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 = 0.0591995, 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 (\text{sech}(x)-i \tanh (x))^5 \, dx &=\int \text{sech}^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{1}{1-x}-\frac{4}{(-1+x)^3}-\frac{4}{(-1+x)^2}\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 = 0.0578758, size = 62, normalized size = 1.48 \[ \frac{11}{4} i \tanh ^4(x)+\frac{1}{2} i \tanh ^2(x)+\frac{5}{4} i \text{sech}^4(x)+\tan ^{-1}(\sinh (x))-i \log (\cosh (x))-\tanh (x) \text{sech}^3(x)-5 \tanh ^3(x) \text{sech}(x)+\tanh (x) \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 82, normalized size = 2. \begin{align*}{\frac{8\,\tanh \left ( x \right ) }{3} \left ({\frac{ \left ({\rm sech} \left (x\right ) \right ) ^{3}}{4}}+{\frac{3\,{\rm sech} \left (x\right )}{8}} \right ) }+2\,\arctan \left ({{\rm e}^{x}} \right ) -{\frac{{\frac{15\,i}{4}} \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{4}}}+{\frac{{\frac{5\,i}{4}} \left ( \sinh \left ( x \right ) \right ) ^{2}}{ \left ( \cosh \left ( x \right ) \right ) ^{2}}}-{\frac{5\,\sinh \left ( x \right ) }{3\, \left ( \cosh \left ( x \right ) \right ) ^{4}}}-5\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{3}}{ \left ( \cosh \left ( x \right ) \right ) ^{4}}}-i\ln \left ( \cosh \left ( x \right ) \right ) +{\frac{i}{2}} \left ( \tanh \left ( x \right ) \right ) ^{2}+{\frac{i}{4}} \left ( \tanh \left ( x \right ) \right ) ^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.64778, size = 317, normalized size = 7.55 \begin{align*} \frac{5}{2} i \, \tanh \left (x\right )^{4} - i \, x - \frac{5 \,{\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} + \frac{3 \, e^{\left (-x\right )} + 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{4 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac{5 \,{\left (e^{\left (-x\right )} - 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} - e^{\left (-7 \, x\right )}\right )}}{2 \,{\left (4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} - \frac{4 i \,{\left (e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1} + \frac{20 i}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} - 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 = 2.09173, 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] time = 0., size = 0, normalized size = 0. \begin{align*} - \int i \tanh ^{5}{\left (x \right )}\, dx - \int 10 \tanh ^{2}{\left (x \right )} \operatorname{sech}^{3}{\left (x \right )}\, dx - \int - 5 \tanh ^{4}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx - \int 5 i \tanh{\left (x \right )} \operatorname{sech}^{4}{\left (x \right )}\, dx - \int - 10 i \tanh ^{3}{\left (x \right )} \operatorname{sech}^{2}{\left (x \right )}\, dx - \int - \operatorname{sech}^{5}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13774, size = 49, normalized size = 1.17 \begin{align*} i \, x - \frac{8 \, e^{\left (3 \, x\right )} - 8 i \, e^{\left (2 \, x\right )} - 8 \, e^{x}}{{\left (e^{x} - i\right )}^{4}} - 2 i \, \log \left (e^{x} - i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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