Optimal. Leaf size=51 \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
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Rubi [A] time = 0.0513079, antiderivative size = 51, 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 = {2310, 2178} \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \frac{e^{-\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{n x^2}\\ &=\frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2}\\ \end{align*}
Mathematica [A] time = 0.0500862, size = 51, normalized size = 1. \[ \frac{e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.165, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}\, dx \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}{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.785426, size = 108, normalized size = 2.12 \begin{align*} \frac{e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left (\frac{e^{\left (-\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{x^{2}}\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}{x^{3} \left (a + b \log{\left (c x^{n} \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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