Optimal. Leaf size=31 \[ -\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
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Rubi [A] time = 0.0145746, antiderivative size = 31, 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 = {2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log \left (1+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (1+e x)}{x} \, dx,x,\left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ &=-\frac{\text{Li}_2\left (-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)}\\ \end{align*}
Mathematica [A] time = 0.0012386, size = 31, normalized size = 1. \[ -\frac{\text{PolyLog}\left (2,-e \left (f^{c (a+b x)}\right )^n\right )}{b c n \log (f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 32, normalized size = 1. \begin{align*} -{\frac{{\it dilog} \left ( 1+e \left ({f}^{c \left ( bx+a \right ) } \right ) ^{n} \right ) }{ncb\ln \left ( f \right ) }} \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}{2} \, b c n x^{2} \log \left (f\right ) + b c n \int \frac{x}{e{\left (f^{b c x}\right )}^{n}{\left (f^{a c}\right )}^{n} + 1}\,{d x} \log \left (f\right ) + x \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 [A] time = 2.19991, size = 63, normalized size = 2.03 \begin{align*} -\frac{{\rm Li}_2\left (-e f^{b c n x + a c n}\right )}{b c n \log \left (f\right )} \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*} - b c e n e^{a c n \log{\left (f \right )}} \log{\left (f \right )} \int \frac{x 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 + x \log{\left (e \left (f^{c \left (a + b x\right )}\right )^{n} + 1 \right )} \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 \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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