Optimal. Leaf size=54 \[ \frac{1}{4 a^8 \left (a^4+x^4\right )}+\frac{1}{8 a^4 \left (a^4+x^4\right )^2}-\frac{\log \left (a^4+x^4\right )}{4 a^{12}}+\frac{\log (x)}{a^{12}} \]
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Rubi [A] time = 0.0302035, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{1}{4 a^8 \left (a^4+x^4\right )}+\frac{1}{8 a^4 \left (a^4+x^4\right )^2}-\frac{\log \left (a^4+x^4\right )}{4 a^{12}}+\frac{\log (x)}{a^{12}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^4+x^4\right )^3} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \left (a^4+x\right )^3} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{a^{12} x}-\frac{1}{a^4 \left (a^4+x\right )^3}-\frac{1}{a^8 \left (a^4+x\right )^2}-\frac{1}{a^{12} \left (a^4+x\right )}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{8 a^4 \left (a^4+x^4\right )^2}+\frac{1}{4 a^8 \left (a^4+x^4\right )}+\frac{\log (x)}{a^{12}}-\frac{\log \left (a^4+x^4\right )}{4 a^{12}}\\ \end{align*}
Mathematica [A] time = 0.0243026, size = 46, normalized size = 0.85 \[ \frac{\frac{2 a^4 x^4+3 a^8}{\left (a^4+x^4\right )^2}-2 \log \left (a^4+x^4\right )+8 \log (x)}{8 a^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{8\,{a}^{4} \left ({a}^{4}+{x}^{4} \right ) ^{2}}}+{\frac{1}{4\,{a}^{8} \left ({a}^{4}+{x}^{4} \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{12}}}-{\frac{\ln \left ({a}^{4}+{x}^{4} \right ) }{4\,{a}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927591, size = 77, normalized size = 1.43 \begin{align*} \frac{3 \, a^{4} + 2 \, x^{4}}{8 \,{\left (a^{16} + 2 \, a^{12} x^{4} + a^{8} x^{8}\right )}} - \frac{\log \left (a^{4} + x^{4}\right )}{4 \, a^{12}} + \frac{\log \left (x^{4}\right )}{4 \, a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9711, size = 181, normalized size = 3.35 \begin{align*} \frac{3 \, a^{8} + 2 \, a^{4} x^{4} - 2 \,{\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )} \log \left (a^{4} + x^{4}\right ) + 8 \,{\left (a^{8} + 2 \, a^{4} x^{4} + x^{8}\right )} \log \left (x\right )}{8 \,{\left (a^{20} + 2 \, a^{16} x^{4} + a^{12} x^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.21095, size = 51, normalized size = 0.94 \begin{align*} \frac{3 a^{4} + 2 x^{4}}{8 a^{16} + 16 a^{12} x^{4} + 8 a^{8} x^{8}} + \frac{\log{\left (x \right )}}{a^{12}} - \frac{\log{\left (a^{4} + x^{4} \right )}}{4 a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05378, size = 76, normalized size = 1.41 \begin{align*} -\frac{\log \left (a^{4} + x^{4}\right )}{4 \, a^{12}} + \frac{\log \left (x^{4}\right )}{4 \, a^{12}} + \frac{6 \, a^{8} + 8 \, a^{4} x^{4} + 3 \, x^{8}}{8 \,{\left (a^{4} + x^{4}\right )}^{2} a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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