Optimal. Leaf size=62 \[ -\frac{\log \left (a e^{2 p x}+b\right )}{4 a b p^2}+\frac{x}{2 a b p}-\frac{x}{2 a p \left (a e^{2 p x}+b\right )} \]
[Out]
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Rubi [A] time = 0.154055, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{\log \left (a e^{2 p x}+b\right )}{4 a b p^2}+\frac{x}{2 a b p}-\frac{x}{2 a p \left (a e^{2 p x}+b\right )} \]
Antiderivative was successfully verified.
[In] Int[x/(b/E^(p*x) + a*E^(p*x))^2,x]
[Out]
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Rubi in Sympy [A] time = 10.1021, size = 54, normalized size = 0.87 \[ \frac{x}{2 b p \left (a + b e^{- 2 p x}\right )} - \frac{\log{\left (a + b e^{- 2 p x} \right )}}{4 a b p^{2}} + \frac{\log{\left (e^{- 2 p x} \right )}}{4 a b p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b/exp(p*x)+a*exp(p*x))**2,x)
[Out]
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Mathematica [A] time = 0.0517093, size = 49, normalized size = 0.79 \[ \frac{\frac{2 p x e^{2 p x}}{a e^{2 p x}+b}-\frac{\log \left (a e^{2 p x}+b\right )}{a}}{4 b p^2} \]
Antiderivative was successfully verified.
[In] Integrate[x/(b/E^(p*x) + a*E^(p*x))^2,x]
[Out]
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Maple [A] time = 0.019, size = 51, normalized size = 0.8 \[ -{\frac{\ln \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }{4\,{p}^{2}ba}}+{\frac{x \left ({{\rm e}^{px}} \right ) ^{2}}{2\,bp \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b/exp(p*x)+a*exp(p*x))^2,x)
[Out]
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Maxima [A] time = 1.36774, size = 69, normalized size = 1.11 \[ \frac{x e^{\left (2 \, p x\right )}}{2 \,{\left (a b p e^{\left (2 \, p x\right )} + b^{2} p\right )}} - \frac{\log \left (\frac{a e^{\left (2 \, p x\right )} + b}{a}\right )}{4 \, a b p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*e^(p*x) + b*e^(-p*x))^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230856, size = 78, normalized size = 1.26 \[ \frac{2 \, a p x e^{\left (2 \, p x\right )} -{\left (a e^{\left (2 \, p x\right )} + b\right )} \log \left (a e^{\left (2 \, p x\right )} + b\right )}{4 \,{\left (a^{2} b p^{2} e^{\left (2 \, p x\right )} + a b^{2} p^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*e^(p*x) + b*e^(-p*x))^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.215637, size = 49, normalized size = 0.79 \[ - \frac{x}{2 a^{2} p e^{2 p x} + 2 a b p} + \frac{x}{2 a b p} - \frac{\log{\left (e^{2 p x} + \frac{b}{a} \right )}}{4 a b p^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b/exp(p*x)+a*exp(p*x))**2,x)
[Out]
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GIAC/XCAS [A] time = 0.198747, size = 100, normalized size = 1.61 \[ \frac{2 \, a p x e^{\left (2 \, p x\right )} - a e^{\left (2 \, p x\right )}{\rm ln}\left (-a e^{\left (2 \, p x\right )} - b\right ) - b{\rm ln}\left (-a e^{\left (2 \, p x\right )} - b\right )}{4 \,{\left (a^{2} b p^{2} e^{\left (2 \, p x\right )} + a b^{2} p^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a*e^(p*x) + b*e^(-p*x))^2,x, algorithm="giac")
[Out]