Optimal. Leaf size=48 \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
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Rubi [A] time = 0.0358402, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2300, 2178} \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{e^{-\frac{a}{b n}} x \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0410216, size = 48, normalized size = 1. \[ \frac{x e^{-\frac{a}{b n}} \left (c x^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.267, size = 241, normalized size = 5. \begin{align*} -{\frac{1}{bn}{{\rm e}^{-{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( x \right ) bn+2\,b\ln \left ( c \right ) +2\,b\ln \left ({x}^{n} \right ) +2\,a}{2\,bn}}}}{\it Ei} \left ( 1,-\ln \left ( x \right ) -{\frac{ib\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-ib\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+ib\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,b\ln \left ( c \right ) +2\,b \left ( \ln \left ({x}^{n} \right ) -n\ln \left ( x \right ) \right ) +2\,a}{2\,bn}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.836507, size = 100, normalized size = 2.08 \begin{align*} \frac{e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left (x e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \log{\left (c x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21367, size = 57, normalized size = 1.19 \begin{align*} \frac{{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x\right )\right ) e^{\left (-\frac{a}{b n}\right )}}{b c^{\left (\frac{1}{n}\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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