Optimal. Leaf size=28 \[ e^x \sin ^{-1}(\tanh (x))-\log \left (e^{2 x}+1\right ) \cosh (x) \sqrt {\text {sech}^2(x)} \]
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Rubi [A] time = 0.08, 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 = {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.
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Rule 12
Rule 260
Rule 2194
Rule 2282
Rule 4844
Rule 6720
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*}
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Mathematica [B] time = 0.97, 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.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^x \sin ^{-1}(\tanh (x)) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.04, size = 26, normalized size = 0.93 \[ {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (\sinh \relax (x)\right ) - \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 29, normalized size = 1.04 \[ \arcsin \left (\frac {e^{\left (2 \, x\right )} - 1}{e^{\left (2 \, x\right )} + 1}\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{x} \arcsin \left (\tanh \relax (x )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 16, normalized size = 0.57 \[ \arcsin \left (\tanh \relax (x)\right ) e^{x} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
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 \[ \int \mathrm {asin}\left (\mathrm {tanh}\relax (x)\right )\,{\mathrm {e}}^x \,d x \]
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 \[ \int e^{x} \operatorname {asin}{\left (\tanh {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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