Optimal. Leaf size=11 \[ \tan ^{-1}(\sinh (x))-i \log (\cosh (x)) \]
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Rubi [A] time = 0.008347, antiderivative size = 11, normalized size of antiderivative = 1., 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 3770
Rule 3475
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*}
Mathematica [A] time = 0.0040074, 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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Maple [A] time = 0.002, size = 11, normalized size = 1. \begin{align*} \arctan \left ( \sinh \left ( x \right ) \right ) -i\ln \left ( \cosh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990642, size = 12, normalized size = 1.09 \begin{align*} \arctan \left (\sinh \left (x\right )\right ) - i \, \log \left (\cosh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08936, size = 32, normalized size = 2.91 \begin{align*} i \, x - 2 i \, \log \left (e^{x} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- i \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13755, size = 24, normalized size = 2.18 \begin{align*} i \, x + 2 \, \arctan \left (e^{x}\right ) - i \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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