Optimal. Leaf size=20 \[ -x-\frac{2 i \cosh (x)}{1-i \sinh (x)} \]
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Rubi [A] time = 0.0468274, antiderivative size = 20, 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 = {4391, 2680, 8} \[ -x-\frac{2 i \cosh (x)}{1-i \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2680
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{(\text{sech}(x)-i \tanh (x))^2} \, dx &=\int \frac{\cosh ^2(x)}{(1-i \sinh (x))^2} \, dx\\ &=-\frac{2 i \cosh (x)}{1-i \sinh (x)}-\int 1 \, dx\\ &=-x-\frac{2 i \cosh (x)}{1-i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0285395, size = 31, normalized size = 1.55 \[ -x+\frac{4 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )-i \sinh \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 29, normalized size = 1.5 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +4\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-1}+\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 = 1.03978, size = 19, normalized size = 0.95 \begin{align*} -x - \frac{4 i}{e^{\left (-x\right )} - i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02849, size = 43, normalized size = 2.15 \begin{align*} -\frac{x e^{x} + i \, x + 4 i}{e^{x} + i} \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 \frac{1}{\left (- i \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13175, size = 16, normalized size = 0.8 \begin{align*} -x - \frac{4 i}{e^{x} + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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