3.55 \(\int \frac{1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^2 f g n \log (F)}+\frac{x}{a^2}+\frac{1}{a f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )} \]

[Out]

x/a^2 + 1/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - Log[a + b*(F^(g*(e + f*
x)))^n]/(a^2*f*g*n*Log[F])

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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]

[Out]

x/a^2 + 1/(a*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F]) - Log[a + b*(F^(g*(e + f*
x)))^n]/(a^2*f*g*n*Log[F])

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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)

[Out]

1/(a*f*g*n*(a + b*(F**(g*(e + f*x)))**n)*log(F)) - log(a + b*(F**(g*(e + f*x)))*
*n)/(a**2*f*g*n*log(F)) + log((F**(g*(e + f*x)))**n)/(a**2*f*g*n*log(F))

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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]

(x + a/(a*f*g*n*Log[F] + b*f*(F^(e*g + f*g*x))^n*g*n*Log[F]) - Log[a + b*(F^(g*(
e + f*x)))^n]/(f*g*n*Log[F]))/a^2

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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)

[Out]

1/g/f/ln(F)/n/a^2*ln((F^(g*(f*x+e)))^n)-ln(a+b*(F^(g*(f*x+e)))^n)/a^2/f/g/n/ln(F
)+1/a/f/(a+b*(F^(g*(f*x+e)))^n)/g/n/ln(F)

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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]

1/(((F^(f*g*x + e*g))^n*a*b*n + a^2*n)*f*g*log(F)) + log(F^(f*g*x + e*g))/(a^2*f
*g*log(F)) - log(((F^(f*g*x + e*g))^n*b + a)/b)/(a^2*f*g*n*log(F))

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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")

[Out]

(F^(f*g*n*x + e*g*n)*b*f*g*n*x*log(F) + a*f*g*n*x*log(F) - (F^(f*g*n*x + e*g*n)*
b + a)*log(F^(f*g*n*x + e*g*n)*b + a) + a)/(F^(f*g*n*x + e*g*n)*a^2*b*f*g*n*log(
F) + a^3*f*g*n*log(F))

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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]

1/(a**2*f*g*n*log(F) + a*b*f*g*n*(F**(g*(e + f*x)))**n*log(F)) + x/a**2 - log(a/
b + (F**(g*(e + f*x)))**n)/(a**2*f*g*n*log(F))

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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]

integrate(((F^((f*x + e)*g))^n*b + a)^(-2), x)