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

Optimal. Leaf size=63 \[ -\frac{\log \left (a f^{2 x}+b\right )}{4 a b \log ^2(f)}-\frac{x}{2 a \log (f) \left (a f^{2 x}+b\right )}+\frac{x}{2 a b \log (f)} \]

[Out]

x/(2*a*b*Log[f]) - x/(2*a*(b + a*f^(2*x))*Log[f]) - Log[b + a*f^(2*x)]/(4*a*b*Lo
g[f]^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.132029, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{\log \left (a f^{2 x}+b\right )}{4 a b \log ^2(f)}-\frac{x}{2 a \log (f) \left (a f^{2 x}+b\right )}+\frac{x}{2 a b \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[x/(b/f^x + a*f^x)^2,x]

[Out]

x/(2*a*b*Log[f]) - x/(2*a*(b + a*f^(2*x))*Log[f]) - Log[b + a*f^(2*x)]/(4*a*b*Lo
g[f]^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.8979, size = 54, normalized size = 0.86 \[ \frac{x}{2 b \left (a + b f^{- 2 x}\right ) \log{\left (f \right )}} + \frac{\log{\left (f^{- 2 x} \right )}}{4 a b \log{\left (f \right )}^{2}} - \frac{\log{\left (a + b f^{- 2 x} \right )}}{4 a b \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*b*(a + b*f**(-2*x))*log(f)) + log(f**(-2*x))/(4*a*b*log(f)**2) - log(a + b*
f**(-2*x))/(4*a*b*log(f)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0511954, size = 48, normalized size = 0.76 \[ \frac{\frac{2 x f^{2 x} \log (f)}{a f^{2 x}+b}-\frac{\log \left (a f^{2 x}+b\right )}{a}}{4 b \log ^2(f)} \]

Antiderivative was successfully verified.

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

[Out]

((2*f^(2*x)*x*Log[f])/(b + a*f^(2*x)) - Log[b + a*f^(2*x)]/a)/(4*b*Log[f]^2)

_______________________________________________________________________________________

Maple [A]  time = 0.025, size = 56, normalized size = 0.9 \[{\frac{x \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}}{2\,b\ln \left ( f \right ) \left ( \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}a+b \right ) }}-{\frac{\ln \left ( \left ({{\rm e}^{x\ln \left ( f \right ) }} \right ) ^{2}a+b \right ) }{4\, \left ( \ln \left ( f \right ) \right ) ^{2}ab}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b/(f^x)+a*f^x)^2,x)

[Out]

1/2/b/ln(f)*x*exp(x*ln(f))^2/(exp(x*ln(f))^2*a+b)-1/4/ln(f)^2/a/b*ln(exp(x*ln(f)
)^2*a+b)

_______________________________________________________________________________________

Maxima [A]  time = 0.750055, size = 73, normalized size = 1.16 \[ \frac{f^{2 \, x} x}{2 \,{\left (a b f^{2 \, x} \log \left (f\right ) + b^{2} \log \left (f\right )\right )}} - \frac{\log \left (\frac{a f^{2 \, x} + b}{a}\right )}{4 \, a b \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*f^(2*x)*x/(a*b*f^(2*x)*log(f) + b^2*log(f)) - 1/4*log((a*f^(2*x) + b)/a)/(a*
b*log(f)^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.28771, size = 82, normalized size = 1.3 \[ \frac{2 \, a f^{2 \, x} x \log \left (f\right ) -{\left (a f^{2 \, x} + b\right )} \log \left (a f^{2 \, x} + b\right )}{4 \,{\left (a^{2} b f^{2 \, x} \log \left (f\right )^{2} + a b^{2} \log \left (f\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*a*f^(2*x)*x*log(f) - (a*f^(2*x) + b)*log(a*f^(2*x) + b))/(a^2*b*f^(2*x)*l
og(f)^2 + a*b^2*log(f)^2)

_______________________________________________________________________________________

Sympy [A]  time = 0.425385, size = 54, normalized size = 0.86 \[ \frac{x}{2 a b \log{\left (f \right )} + 2 b^{2} f^{- 2 x} \log{\left (f \right )}} - \frac{x}{2 a b \log{\left (f \right )}} - \frac{\log{\left (\frac{a}{b} + f^{- 2 x} \right )}}{4 a b \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*a*b*log(f) + 2*b**2*f**(-2*x)*log(f)) - x/(2*a*b*log(f)) - log(a/b + f**(-2
*x))/(4*a*b*log(f)**2)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (a f^{x} + \frac{b}{f^{x}}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(a*f^x + b/f^x)^2, x)