Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}-\frac{1}{2 a^4 x^2} \]
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Rubi [A] time = 0.0121596, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {275, 325, 206} \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}-\frac{1}{2 a^4 x^2} \]
Antiderivative was successfully verified.
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Rule 275
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a^4-x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a^4-x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^4 x^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^4-x^2} \, dx,x,x^2\right )}{2 a^4}\\ &=-\frac{1}{2 a^4 x^2}+\frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}\\ \end{align*}
Mathematica [A] time = 0.0068284, size = 50, normalized size = 1.92 \[ -\frac{1}{2 a^4 x^2}+\frac{\log \left (a^2+x^2\right )}{4 a^6}-\frac{\log (a-x)}{4 a^6}-\frac{\log (a+x)}{4 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 43, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{6}}}+{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{6}}}-{\frac{\ln \left ( a+x \right ) }{4\,{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924299, size = 50, normalized size = 1.92 \begin{align*} \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{\log \left (-a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{1}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8199, size = 89, normalized size = 3.42 \begin{align*} \frac{x^{2} \log \left (a^{2} + x^{2}\right ) - x^{2} \log \left (-a^{2} + x^{2}\right ) - 2 \, a^{2}}{4 \, a^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.347397, size = 34, normalized size = 1.31 \begin{align*} - \frac{1}{2 a^{4} x^{2}} - \frac{\frac{\log{\left (- a^{2} + x^{2} \right )}}{4} - \frac{\log{\left (a^{2} + x^{2} \right )}}{4}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07005, size = 51, normalized size = 1.96 \begin{align*} \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{6}} - \frac{1}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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