Optimal. Leaf size=22 \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.0235673, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2302, 30} \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=-\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0038617, size = 22, normalized size = 1. \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,bn \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09555, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.792293, size = 150, normalized size = 6.82 \begin{align*} -\frac{1}{2 \,{\left (b^{3} n^{3} \log \left (x\right )^{2} + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n + 2 \,{\left (b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28818, size = 28, normalized size = 1.27 \begin{align*} -\frac{1}{2 \,{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{2} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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