3.58 \(\int \frac{1}{\left (b f^{-x}+a f^x\right )^2} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{2 a \log (f) \left (a f^{2 x}+b\right )} \]

[Out]

-1/(2*a*(b + a*f^(2*x))*Log[f])

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Rubi [A]  time = 0.0348567, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{1}{2 a \log (f) \left (a f^{2 x}+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[(b/f^x + a*f^x)^(-2),x]

[Out]

-1/(2*a*(b + a*f^(2*x))*Log[f])

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Rubi in Sympy [A]  time = 8.54269, size = 15, normalized size = 0.68 \[ \frac{1}{2 b \left (a + b f^{- 2 x}\right ) \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b/(f**x)+a*f**x)**2,x)

[Out]

1/(2*b*(a + b*f**(-2*x))*log(f))

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Mathematica [A]  time = 0.0213169, size = 23, normalized size = 1.05 \[ -\frac{1}{2 a^2 f^{2 x} \log (f)+2 a b \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[(b/f^x + a*f^x)^(-2),x]

[Out]

-(2*a*b*Log[f] + 2*a^2*f^(2*x)*Log[f])^(-1)

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Maple [A]  time = 0.004, size = 21, normalized size = 1. \[ -{\frac{1}{2\,\ln \left ( f \right ) a \left ( a \left ({f}^{x} \right ) ^{2}+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b/(f^x)+a*f^x)^2,x)

[Out]

-1/2/ln(f)/a/(a*(f^x)^2+b)

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Maxima [A]  time = 0.752802, size = 31, normalized size = 1.41 \[ \frac{1}{2 \,{\left (a b + \frac{b^{2}}{f^{2 \, x}}\right )} \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*f^x + b/f^x)^(-2),x, algorithm="maxima")

[Out]

1/2/((a*b + b^2/f^(2*x))*log(f))

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Fricas [A]  time = 0.251931, size = 28, normalized size = 1.27 \[ -\frac{1}{2 \,{\left (a^{2} f^{2 \, x} \log \left (f\right ) + a b \log \left (f\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*f^x + b/f^x)^(-2),x, algorithm="fricas")

[Out]

-1/2/(a^2*f^(2*x)*log(f) + a*b*log(f))

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Sympy [A]  time = 0.227472, size = 22, normalized size = 1. \[ \frac{1}{2 a b \log{\left (f \right )} + 2 b^{2} f^{- 2 x} \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b/(f**x)+a*f**x)**2,x)

[Out]

1/(2*a*b*log(f) + 2*b**2*f**(-2*x)*log(f))

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GIAC/XCAS [A]  time = 0.236256, size = 27, normalized size = 1.23 \[ -\frac{1}{2 \,{\left (a f^{2 \, x} + b\right )} a{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*f^x + b/f^x)^(-2),x, algorithm="giac")

[Out]

-1/2/((a*f^(2*x) + b)*a*ln(f))