3.23 \(\int \frac{x}{\left (b e^{-p x}+a e^{p x}\right )^2} \, dx\)

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

1/4*(2*a*p*x*e^(2*p*x) - a*e^(2*p*x)*ln(-a*e^(2*p*x) - b) - b*ln(-a*e^(2*p*x) -
b))/(a^2*b*p^2*e^(2*p*x) + a*b^2*p^2)