Optimal. Leaf size=40 \[ \frac{x}{a}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \]
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Rubi [A] time = 0.0300728, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2282, 266, 36, 29, 31} \[ \frac{x}{a}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)}\\ &=\frac{x}{a}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0061125, size = 40, normalized size = 1. \[ \frac{x}{a}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 65, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{\ln \left ( F \right ) afgn}}-{\frac{\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{\ln \left ( F \right ) afgn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06782, size = 89, normalized size = 2.22 \begin{align*} -\frac{\log \left ({\left (F^{f g x + e g}\right )}^{n} b + a\right )}{a f g n \log \left (F\right )} + \frac{\log \left ({\left (F^{f g x + e g}\right )}^{n}\right )}{a f g n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79227, size = 92, normalized size = 2.3 \begin{align*} \frac{f g n x \log \left (F\right ) - \log \left (F^{f g n x + e g n} b + a\right )}{a f g n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.200935, size = 27, normalized size = 0.68 \begin{align*} \frac{x}{a} - \frac{\log{\left (\frac{a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a f g n \log{\left (F \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 \frac{1}{{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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