Optimal. Leaf size=18 \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0246855, antiderivative size = 18, 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, 29} \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=\frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0158856, size = 18, normalized size = 1. \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05798, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.827532, size = 51, normalized size = 2.83 \begin{align*} \frac{\log \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.98887, size = 32, normalized size = 1.78 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac{\log{\left (x \right )}}{a + b \log{\left (c \right )}} & \text{for}\: n = 0 \\\frac{\log{\left (\frac{a}{b} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23544, size = 61, normalized size = 3.39 \begin{align*} \frac{\log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + a\right )}^{2}\right )}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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