac {1}{-1+\sqrt {2}+x} \, dx+\frac {1}{512} \left (-2+7 \sqrt {2}\right ) \int \frac {1}{1-\sqrt {2}+x} \, dx+\frac {1}{512} \left (2+7 \sqrt {2}\right ) \int \frac {1}{-1-\sqrt {2}+x} \, dx-\frac {1}{512} \left (2+7 \sqrt {2}\right ) \int \frac {1}{1+\sqrt {2}+x} \, dx \\ & = \frac {1}{64 (1-x)}-\frac {1}{64 (1+x)}+\frac {41+17 x}{256 \left (1-2 x-x^2\right )}-\frac {41-17 x}{256 \left (1+2 x-x^2\right )}+\frac {5}{32} \tanh ^{-1}(x)+\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (1-\sqrt {2}-x\right )+\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )-\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (1-\sqrt {2}+x\right )-\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )+\frac {17}{128} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 x\right )+\frac {17}{128} \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 x\right ) \\ & = \frac {1}{64 (1-x)}-\frac {1}{64 (1+x)}+\frac {41+17 x}{256 \left (1-2 x-x^2\right )}-\frac {41-17 x}{256 \left (1+2 x-x^2\right )}-\frac {17 \tanh ^{-1}\left (\frac {1-x}{\sqrt {2}}\right )}{256 \sqrt {2}}+\frac {5}{32} \tanh ^{-1}(x)+\frac {17 \tanh ^{-1}\left (\frac {1+x}{\sqrt {2}}\right )}{256 \sqrt {2}}+\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (1-\sqrt {2}-x\right )+\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )-\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (1-\sqrt {2}+x\right )-\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {-\frac {8 x \left (103-140 x^2+21 x^4\right )}{-1+7 x^2-7 x^4+x^6}-80 \log (1-x)-\left (4+3 \sqrt {2}\right ) \log \left (-1+\sqrt {2}-x\right )+\left (4-3 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )+80 \log (1+x)+\left (4+3 \sqrt {2}\right ) \log \left (-1+\sqrt {2}+x\right )+\left (-4+3 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )}{1024} \]
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Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27
method | result | size |
default | \(-\frac {1}{64 \left (x +1\right )}+\frac {5 \ln \left (x +1\right )}{64}-\frac {\frac {17 x}{2}+\frac {41}{2}}{128 \left (x^{2}+2 x -1\right )}-\frac {\ln \left (x^{2}+2 x -1\right )}{256}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{512}-\frac {1}{64 \left (x -1\right )}-\frac {5 \ln \left (x -1\right )}{64}+\frac {-\frac {17 x}{2}+\frac {41}{2}}{128 x^{2}-256 x -128}+\frac {\ln \left (x^{2}-2 x -1\right )}{256}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x -2\right ) \sqrt {2}}{4}\right )}{512}\) | \(116\) |
risch | \(\frac {-\frac {21}{128} x^{5}+\frac {35}{32} x^{3}-\frac {103}{128} x}{x^{6}-7 x^{4}+7 x^{2}-1}+\frac {\ln \left (x -1+\sqrt {2}\right )}{256}+\frac {3 \ln \left (x -1+\sqrt {2}\right ) \sqrt {2}}{1024}+\frac {\ln \left (x -1-\sqrt {2}\right )}{256}-\frac {3 \ln \left (x -1-\sqrt {2}\right ) \sqrt {2}}{1024}-\frac {\ln \left (1+x +\sqrt {2}\right )}{256}+\frac {3 \ln \left (1+x +\sqrt {2}\right ) \sqrt {2}}{1024}-\frac {\ln \left (1+x -\sqrt {2}\right )}{256}-\frac {3 \ln \left (1+x -\sqrt {2}\right ) \sqrt {2}}{1024}+\frac {5 \ln \left (x +1\right )}{64}-\frac {5 \ln \left (x -1\right )}{64}\) | \(138\) |
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (71) = 142\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.45 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {168 \, x^{5} - 1120 \, x^{3} - 3 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} + 2 \, \sqrt {2} {\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) - 3 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} + 2 \, \sqrt {2} {\left (x - 1\right )} - 2 \, x + 3}{x^{2} - 2 \, x - 1}\right ) + 4 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) - 4 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 80 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 80 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 824 \, x}{1024 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (75) = 150\).
Time = 0.81 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.25 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {- 21 x^{5} + 140 x^{3} - 103 x}{128 x^{6} - 896 x^{4} + 896 x^{2} - 128} - \frac {5 \log {\left (x - 1 \right )}}{64} + \frac {5 \log {\left (x + 1 \right )}}{64} + \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {8071264001}{202624020} - \frac {471550901878784 \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {1299552375287054336 \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} + \frac {8071264001 \sqrt {2}}{270165360} \right )} + \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right ) \log {\left (x - \frac {8071264001 \sqrt {2}}{270165360} - \frac {8071264001}{202624020} + \frac {1299552375287054336 \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right )^{5}}{50656005} - \frac {471550901878784 \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right )^{3}}{2979765} \right )} + \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {8071264001 \sqrt {2}}{270165360} + \frac {1299552375287054336 \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} - \frac {471550901878784 \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {8071264001}{202624020} \right )} + \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {471550901878784 \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {1299552375287054336 \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} + \frac {8071264001}{202624020} + \frac {8071264001 \sqrt {2}}{270165360} \right )} \]
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Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {3}{1024} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} + 1}{x + \sqrt {2} + 1}\right ) - \frac {3}{1024} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} - 1}{x + \sqrt {2} - 1}\right ) - \frac {21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac {1}{256} \, \log \left (x^{2} + 2 \, x - 1\right ) + \frac {1}{256} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac {5}{64} \, \log \left (x + 1\right ) - \frac {5}{64} \, \log \left (x - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {3}{1024} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + 2 \right |}}\right ) - \frac {3}{1024} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} - 2 \right |}}\right ) - \frac {21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac {1}{256} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) + \frac {1}{256} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac {5}{64} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {5}{64} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 10.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{32}-\frac {\frac {21\,x^5}{128}-\frac {35\,x^3}{32}+\frac {103\,x}{128}}{x^6-7\,x^4+7\,x^2-1}-\mathrm {atan}\left (\frac {x\,940311{}\mathrm {i}}{134217728\,\left (\frac {275445\,\sqrt {2}}{134217728}-\frac {389421}{134217728}\right )}-\frac {\sqrt {2}\,x\,332433{}\mathrm {i}}{67108864\,\left (\frac {275445\,\sqrt {2}}{134217728}-\frac {389421}{134217728}\right )}\right )\,\left (\frac {\sqrt {2}\,3{}\mathrm {i}}{512}-\frac {1}{128}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,940311{}\mathrm {i}}{134217728\,\left (\frac {275445\,\sqrt {2}}{134217728}+\frac {389421}{134217728}\right )}+\frac {\sqrt {2}\,x\,332433{}\mathrm {i}}{67108864\,\left (\frac {275445\,\sqrt {2}}{134217728}+\frac {389421}{134217728}\right )}\right )\,\left (\frac {\sqrt {2}\,3{}\mathrm {i}}{512}+\frac {1}{128}{}\mathrm {i}\right ) \]
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