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 = {2320, 3855}
\begin {gather*} \text {ArcTan}\left (\sinh \left (e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 3855
Rubi steps
\begin {align*} \int e^x \text {sech}\left (e^x\right ) \, dx &=\text {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 = 0.00, size = 5, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sinh \left (e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 5, normalized size = 1.00
method | result | size |
derivativedivides | \(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\) | \(5\) |
default | \(\arctan \left (\sinh \left ({\mathrm e}^{x}\right )\right )\) | \(5\) |
risch | \(i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}+i\right )-i \ln \left ({\mathrm e}^{{\mathrm e}^{x}}-i\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.66, size = 4, normalized size = 0.80 \begin {gather*} \arctan \left (\sinh \left (e^{x}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16 vs.
\(2 (4) = 8\).
time = 0.98, size = 16, normalized size = 3.20 \begin {gather*} 2 \, \arctan \left (\cosh \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sinh \left (\cosh \left (x\right ) + \sinh \left (x\right )\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 {sech}{\left (e^{x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 6, normalized size = 1.20 \begin {gather*} 2 \, \arctan \left (e^{\left (e^{x}\right )}\right ) \end {gather*}
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 \begin {gather*} 2\,\mathrm {atan}\left ({\mathrm {e}}^{{\mathrm {e}}^x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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