Optimal. Leaf size=27 \[ -i x-\frac{1}{2} \tanh ^{-1}(\cosh (x))+\frac{1}{2} \coth (x) (-\text{csch}(x)+2 i) \]
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Rubi [A] time = 0.056028, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ -i x-\frac{1}{2} \tanh ^{-1}(\cosh (x))+\frac{1}{2} \coth (x) (-\text{csch}(x)+2 i) \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3881
Rule 3770
Rubi steps
\begin{align*} \int \frac{\coth ^4(x)}{i+\text{csch}(x)} \, dx &=\int \coth ^2(x) (-i+\text{csch}(x)) \, dx\\ &=\frac{1}{2} \coth (x) (2 i-\text{csch}(x))+\frac{1}{2} \int (-2 i+\text{csch}(x)) \, dx\\ &=-i x+\frac{1}{2} \coth (x) (2 i-\text{csch}(x))+\frac{1}{2} \int \text{csch}(x) \, dx\\ &=-i x-\frac{1}{2} \tanh ^{-1}(\cosh (x))+\frac{1}{2} \coth (x) (2 i-\text{csch}(x))\\ \end{align*}
Mathematica [B] time = 0.0360378, size = 65, normalized size = 2.41 \[ -i x+\frac{1}{2} i \tanh \left (\frac{x}{2}\right )+\frac{1}{2} i \coth \left (\frac{x}{2}\right )-\frac{1}{8} \text{csch}^2\left (\frac{x}{2}\right )-\frac{1}{8} \text{sech}^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\tanh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 61, normalized size = 2.3 \begin{align*}{\frac{i}{2}}\tanh \left ({\frac{x}{2}} \right ) +{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+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 [B] time = 1.02958, size = 74, normalized size = 2.74 \begin{align*} -i \, x + \frac{e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )} - 2 i}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6225, size = 254, normalized size = 9.41 \begin{align*} \frac{-2 i \, x e^{\left (4 \, x\right )} +{\left (4 i \, x + 4 i\right )} e^{\left (2 \, x\right )} -{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + 1\right ) +{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i \, x - 2 \, e^{\left (3 \, x\right )} - 2 \, e^{x} - 4 i}{2 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \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 \frac{\coth ^{4}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16535, size = 65, normalized size = 2.41 \begin{align*} \frac{e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + 2 i}{{\left (i \, e^{\left (2 \, x\right )} - i\right )}^{2}} - i \, \log \left (-i \, e^{x}\right ) - \frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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