Optimal. Leaf size=42 \[ -\frac{4 i}{1+i \sinh (x)}+\frac{2 i}{(1+i \sinh (x))^2}-i \log (-\sinh (x)+i) \]
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Rubi [A] time = 0.0554388, antiderivative size = 42, 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} \[ -\frac{4 i}{1+i \sinh (x)}+\frac{2 i}{(1+i \sinh (x))^2}-i \log (-\sinh (x)+i) \]
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))^5} \, dx &=\int \frac{\cosh ^5(x)}{(1+i \sinh (x))^5} \, dx\\ &=-\left (i \operatorname{Subst}\left (\int \frac{(1-x)^2}{(1+x)^3} \, dx,x,i \sinh (x)\right )\right )\\ &=-\left (i \operatorname{Subst}\left (\int \left (\frac{4}{(1+x)^3}-\frac{4}{(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))^2}-\frac{4 i}{1+i \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0986096, size = 45, normalized size = 1.07 \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-i \log (\cosh (x))+\frac{4 \sinh (x)-2 i}{\left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 68, normalized size = 1.6 \begin{align*} i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{8\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-4}}-2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) -{8\,i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+16\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-3}+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 = 1.0906, size = 81, normalized size = 1.93 \begin{align*} -i \, x - \frac{8 \, e^{\left (-x\right )} - 8 i \, e^{\left (-2 \, x\right )} - 8 \, e^{\left (-3 \, x\right )}}{-4 i \, e^{\left (-x\right )} - 6 \, e^{\left (-2 \, x\right )} + 4 i \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, 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 = 2.14075, size = 274, normalized size = 6.52 \begin{align*} \frac{i \, x e^{\left (4 \, x\right )} + 4 \,{\left (x - 2\right )} e^{\left (3 \, x\right )} +{\left (-6 i \, x + 8 i\right )} e^{\left (2 \, x\right )} - 4 \,{\left (x - 2\right )} e^{x} +{\left (-2 i \, e^{\left (4 \, x\right )} - 8 \, e^{\left (3 \, x\right )} + 12 i \, e^{\left (2 \, x\right )} + 8 \, e^{x} - 2 i\right )} \log \left (e^{x} - i\right ) + i \, x}{e^{\left (4 \, x\right )} - 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} + 1} \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 = 1.16191, size = 54, normalized size = 1.29 \begin{align*} -\frac{8 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )} - 8 \, e^{x}}{{\left (e^{x} - i\right )}^{4}} + i \, \log \left (i \, e^{x}\right ) - 2 i \, \log \left (e^{x} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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