Optimal. Leaf size=63 \[ \frac{\text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
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Rubi [A] time = 0.0382002, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2531, 2282, 6589} \[ \frac{\text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=-\frac{x \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{\int \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right ) \, dx}{b c n \log (f)}\\ &=-\frac{x \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-e x^n\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^2 c^2 n \log ^2(f)}\\ &=-\frac{x \text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}+\frac{\text{Li}_3\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}\\ \end{align*}
Mathematica [A] time = 0.0047085, size = 63, normalized size = 1. \[ \frac{\text{PolyLog}\left (3,-e \left (f^{c (a+b x)}\right )^n\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac{x \text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 282, normalized size = 4.5 \begin{align*}{\frac{{x}^{2}\ln \left ( 1+e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n} \right ) }{2}}-{\frac{\ln \left ( 1+e{f}^{bcnx}{{\rm e}^{-n \left ( \ln \left ( f \right ) bcx-\ln \left ({f}^{c \left ( bx+a \right ) } \right ) \right ) }} \right ){x}^{2}}{2}}-{\frac{{\it polylog} \left ( 2,-e{f}^{bcnx}{{\rm e}^{-n \left ( \ln \left ( f \right ) bcx-\ln \left ({f}^{c \left ( bx+a \right ) } \right ) \right ) }} \right ) \ln \left ({f}^{c \left ( bx+a \right ) } \right ) }{{c}^{2}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}n}}+{\frac{{\it polylog} \left ( 3,-e{f}^{bcnx}{{\rm e}^{-n \left ( \ln \left ( f \right ) bcx-\ln \left ({f}^{c \left ( bx+a \right ) } \right ) \right ) }} \right ) }{{c}^{2}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}{n}^{2}}}-{\frac{{\it dilog} \left ( 1+e{f}^{bcnx}{{\rm e}^{-n \left ( \ln \left ( f \right ) bcx-\ln \left ({f}^{c \left ( bx+a \right ) } \right ) \right ) }} \right ) x}{ncb\ln \left ( f \right ) }}+{\frac{{\it dilog} \left ( 1+e{f}^{bcnx}{{\rm e}^{-n \left ( \ln \left ( f \right ) bcx-\ln \left ({f}^{c \left ( bx+a \right ) } \right ) \right ) }} \right ) \ln \left ({f}^{c \left ( bx+a \right ) } \right ) }{{c}^{2}{b}^{2} \left ( \ln \left ( f \right ) \right ) ^{2}n}} \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}{6} \, b c n x^{3} \log \left (f\right ) + b c n \int \frac{x^{2}}{2 \,{\left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + 1\right )}}\,{d x} \log \left (f\right ) + \frac{1}{2} \, x^{2} \log \left (e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.21496, size = 147, normalized size = 2.33 \begin{align*} -\frac{b c n x{\rm Li}_2\left (-e f^{b c n x + a c n}\right ) \log \left (f\right ) -{\rm polylog}\left (3, -e f^{b c n x + a c n}\right )}{b^{2} c^{2} n^{2} \log \left (f\right )^{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*} - \frac{b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x^{2} 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}{2} + \frac{x^{2} \log{\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )}}{2} \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 \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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