Optimal. Leaf size=132 \[ \frac{3 x^2 \text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-e \left (f^{c (a+b x)}\right )^n\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
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Rubi [A] time = 0.0969669, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2531, 6609, 2282, 6589} \[ \frac{3 x^2 \text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-e \left (f^{c (a+b x)}\right )^n\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=-\frac{x^3 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{3 \int x^2 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right ) \, dx}{b c n \log (f)}\\ &=-\frac{x^3 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 \int x \text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right ) \, dx}{b^2 c^2 n^2 \log ^2(f)}\\ &=-\frac{x^3 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \int \text{Li}_4\left (-e \left (f^{c (a+b x)}\right )^n\right ) \, dx}{b^3 c^3 n^3 \log ^3(f)}\\ &=-\frac{x^3 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-e x^n\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^4 c^4 n^3 \log ^4(f)}\\ &=-\frac{x^3 \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{3 x^2 \text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{Li}_4\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{Li}_5\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^4 c^4 n^4 \log ^4(f)}\\ \end{align*}
Mathematica [A] time = 0.0133012, size = 132, normalized size = 1. \[ \frac{3 x^2 \text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{6 x \text{PolyLog}\left (4,-e \left (f^{c (a+b x)}\right )^n\right )}{b^3 c^3 n^3 \log ^3(f)}+\frac{6 \text{PolyLog}\left (5,-e \left (f^{c (a+b x)}\right )^n\right )}{b^4 c^4 n^4 \log ^4(f)}-\frac{x^3 \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 645, normalized size = 4.9 \begin{align*} \text{result too large to display} \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}{20} \, b c n x^{5} \log \left (f\right ) + \frac{1}{4} \, x^{4} \log \left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + 1\right ) + b c n \int \frac{x^{4}}{4 \,{\left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + 1\right )}}\,{d x} \log \left (f\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.19526, size = 323, normalized size = 2.45 \begin{align*} -\frac{b^{3} c^{3} n^{3} x^{3}{\rm Li}_2\left (-e f^{b c n x + a c n}\right ) \log \left (f\right )^{3} - 3 \, b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2}{\rm polylog}\left (3, -e f^{b c n x + a c n}\right ) + 6 \, b c n x \log \left (f\right ){\rm polylog}\left (4, -e f^{b c n x + a c n}\right ) - 6 \,{\rm polylog}\left (5, -e f^{b c n x + a c n}\right )}{b^{4} c^{4} n^{4} \log \left (f\right )^{4}} \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*} - \frac{b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x^{4} e^{b c n x \log{\left (f \right )}}}{e e^{a c n \log{\left (f \right )}} e^{b c n x \log{\left (f \right )}} + 1}\, dx}{4} + \frac{x^{4} \log{\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )}}{4} \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 x^{3} \log \left (e{\left (f^{{\left (b x + a\right )} c}\right )}^{n} + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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