Optimal. Leaf size=14 \[ -i \log (\cosh (x)+i \sinh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.49, size = 13, normalized size = 0.93 \[ i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 13, normalized size = 0.93 \[ i \, x - i \, \log \left (e^{\left (2 \, x\right )} - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 13, normalized size = 0.93 \[ -i \ln \left (\cosh \relax (x )+i \sinh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 10, normalized size = 0.71 \[ -i \, \log \left (\cosh \relax (x) + i \, \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 16, normalized size = 1.14 \[ x\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^{2\,x}-\mathrm {i}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 1.00 \[ x \left (-2 - i\right ) + \log {\left (e^{2 x} - i \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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