Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {1}{a^4 x}-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^5}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^5} \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {331, 304, 209, 212} \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^5}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^5}-\frac {1}{a^4 x} \]
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Rule 209
Rule 212
Rule 304
Rule 331
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a^4 x}+\frac {\int \frac {x^2}{a^4-x^4} \, dx}{a^4} \\ & = -\frac {1}{a^4 x}+\frac {\int \frac {1}{a^2-x^2} \, dx}{2 a^4}-\frac {\int \frac {1}{a^2+x^2} \, dx}{2 a^4} \\ & = -\frac {1}{a^4 x}-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^5}+\frac {\text {arctanh}\left (\frac {x}{a}\right )}{2 a^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {1}{a^4 x}-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^5}-\frac {\log (a-x)}{4 a^5}+\frac {\log (a+x)}{4 a^5} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {1}{a^{4} x}-\frac {\ln \left (a -x \right )}{4 a^{5}}-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{5}}+\frac {\ln \left (a +x \right )}{4 a^{5}}\) | \(41\) |
risch | \(-\frac {1}{a^{4} x}-\frac {\arctan \left (\frac {x}{a}\right )}{2 a^{5}}-\frac {\ln \left (-a +x \right )}{4 a^{5}}+\frac {\ln \left (a +x \right )}{4 a^{5}}\) | \(41\) |
parallelrisch | \(-\frac {-i \ln \left (-i a +x \right ) x +i \ln \left (i a +x \right ) x +\ln \left (-a +x \right ) x -\ln \left (a +x \right ) x +4 a}{4 a^{5} x}\) | \(50\) |
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {2 \, x \arctan \left (\frac {x}{a}\right ) - x \log \left (a + x\right ) + x \log \left (-a + x\right ) + 4 \, a}{4 \, a^{5} x} \]
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Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=- \frac {1}{a^{4} x} - \frac {\frac {\log {\left (- a + x \right )}}{4} - \frac {\log {\left (a + x \right )}}{4} - \frac {i \log {\left (- i a + x \right )}}{4} + \frac {i \log {\left (i a + x \right )}}{4}}{a^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{5}} + \frac {\log \left (a + x\right )}{4 \, a^{5}} - \frac {\log \left (-a + x\right )}{4 \, a^{5}} - \frac {1}{a^{4} x} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=-\frac {\arctan \left (\frac {x}{a}\right )}{2 \, a^{5}} + \frac {\log \left ({\left | a + x \right |}\right )}{4 \, a^{5}} - \frac {\log \left ({\left | -a + x \right |}\right )}{4 \, a^{5}} - \frac {1}{a^{4} x} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a^4-x^4\right )} \, dx=\frac {\mathrm {atanh}\left (\frac {x}{a}\right )}{2\,a^5}-\frac {\mathrm {atan}\left (\frac {x}{a}\right )}{2\,a^5}-\frac {1}{a^4\,x} \]
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