3.705 \(\int e^x \sin ^{-1}(\tanh (x)) \, dx\)

Optimal. Leaf size=28 \[ e^x \sin ^{-1}(\tanh (x))-\log \left (e^{2 x}+1\right ) \cosh (x) \sqrt{\text{sech}^2(x)} \]

[Out]

E^x*ArcSin[Tanh[x]] - Cosh[x]*Log[1 + E^(2*x)]*Sqrt[Sech[x]^2]

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Rubi [B]  time = 0.0870904, antiderivative size = 67, normalized size of antiderivative = 2.39, number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714 \[ -e^{-x} \sqrt{\frac{e^{2 x}}{\left (e^{2 x}+1\right )^2}} \left (e^{2 x}+1\right ) \log \left (e^{2 x}+1\right )-e^x \sin ^{-1}\left (\frac{1-e^{2 x}}{e^{2 x}+1}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Int[E^x*ArcSin[Tanh[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.42831, size = 42, normalized size = 1.5 \[ - \sqrt{\frac{e^{2 x}}{\left (e^{2 x} + 1\right )^{2}}} \left (e^{2 x} + 1\right ) e^{- x} \log{\left (e^{2 x} + 1 \right )} + e^{x} \operatorname{asin}{\left (\tanh{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)*asin(tanh(x)),x)

[Out]

-sqrt(exp(2*x)/(exp(2*x) + 1)**2)*(exp(2*x) + 1)*exp(-x)*log(exp(2*x) + 1) + exp
(x)*asin(tanh(x))

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Mathematica [B]  time = 0.851244, size = 64, normalized size = 2.29 \[ e^x \sin ^{-1}\left (\frac{e^{2 x}-1}{e^{2 x}+1}\right )-e^{-x} \sqrt{\frac{e^{2 x}}{\left (e^{2 x}+1\right )^2}} \left (e^{2 x}+1\right ) \log \left (e^{2 x}+1\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[E^x*ArcSin[Tanh[x]],x]

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{{\rm e}^{x}}\arcsin \left ( \tanh \left ( x \right ) \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*arcsin(tanh(x)),x)

[Out]

int(exp(x)*arcsin(tanh(x)),x)

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Maxima [A]  time = 1.63831, size = 22, normalized size = 0.79 \[ \arcsin \left (\tanh \left (x\right )\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(tanh(x))*e^x,x, algorithm="maxima")

[Out]

arcsin(tanh(x))*e^x - log(e^(2*x) + 1)

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Fricas [A]  time = 0.227128, size = 35, normalized size = 1.25 \[{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\sinh \left (x\right )\right ) - \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(tanh(x))*e^x,x, algorithm="fricas")

[Out]

(cosh(x) + sinh(x))*arctan(sinh(x)) - log(2*cosh(x)/(cosh(x) - sinh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int e^{x} \operatorname{asin}{\left (\tanh{\left (x \right )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*asin(tanh(x)),x)

[Out]

Integral(exp(x)*asin(tanh(x)), x)

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GIAC/XCAS [A]  time = 0.203641, size = 39, normalized size = 1.39 \[ \arcsin \left (\frac{e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right ) e^{x} -{\rm ln}\left (e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arcsin(tanh(x))*e^x,x, algorithm="giac")

[Out]

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