Optimal. Leaf size=23 \[ \frac{1}{4} i \coth ^4(x)-\frac{\text{csch}^3(x)}{3}-\text{csch}(x) \]
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Rubi [A] time = 0.0768823, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2607, 30, 2606} \[ \frac{1}{4} i \coth ^4(x)-\frac{\text{csch}^3(x)}{3}-\text{csch}(x) \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rubi steps
\begin{align*} \int \frac{\coth ^5(x)}{i+\sinh (x)} \, dx &=-\left (i \int \coth ^3(x) \text{csch}^2(x) \, dx\right )+\int \coth ^3(x) \text{csch}(x) \, dx\\ &=i \operatorname{Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )+i \operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,-i \text{csch}(x)\right )\\ &=\frac{1}{4} i \coth ^4(x)-\text{csch}(x)-\frac{\text{csch}^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0096195, size = 33, normalized size = 1.43 \[ \frac{1}{4} i \text{csch}^4(x)-\frac{\text{csch}^3(x)}{3}+\frac{1}{2} i \text{csch}^2(x)-\text{csch}(x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 68, normalized size = 3. \begin{align*}{\frac{3}{8}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{i}{64}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}+{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+{\frac{i}{16}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+{{\frac{i}{16}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{3}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{24} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{{\frac{i}{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 = 1.06873, size = 277, normalized size = 12.04 \begin{align*} \frac{2 \, e^{\left (-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{2 i \, 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{10 \, 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{10 \, 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 i \, 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} - \frac{2 \, e^{\left (-7 \, 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 = 1.97264, size = 176, normalized size = 7.65 \begin{align*} -\frac{6 \, e^{\left (7 \, x\right )} - 6 i \, e^{\left (6 \, x\right )} - 10 \, e^{\left (5 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 6 i \, e^{\left (2 \, x\right )} - 6 \, e^{x}}{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 = 0.754139, size = 71, normalized size = 3.09 \begin{align*} \frac{- 2 e^{7 x} + 2 i e^{6 x} + \frac{10 e^{5 x}}{3} - \frac{10 e^{3 x}}{3} + 2 i e^{2 x} + 2 e^{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 [B] time = 1.13416, size = 69, normalized size = 3. \begin{align*} \frac{6 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 6 i \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, e^{\left (-x\right )} - 8 \, e^{x} + 12 i}{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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