Optimal. Leaf size=26 \[ -\frac{2 i}{1-i \sinh (x)}-i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.0535539, antiderivative size = 26, 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 (\text{sech}(x)+i \tanh (x))^3 \, dx &=\int \text{sech}^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{2}{(-1+x)^2}+\frac{1}{-1+x}\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 = 0.0270304, size = 39, normalized size = 1.5 \[ \frac{1}{2} i \tanh ^2(x)-\frac{3}{2} i \text{sech}^2(x)-\tan ^{-1}(\sinh (x))-i \log (\cosh (x))+2 \tanh (x) \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 45, normalized size = 1.7 \begin{align*} -{\rm sech} \left (x\right )\tanh \left ( x \right ) -2\,\arctan \left ({{\rm e}^{x}} \right ) +{\frac{{\frac{3\,i}{2}} \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}+3\,{\frac{\sinh \left ( x \right ) }{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}-i\ln \left ( \cosh \left ( x \right ) \right ) +{\frac{i}{2}} \left ( \tanh \left ( x \right ) \right ) ^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60457, size = 99, normalized size = 3.81 \begin{align*} \frac{3}{2} i \, \tanh \left (x\right )^{2} - i \, x + \frac{4 \,{\left (e^{\left (-x\right )} - e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} - \frac{2 i \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \arctan \left (e^{\left (-x\right )}\right ) - i \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47527, size = 142, normalized size = 5.46 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i \tanh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1199, size = 28, normalized size = 1.08 \begin{align*} i \, x + \frac{4 \, e^{x}}{{\left (e^{x} + i\right )}^{2}} - 2 i \, \log \left (e^{x} + i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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