Optimal. Leaf size=46 \[ \frac{\coth ^4(x)}{4 a}-\frac{3 \tanh ^{-1}(\cosh (x))}{8 a}-\frac{\coth ^3(x) \text{csch}(x)}{4 a}-\frac{3 \coth (x) \text{csch}(x)}{8 a} \]
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Rubi [A] time = 0.106865, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2706, 2607, 30, 2611, 3770} \[ \frac{\coth ^4(x)}{4 a}-\frac{3 \tanh ^{-1}(\cosh (x))}{8 a}-\frac{\coth ^3(x) \text{csch}(x)}{4 a}-\frac{3 \coth (x) \text{csch}(x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{a+a \cosh (x)} \, dx &=\frac{\int \coth ^4(x) \text{csch}(x) \, dx}{a}-\frac{\int \coth ^3(x) \text{csch}^2(x) \, dx}{a}\\ &=-\frac{\coth ^3(x) \text{csch}(x)}{4 a}+\frac{3 \int \coth ^2(x) \text{csch}(x) \, dx}{4 a}+\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )}{a}\\ &=\frac{\coth ^4(x)}{4 a}-\frac{3 \coth (x) \text{csch}(x)}{8 a}-\frac{\coth ^3(x) \text{csch}(x)}{4 a}+\frac{3 \int \text{csch}(x) \, dx}{8 a}\\ &=-\frac{3 \tanh ^{-1}(\cosh (x))}{8 a}+\frac{\coth ^4(x)}{4 a}-\frac{3 \coth (x) \text{csch}(x)}{8 a}-\frac{\coth ^3(x) \text{csch}(x)}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0990674, size = 60, normalized size = 1.3 \[ \frac{-2 \coth ^2\left (\frac{x}{2}\right )+\text{sech}^2\left (\frac{x}{2}\right )-12 \cosh ^2\left (\frac{x}{2}\right ) \left (\log \left (\cosh \left (\frac{x}{2}\right )\right )-\log \left (\sinh \left (\frac{x}{2}\right )\right )\right )-8}{16 a (\cosh (x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 45, normalized size = 1. \begin{align*}{\frac{1}{32\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}+{\frac{3}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{3}{8\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08625, size = 139, normalized size = 3.02 \begin{align*} -\frac{5 \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{4 \,{\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} - \frac{3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} + \frac{3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01018, size = 2071, normalized size = 45.02 \begin{align*} \text{result too large to display} \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*} \frac{\int \frac{\coth ^{3}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17454, size = 127, normalized size = 2.76 \begin{align*} -\frac{3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} + \frac{3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} - \frac{3 \, e^{\left (-x\right )} + 3 \, e^{x} - 2}{16 \, a{\left (e^{\left (-x\right )} + e^{x} - 2\right )}} + \frac{9 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} + 4 \, e^{x} - 12}{32 \, a{\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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