Integrand size = 11, antiderivative size = 58 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {1}{24 \left (4-x^2\right )^3}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{128 \left (4-x^2\right )}+\frac {\log (x)}{256}-\frac {1}{512} \log \left (4-x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {272, 46} \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {1}{128 \left (4-x^2\right )}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{24 \left (4-x^2\right )^3}-\frac {1}{512} \log \left (4-x^2\right )+\frac {\log (x)}{256} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-4+x)^4 x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{4 (-4+x)^4}-\frac {1}{16 (-4+x)^3}+\frac {1}{64 (-4+x)^2}-\frac {1}{256 (-4+x)}+\frac {1}{256 x}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{24 \left (4-x^2\right )^3}+\frac {1}{64 \left (4-x^2\right )^2}+\frac {1}{128 \left (4-x^2\right )}+\frac {\log (x)}{256}-\frac {1}{512} \log \left (4-x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {-\frac {4 \left (88-30 x^2+3 x^4\right )}{\left (-4+x^2\right )^3}+6 \log (x)-3 \log \left (4-x^2\right )}{1536} \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {-\frac {1}{128} x^{4}+\frac {5}{64} x^{2}-\frac {11}{48}}{\left (x^{2}-4\right )^{3}}+\frac {\ln \left (x \right )}{256}-\frac {\ln \left (x^{2}-4\right )}{512}\) | \(34\) |
norman | \(\frac {-\frac {1}{128} x^{4}+\frac {5}{64} x^{2}-\frac {11}{48}}{\left (x^{2}-4\right )^{3}}+\frac {\ln \left (x \right )}{256}-\frac {\ln \left (-2+x \right )}{512}-\frac {\ln \left (2+x \right )}{512}\) | \(38\) |
meijerg | \(\frac {11}{3072}+\frac {\ln \left (x \right )}{256}-\frac {\ln \left (2\right )}{256}+\frac {i \pi }{512}+\frac {x^{2} \left (\frac {11}{16} x^{4}-\frac {27}{4} x^{2}+18\right )}{12288 \left (1-\frac {x^{2}}{4}\right )^{3}}-\frac {\ln \left (1-\frac {x^{2}}{4}\right )}{512}\) | \(51\) |
default | \(\frac {1}{1536 \left (2+x \right )^{3}}+\frac {3}{2048 \left (2+x \right )^{2}}+\frac {11}{4096 \left (2+x \right )}-\frac {\ln \left (2+x \right )}{512}+\frac {\ln \left (x \right )}{256}-\frac {1}{1536 \left (-2+x \right )^{3}}+\frac {3}{2048 \left (-2+x \right )^{2}}-\frac {11}{4096 \left (-2+x \right )}-\frac {\ln \left (-2+x \right )}{512}\) | \(60\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{6}-3 \ln \left (-2+x \right ) x^{6}-3 \ln \left (2+x \right ) x^{6}-352-72 x^{4} \ln \left (x \right )+36 \ln \left (-2+x \right ) x^{4}+36 \ln \left (2+x \right ) x^{4}-12 x^{4}+288 x^{2} \ln \left (x \right )-144 \ln \left (-2+x \right ) x^{2}-144 \ln \left (2+x \right ) x^{2}+120 x^{2}-384 \ln \left (x \right )+192 \ln \left (-2+x \right )+192 \ln \left (2+x \right )}{1536 \left (x^{2}-4\right )^{3}}\) | \(113\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=-\frac {12 \, x^{4} - 120 \, x^{2} + 3 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \left (x^{2} - 4\right ) - 6 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \left (x\right ) + 352}{1536 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {- 3 x^{4} + 30 x^{2} - 88}{384 x^{6} - 4608 x^{4} + 18432 x^{2} - 24576} + \frac {\log {\left (x \right )}}{256} - \frac {\log {\left (x^{2} - 4 \right )}}{512} \]
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Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=-\frac {3 \, x^{4} - 30 \, x^{2} + 88}{384 \, {\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} - \frac {1}{512} \, \log \left (x^{2} - 4\right ) + \frac {1}{512} \, \log \left (x^{2}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {11 \, x^{6} - 156 \, x^{4} + 768 \, x^{2} - 1408}{3072 \, {\left (x^{2} - 4\right )}^{3}} + \frac {1}{512} \, \log \left (x^{2}\right ) - \frac {1}{512} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \left (-4+x^2\right )^4} \, dx=\frac {\ln \left (x\right )}{256}-\frac {\ln \left (x^2-4\right )}{512}-\frac {\frac {x^4}{128}-\frac {5\,x^2}{64}+\frac {11}{48}}{x^6-12\,x^4+48\,x^2-64} \]
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