Optimal. Leaf size=22 \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]
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Rubi [A] time = 0.0083182, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^3+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \left (a^3+x\right )} \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )}{3 a^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^3+x} \, dx,x,x^3\right )}{3 a^3}\\ &=\frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0044086, size = 22, normalized size = 1. \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 34, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{3\,{a}^{3}}}-{\frac{\ln \left ( a+x \right ) }{3\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.922174, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (a^{3} + x^{3}\right )}{3 \, a^{3}} + \frac{\log \left (x^{3}\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01846, size = 51, normalized size = 2.32 \begin{align*} -\frac{\log \left (a^{3} + x^{3}\right ) - 3 \, \log \left (x\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.17946, size = 19, normalized size = 0.86 \begin{align*} \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (a^{3} + x^{3} \right )}}{3 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05538, size = 30, normalized size = 1.36 \begin{align*} -\frac{\log \left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{3}} + \frac{\log \left ({\left | x \right |}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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