Optimal. Leaf size=13 \[ -i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.029448, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3159, 2667, 31} \[ -i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 3159
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\text{sech}(x)+i \tanh (x)} \, dx &=\int \frac{\cosh (x)}{1+i \sinh (x)} \, dx\\ &=-\left (i \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,i \sinh (x)\right )\right )\\ &=-i \log (i-\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0167274, size = 17, normalized size = 1.31 \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-i \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 33, normalized size = 2.5 \begin{align*} i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) +i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05101, size = 20, normalized size = 1.54 \begin{align*} -i \, x - 2 i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.48037, size = 32, normalized size = 2.46 \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 [B] time = 0.667332, size = 22, normalized size = 1.69 \begin{align*} - i x - i \log{\left (i \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )} \right )} + i \log{\left (\tanh{\left (x \right )} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13534, size = 18, normalized size = 1.38 \begin{align*} i \, x - 2 i \, \log \left (i \, e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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