Optimal. Leaf size=76 \[ \frac{4 x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac{x^4}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0785481, 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} \[ \frac{4 x^4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac{x^4}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{x^4}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac{4 \int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx}{b n}\\ &=-\frac{x^4}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (4 x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2}\\ &=\frac{4 e^{-\frac{4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac{x^4}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.118405, size = 70, normalized size = 0.92 \[ \frac{x^4 \left (4 e^{-\frac{4 a}{b n}} \left (c x^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac{b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.747, size = 0, normalized size = 0. \begin{align*} \int{\frac{{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{x^{4}}{b^{2} n \log \left (c\right ) + b^{2} n \log \left (x^{n}\right ) + a b n} + 4 \, \int \frac{x^{3}}{b^{2} n \log \left (c\right ) + b^{2} n \log \left (x^{n}\right ) + a b n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.833669, size = 257, normalized size = 3.38 \begin{align*} -\frac{{\left (b n x^{4} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} - 4 \,{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \logintegral \left (x^{4} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{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{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 [B] time = 1.28218, size = 352, normalized size = 4.63 \begin{align*} -\frac{b n x^{4}}{b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}} + \frac{4 \, b n{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac{4 \, a}{b n}\right )} \log \left (x\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{4}{n}}} + \frac{4 \, b{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac{4 \, a}{b n}\right )} \log \left (c\right )}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{4}{n}}} + \frac{4 \, a{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x\right )\right ) e^{\left (-\frac{4 \, a}{b n}\right )}}{{\left (b^{3} n^{3} \log \left (x\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} c^{\frac{4}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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