Optimal. Leaf size=74 \[ -\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}+\frac{x}{a^2}+\frac{1}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
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Rubi [A] time = 0.0494327, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2282, 266, 44} \[ -\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}+\frac{x}{a^2}+\frac{1}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^n\right )^2} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}\\ &=\frac{x}{a^2}+\frac{1}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0588887, size = 62, normalized size = 0.84 \[ \frac{\frac{a}{a+b \left (F^{g (e+f x)}\right )^n}-\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )+f g n x \log (F)}{a^2 f g n \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 99, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ){a}^{2}}}-{\frac{\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ){a}^{2}}}+{\frac{1}{af \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) gn\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02679, size = 135, normalized size = 1.82 \begin{align*} \frac{1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \left (F\right )} + \frac{\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \left (F\right )} - \frac{\log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} 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 = 1.56327, size = 248, normalized size = 3.35 \begin{align*} \frac{F^{f g n x + e g n} b f g n x \log \left (F\right ) + a f g n x \log \left (F\right ) -{\left (F^{f g n x + e g n} b + a\right )} \log \left (F^{f g n x + e g n} b + a\right ) + a}{F^{f g n x + e g n} a^{2} b f g n \log \left (F\right ) + a^{3} 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 = 0.202329, size = 66, normalized size = 0.89 \begin{align*} \frac{1}{a^{2} f g n \log{\left (F \right )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} 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 ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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