Optimal. Leaf size=3 \[ \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3770} \[ \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \text {sech}(x) \, dx &=\tan ^{-1}(\sinh (x))\\ \end {align*}
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Mathematica [B] time = 0.00, size = 9, normalized size = 3.00 \[ 2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \text {sech}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.24, size = 8, normalized size = 2.67 \[ 2 \, \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 5, normalized size = 1.67 \[ 2 \, \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 4, normalized size = 1.33
method | result | size |
lookup | \(\arctan \left (\sinh \relax (x )\right )\) | \(4\) |
default | \(\arctan \left (\sinh \relax (x )\right )\) | \(4\) |
risch | \(i \ln \left ({\mathrm e}^{x}+i\right )-i \ln \left ({\mathrm e}^{x}-i\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 3, normalized size = 1.00 \[ \arctan \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 5, normalized size = 1.67 \[ 2\,\mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
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 \[ \int \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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