Optimal. Leaf size=39 \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]
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Rubi [A] time = 0.0259307, antiderivative size = 39, 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 = {2279, 2392, 2391} \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2392
Rule 2391
Rubi steps
\begin{align*} \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (\pi +b x)}{x} \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=x \log (\pi )+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{\pi }\right )}{x} \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=x \log (\pi )-\frac{\text{Li}_2\left (-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0086595, size = 39, normalized size = 1. \[ x \log (\pi )-\frac{\text{PolyLog}\left (2,-\frac{b \left (F^{e (c+d x)}\right )^n}{\pi }\right )}{d e n \log (F)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 138, normalized size = 3.5 \begin{align*} -{\frac{1}{nde\ln \left ( F \right ) }\ln \left ( -{\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}}{\pi }} \right ) \ln \left ({\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi }{\pi }} \right ) }+{\frac{\ln \left ( b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi \right ) }{nde\ln \left ( F \right ) }\ln \left ( -{\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}}{\pi }} \right ) }-{\frac{1}{nde\ln \left ( F \right ) }{\it dilog} \left ({\frac{b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n}+\pi }{\pi }} \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} \, d e n x^{2} \log \left (F\right ) + \pi d e n \int \frac{x}{\pi +{\left (F^{d e x}\right )}^{n}{\left (F^{c e}\right )}^{n} b}\,{d x} \log \left (F\right ) + x \log \left (\pi +{\left (F^{d e x}\right )}^{n}{\left (F^{c e}\right )}^{n} b\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14814, size = 250, normalized size = 6.41 \begin{align*} \frac{{\left (d e n x + c e n\right )} \log \left (\pi + F^{d e n x + c e n} b\right ) \log \left (F\right ) -{\left (d e n x + c e n\right )} \log \left (F\right ) \log \left (\frac{\pi + F^{d e n x + c e n} b}{\pi }\right ) -{\rm Li}_2\left (-\frac{\pi + F^{d e n x + c e n} b}{\pi } + 1\right )}{d e 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 d e n e^{c e n \log{\left (F \right )}} \log{\left (F \right )} \int \frac{x e^{d e n x \log{\left (F \right )}}}{b e^{c e n \log{\left (F \right )}} e^{d e n x \log{\left (F \right )}} + \pi }\, dx + x \log{\left (b \left (F^{e \left (c + d x\right )}\right )^{n} + \pi \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 (\pi +{\left (F^{{\left (d x + c\right )} e}\right )}^{n} b\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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