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 = 0.108203, 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 \[ -\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}+\frac{x}{a^2}+\frac{1}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(F^(g*(e + f*x)))^n)^(-2),x]
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Rubi in Sympy [A] time = 20.2427, size = 75, normalized size = 1.01 \[ \frac{1}{a f g n \left (a + b \left (F^{g \left (e + f x\right )}\right )^{n}\right ) \log{\left (F \right )}} - \frac{\log{\left (a + b \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} + \frac{\log{\left (\left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(F**(g*(f*x+e)))**n)**2,x)
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Mathematica [A] time = 0.192189, size = 68, normalized size = 0.92 \[ \frac{\frac{a}{a f g n \log (F)+b f g n \log (F) \left (F^{e g+f g x}\right )^n}-\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)}+x}{a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(F^(g*(e + f*x)))^n)^(-2),x]
[Out]
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Maple [A] time = 0.003, size = 99, normalized size = 1.3 \[{\frac{\ln \left ( \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ){a}^{2}}}-{\frac{\ln \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) }{ngf\ln \left ( F \right ){a}^{2}}}+{\frac{1}{af \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) gn\ln \left ( F \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(F^(g*(f*x+e)))^n)^2,x)
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Maxima [A] time = 0.773008, size = 135, normalized size = 1.82 \[ \frac{1}{{\left ({\left (F^{f g x + e g}\right )}^{n} a b n + a^{2} n\right )} f g \log \left (F\right )} + \frac{\log \left (F^{f g x + e g}\right )}{a^{2} f g \log \left (F\right )} - \frac{\log \left (\frac{{\left (F^{f g x + e g}\right )}^{n} b + a}{b}\right )}{a^{2} f g n \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^(-2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27551, size = 135, normalized size = 1.82 \[ \frac{F^{f g n x + e g n} b f g n x \log \left (F\right ) + a f g n x \log \left (F\right ) -{\left (F^{f g n x + e g n} b + a\right )} \log \left (F^{f g n x + e g n} b + a\right ) + a}{F^{f g n x + e g n} a^{2} b f g n \log \left (F\right ) + a^{3} f g n \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^(-2),x, algorithm="fricas")
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Sympy [A] time = 0.427752, size = 66, normalized size = 0.89 \[ \frac{1}{a^{2} f g n \log{\left (F \right )} + a b f g n \left (F^{g \left (e + f x\right )}\right )^{n} \log{\left (F \right )}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + \left (F^{g \left (e + f x\right )}\right )^{n} \right )}}{a^{2} f g n \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(F**(g*(f*x+e)))**n)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^(-2),x, algorithm="giac")
[Out]