Optimal. Leaf size=76 \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0777883, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{2 \int \frac{1}{x^3 \left (a+b \log \left (c x^n\right )\right )} \, dx}{b n}\\ &=-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}-\frac{\left (2 \left (c x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2 x^2}\\ &=-\frac{2 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^2 n^2 x^2}-\frac{1}{b n x^2 \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.0910069, size = 80, normalized size = 1.05 \[ -\frac{2 e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right ) \text{Ei}\left (-\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+b n}{b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.725, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, 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*} -\frac{1}{b^{2} n x^{2} \log \left (x^{n}\right ) +{\left (b^{2} n \log \left (c\right ) + a b n\right )} x^{2}} - 2 \, \int \frac{1}{b^{2} n x^{3} \log \left (x^{n}\right ) +{\left (b^{2} n \log \left (c\right ) + a b n\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.952235, size = 247, normalized size = 3.25 \begin{align*} -\frac{2 \,{\left (b n x^{2} \log \left (x\right ) + b x^{2} \log \left (c\right ) + a x^{2}\right )} 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}{b^{3} n^{3} x^{2} \log \left (x\right ) + b^{3} n^{2} x^{2} \log \left (c\right ) + a b^{2} n^{2} x^{2}} \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 )^{2}}\, 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 )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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