Optimal. Leaf size=38 \[ x+\frac{2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac{2 i \cosh (x)}{1+i \sinh (x)} \]
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Rubi [A] time = 0.0774291, antiderivative size = 38, 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, 2680, 8} \[ x+\frac{2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac{2 i \cosh (x)}{1+i \sinh (x)} \]
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))^4} \, dx &=\int \frac{\cosh ^4(x)}{(1+i \sinh (x))^4} \, dx\\ &=\frac{2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\int \frac{\cosh ^2(x)}{(1+i \sinh (x))^2} \, dx\\ &=\frac{2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac{2 i \cosh (x)}{1+i \sinh (x)}+\int 1 \, dx\\ &=x+\frac{2 i \cosh ^3(x)}{3 (1+i \sinh (x))^3}-\frac{2 i \cosh (x)}{1+i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0745308, size = 75, normalized size = 1.97 \[ \frac{3 (3 x+8 i) \cosh \left (\frac{x}{2}\right )-(3 x+16 i) \cosh \left (\frac{3 x}{2}\right )+6 i \sinh \left (\frac{x}{2}\right ) (2 x+x \cosh (x)+4 i)}{6 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.116, size = 41, normalized size = 1.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{8\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}-{\frac{16}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}}-\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 = 1.0918, size = 54, normalized size = 1.42 \begin{align*} x - \frac{24 \, e^{\left (-x\right )} - 24 i \, e^{\left (-2 \, x\right )} + 16 i}{9 \, e^{\left (-x\right )} - 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05434, size = 154, normalized size = 4.05 \begin{align*} \frac{3 \, x e^{\left (3 \, x\right )} +{\left (-9 i \, x - 24 i\right )} e^{\left (2 \, x\right )} - 3 \,{\left (3 \, x + 8\right )} e^{x} + 3 i \, x + 16 i}{3 \, e^{\left (3 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} + 3 i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13871, size = 30, normalized size = 0.79 \begin{align*} x - \frac{24 i \, e^{\left (2 \, x\right )} + 24 \, e^{x} - 16 i}{3 \,{\left (e^{x} - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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