Optimal. Leaf size=5 \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 3770} \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 3770
Rubi steps
\begin {align*} \int e^x \text {sech}\left (e^x\right ) \, dx &=\operatorname {Subst}\left (\int \text {sech}(x) \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (\sinh \left (e^x\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 5, normalized size = 1.00 \[ \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 16, normalized size = 3.20 \[ 2 \, \arctan \left (\cosh \left (\cosh \relax (x) + \sinh \relax (x)\right ) + \sinh \left (\cosh \relax (x) + \sinh \relax (x)\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 6, normalized size = 1.20 \[ 2 \, \arctan \left (e^{\left (e^{x}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 5, normalized size = 1.00 \[ \arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 4, normalized size = 0.80 \[ \arctan \left (\sinh \left (e^{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 6, normalized size = 1.20 \[ 2\,\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x}\right ) \]
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 \[ \int e^{x} \operatorname {sech}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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