Optimal. Leaf size=11 \[ i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.0301713, antiderivative size = 11, 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\\ &=i \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,-i \sinh (x)\right )\\ &=i \log (i+\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0173788, 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 [B] time = 0.045, size = 33, normalized size = 3. \begin{align*} -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04957, size = 20, normalized size = 1.82 \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 = 2.05302, size = 34, normalized size = 3.09 \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 = 0.663409, size = 22, normalized size = 2. \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 = 1.17548, size = 18, normalized size = 1.64 \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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