Optimal. Leaf size=28 \[ e^x \sin ^{-1}(\tanh (x))-\cosh (x) \log \left (1+e^{2 x}\right ) \sqrt {\text {sech}^2(x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {2225, 4928,
6852, 2320, 12, 266} \begin {gather*} e^x \text {ArcSin}(\tanh (x))-\log \left (e^{2 x}+1\right ) \cosh (x) \sqrt {\text {sech}^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 2225
Rule 2320
Rule 4928
Rule 6852
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 ) \text {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 ) \text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(28)=56\).
time = 0.56, size = 64, normalized size = 2.29 \begin {gather*} e^x \sin ^{-1}\left (\frac {-1+e^{2 x}}{1+e^{2 x}}\right )-e^{-x} \sqrt {\frac {e^{2 x}}{\left (1+e^{2 x}\right )^2}} \left (1+e^{2 x}\right ) \log \left (1+e^{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{x} \arcsin \left (\tanh \left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.69, size = 16, normalized size = 0.57 \begin {gather*} \arcsin \left (\tanh \left (x\right )\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 26, normalized size = 0.93 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \operatorname {asin}{\left (\tanh {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 29, normalized size = 1.04 \begin {gather*} \arcsin \left (\frac {e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \mathrm {asin}\left (\mathrm {tanh}\left (x\right )\right )\,{\mathrm {e}}^x \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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