3.102 \(\int e^{-x} \cot ^{-1}\left (e^x\right ) \, dx\)

Optimal. Leaf size=27 \[ -x+\frac{1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

[Out]

-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2

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Rubi [A]  time = 0.0389215, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -x+\frac{1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcCot[E^x]/E^x,x]

[Out]

-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2

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Rubi in Sympy [A]  time = 2.94892, size = 26, normalized size = 0.96 \[ \frac{\log{\left (e^{2 x} + 1 \right )}}{2} - \frac{\log{\left (e^{2 x} \right )}}{2} - e^{- x} \operatorname{acot}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acot(exp(x))/exp(x),x)

[Out]

log(exp(2*x) + 1)/2 - log(exp(2*x))/2 - exp(-x)*acot(exp(x))

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Mathematica [A]  time = 0.0164087, size = 24, normalized size = 0.89 \[ \frac{1}{2} \log \left (e^{-2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCot[E^x]/E^x,x]

[Out]

-(ArcCot[E^x]/E^x) + Log[1 + E^(-2*x)]/2

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Maple [A]  time = 0.01, size = 25, normalized size = 0.9 \[ -{\frac{{\rm arccot} \left ({{\rm e}^{x}}\right )}{{{\rm e}^{x}}}}+{\frac{\ln \left ( \left ({{\rm e}^{x}} \right ) ^{2}+1 \right ) }{2}}-\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccot(exp(x))/exp(x),x)

[Out]

-arccot(exp(x))/exp(x)+1/2*ln(exp(x)^2+1)-ln(exp(x))

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Maxima [A]  time = 1.38409, size = 26, normalized size = 0.96 \[ -\operatorname{arccot}\left (e^{x}\right ) e^{\left (-x\right )} + \frac{1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccot(e^x)*e^(-x),x, algorithm="maxima")

[Out]

-arccot(e^x)*e^(-x) + 1/2*log(e^(-2*x) + 1)

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Fricas [A]  time = 0.228226, size = 38, normalized size = 1.41 \[ -\frac{1}{2} \,{\left (2 \, x e^{x} - e^{x} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, \operatorname{arccot}\left (e^{x}\right )\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccot(e^x)*e^(-x),x, algorithm="fricas")

[Out]

-1/2*(2*x*e^x - e^x*log(e^(2*x) + 1) + 2*arccot(e^x))*e^(-x)

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Sympy [A]  time = 22.7683, size = 19, normalized size = 0.7 \[ - x + \frac{\log{\left (e^{2 x} + 1 \right )}}{2} - e^{- x} \operatorname{acot}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acot(exp(x))/exp(x),x)

[Out]

-x + log(exp(2*x) + 1)/2 - exp(-x)*acot(exp(x))

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GIAC/XCAS [A]  time = 0.208363, size = 28, normalized size = 1.04 \[ -\arctan \left (e^{\left (-x\right )}\right ) e^{\left (-x\right )} + \frac{1}{2} \,{\rm ln}\left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccot(e^x)*e^(-x),x, algorithm="giac")

[Out]

-arctan(e^(-x))*e^(-x) + 1/2*ln(e^(-2*x) + 1)