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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]