Optimal. Leaf size=11 \[ \tan ^{-1}(\sinh (x))-i \log (\cosh (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3770, 3475} \[ \tan ^{-1}(\sinh (x))-i \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (\text {sech}(x)-i \tanh (x)) \, dx &=-(i \int \tanh (x) \, dx)+\int \text {sech}(x) \, dx\\ &=\tan ^{-1}(\sinh (x))-i \log (\cosh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.55 \[ 2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-i \log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 11, normalized size = 1.00 \[ i \, x - 2 i \, \log \left (e^{x} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 18, normalized size = 1.64 \[ i \, x + 2 \, \arctan \left (e^{x}\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 11, normalized size = 1.00 \[ \arctan \left (\sinh \relax (x )\right )-i \ln \left (\cosh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 9, normalized size = 0.82 \[ \arctan \left (\sinh \relax (x)\right ) - i \, \log \left (\cosh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 14, normalized size = 1.27 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,2{}\mathrm {i} \]
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 \[ \int \left (- i \tanh {\relax (x )} + \operatorname {sech}{\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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