Optimal. Leaf size=40 \[ \frac{4 i}{1-i \sinh (x)}-\frac{2 i}{(1-i \sinh (x))^2}+i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.0551645, antiderivative size = 40, 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\\ &=i \operatorname{Subst}\left (\int \frac{(1-x)^2}{(1+x)^3} \, dx,x,-i \sinh (x)\right )\\ &=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 )\\ &=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 = 0.0983803, size = 45, normalized size = 1.12 \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+i \log (\cosh (x))+\frac{4 \sinh (x)+2 i}{\left (\cosh \left (\frac{x}{2}\right )-i \sinh \left (\frac{x}{2}\right )\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 68, normalized size = 1.7 \begin{align*} -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +{8\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}+2\,i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) -{8\,i \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-4}}+16\, \left ( \tanh \left ( x/2 \right ) +i \right ) ^{-3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08159, size = 81, normalized size = 2.02 \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 = 2.16657, size = 273, normalized size = 6.82 \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 = 1.15358, size = 54, normalized size = 1.35 \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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