Optimal. Leaf size=28 \[ \frac{2 i}{1+i \sinh (x)}+i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.0521178, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4391, 2667, 43} \[ \frac{2 i}{1+i \sinh (x)}+i \log (-\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 4391
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{(\text{sech}(x)+i \tanh (x))^3} \, dx &=\int \frac{\cosh ^3(x)}{(1+i \sinh (x))^3} \, dx\\ &=-\left (i \operatorname{Subst}\left (\int \frac{1-x}{(1+x)^2} \, dx,x,i \sinh (x)\right )\right )\\ &=-\left (i \operatorname{Subst}\left (\int \left (\frac{1}{-1-x}+\frac{2}{(1+x)^2}\right ) \, dx,x,i \sinh (x)\right )\right )\\ &=i \log (i-\sinh (x))+\frac{2 i}{1+i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0381816, size = 40, normalized size = 1.43 \[ -2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+i \log (\cosh (x))+\frac{2 i}{\left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 56, normalized size = 2. \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 ) -{4\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}-4\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}-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 = 1.05399, size = 45, normalized size = 1.61 \begin{align*} i \, x - \frac{4 \, e^{\left (-x\right )}}{2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} + 2 i \, \log \left (e^{\left (-x\right )} + i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.2852, size = 142, normalized size = 5.07 \begin{align*} \frac{-i \, x e^{\left (2 \, x\right )} - 2 \,{\left (x - 2\right )} e^{x} +{\left (2 i \, e^{\left (2 \, x\right )} + 4 \, e^{x} - 2 i\right )} \log \left (e^{x} - i\right ) + i \, x}{e^{\left (2 \, x\right )} - 2 i \, e^{x} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 30.0508, size = 513, normalized size = 18.32 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15608, size = 36, normalized size = 1.29 \begin{align*} \frac{4 \, e^{x}}{{\left (e^{x} - i\right )}^{2}} - i \, \log \left (i \, e^{x}\right ) + 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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