Optimal. Leaf size=14 \[ -i \log (\cosh (x)+i \sinh (x)) \]
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Rubi [A] time = 0.0266594, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3133} \[ -i \log (\cosh (x)+i \sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3133
Rubi steps
\begin{align*} \int \frac{\cosh (x)-i \sinh (x)}{\cosh (x)+i \sinh (x)} \, dx &=-i \log (\cosh (x)+i \sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0337679, size = 15, normalized size = 1.07 \[ \tan ^{-1}(\tanh (x))-\frac{1}{2} i \log (\cosh (2 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 13, normalized size = 0.9 \begin{align*} -i\ln \left ( \cosh \left ( x \right ) +i\sinh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0486, size = 14, normalized size = 1. \begin{align*} -i \, \log \left (\cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32311, size = 35, normalized size = 2.5 \begin{align*} i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialDivisionFailed} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11476, size = 18, normalized size = 1.29 \begin{align*} i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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