Optimal. Leaf size=22 \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]
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Rubi [A] time = 0.0083307, 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} \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]
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^5+x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \left (a^5+x\right )} \, dx,x,x^5\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )}{5 a^5}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^5+x} \, dx,x,x^5\right )}{5 a^5}\\ &=\frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5}\\ \end{align*}
Mathematica [A] time = 0.0039255, size = 22, normalized size = 1. \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 49, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ({a}^{4}-{a}^{3}x+{a}^{2}{x}^{2}-a{x}^{3}+{x}^{4} \right ) }{5\,{a}^{5}}}+{\frac{\ln \left ( x \right ) }{{a}^{5}}}-{\frac{\ln \left ( a+x \right ) }{5\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927172, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac{\log \left (x^{5}\right )}{5 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89974, size = 51, normalized size = 2.32 \begin{align*} -\frac{\log \left (a^{5} + x^{5}\right ) - 5 \, \log \left (x\right )}{5 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.224743, size = 19, normalized size = 0.86 \begin{align*} \frac{\log{\left (x \right )}}{a^{5}} - \frac{\log{\left (a^{5} + x^{5} \right )}}{5 a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06726, size = 30, normalized size = 1.36 \begin{align*} -\frac{\log \left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac{\log \left ({\left | x \right |}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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