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]
________________________________________________________________________________________
Rubi [A] time = 0.0769321, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {2194, 4844, 6720, 2282, 12, 260} \[ e^x \sin ^{-1}(\tanh (x))-\log \left (e^{2 x}+1\right ) \cosh (x) \sqrt{\text{sech}^2(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2194
Rule 4844
Rule 6720
Rule 2282
Rule 12
Rule 260
Rubi steps
\begin{align*} \int e^x \sin ^{-1}(\tanh (x)) \, dx &=e^x \sin ^{-1}(\tanh (x))-\int e^x \sqrt{\text{sech}^2(x)} \, dx\\ &=e^x \sin ^{-1}(\tanh (x))-\left (\cosh (x) \sqrt{\text{sech}^2(x)}\right ) \int e^x \text{sech}(x) \, dx\\ &=e^x \sin ^{-1}(\tanh (x))-\left (\cosh (x) \sqrt{\text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{2 x}{1+x^2} \, dx,x,e^x\right )\\ &=e^x \sin ^{-1}(\tanh (x))-\left (2 \cosh (x) \sqrt{\text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,e^x\right )\\ &=e^x \sin ^{-1}(\tanh (x))-\cosh (x) \log \left (1+e^{2 x}\right ) \sqrt{\text{sech}^2(x)}\\ \end{align*}
Mathematica [B] time = 0.837499, 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]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{x}}\arcsin \left ( \tanh \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.64726, size = 22, normalized size = 0.79 \begin{align*} \arcsin \left (\tanh \left (x\right )\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21292, size = 100, normalized size = 3.57 \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{x} \operatorname{asin}{\left (\tanh{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07862, size = 51, normalized size = 1.82 \begin{align*} \arcsin \left (\frac{e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} + 1} - \frac{1}{e^{\left (2 \, x\right )} + 1}\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]