Optimal. Leaf size=27 \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]
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Rubi [A] time = 0.0427058, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2707, 43} \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 43
Rubi steps
\begin{align*} \int \frac{\coth ^5(x)}{(i+\sinh (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{(i-x)^2}{x^5} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{x^5}-\frac{2 i}{x^4}+\frac{1}{x^3}\right ) \, dx,x,\sinh (x)\right )\\ &=-\frac{1}{2} \text{csch}^2(x)+\frac{2}{3} i \text{csch}^3(x)+\frac{\text{csch}^4(x)}{4}\\ \end{align*}
Mathematica [A] time = 0.0123971, size = 27, normalized size = 1. \[ \frac{\text{csch}^4(x)}{4}+\frac{2}{3} i \text{csch}^3(x)-\frac{\text{csch}^2(x)}{2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 68, normalized size = 2.5 \begin{align*}{\frac{i}{4}}\tanh \left ({\frac{x}{2}} \right ) +{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}-{\frac{3}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{3}{16} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{{\frac{i}{4}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{{\frac{i}{12}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02155, size = 231, normalized size = 8.56 \begin{align*} \frac{2 \, e^{\left (-2 \, x\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{16 i \, e^{\left (-3 \, x\right )}}{3 \,{\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{8 \, e^{\left (-4 \, x\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{16 i \, e^{\left (-5 \, x\right )}}{3 \,{\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{2 \, e^{\left (-6 \, x\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95788, size = 166, normalized size = 6.15 \begin{align*} -\frac{6 \, e^{\left (6 \, x\right )} - 16 i \, e^{\left (5 \, x\right )} - 24 \, e^{\left (4 \, x\right )} + 16 i \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )}}{3 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.74758, size = 66, normalized size = 2.44 \begin{align*} \frac{- 2 e^{6 x} + \frac{16 i e^{5 x}}{3} + 8 e^{4 x} - \frac{16 i e^{3 x}}{3} - 2 e^{2 x}}{e^{8 x} - 4 e^{6 x} + 6 e^{4 x} - 4 e^{2 x} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10279, size = 51, normalized size = 1.89 \begin{align*} -\frac{6 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 16 i \, e^{\left (-x\right )} - 16 i \, e^{x} - 12}{3 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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