Optimal. Leaf size=46 \[ \frac{\text{csch}^4(x)}{4 a}+\frac{\tanh ^{-1}(\cosh (x))}{8 a}-\frac{\coth (x) \text{csch}^3(x)}{4 a}-\frac{\coth (x) \text{csch}(x)}{8 a} \]
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Rubi [A] time = 0.194662, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {3872, 2835, 2606, 30, 2611, 3768, 3770} \[ \frac{\text{csch}^4(x)}{4 a}+\frac{\tanh ^{-1}(\cosh (x))}{8 a}-\frac{\coth (x) \text{csch}^3(x)}{4 a}-\frac{\coth (x) \text{csch}(x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2835
Rule 2606
Rule 30
Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{csch}^3(x)}{a+a \text{sech}(x)} \, dx &=-\int \frac{\coth (x) \text{csch}^2(x)}{-a-a \cosh (x)} \, dx\\ &=\frac{\int \coth ^2(x) \text{csch}^3(x) \, dx}{a}-\frac{\int \coth (x) \text{csch}^4(x) \, dx}{a}\\ &=-\frac{\coth (x) \text{csch}^3(x)}{4 a}+\frac{\int \text{csch}^3(x) \, dx}{4 a}+\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,-i \text{csch}(x)\right )}{a}\\ &=-\frac{\coth (x) \text{csch}(x)}{8 a}-\frac{\coth (x) \text{csch}^3(x)}{4 a}+\frac{\text{csch}^4(x)}{4 a}-\frac{\int \text{csch}(x) \, dx}{8 a}\\ &=\frac{\tanh ^{-1}(\cosh (x))}{8 a}-\frac{\coth (x) \text{csch}(x)}{8 a}-\frac{\coth (x) \text{csch}^3(x)}{4 a}+\frac{\text{csch}^4(x)}{4 a}\\ \end{align*}
Mathematica [A] time = 0.154225, size = 59, normalized size = 1.28 \[ \frac{\cosh ^2\left (\frac{x}{2}\right ) \text{sech}(x) \left (-2 \text{csch}^2\left (\frac{x}{2}\right )+\text{sech}^4\left (\frac{x}{2}\right )-4 \log \left (\sinh \left (\frac{x}{2}\right )\right )+4 \log \left (\cosh \left (\frac{x}{2}\right )\right )\right )}{16 (a \text{sech}(x)+a)} \]
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{1}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12578, size = 134, normalized size = 2.91 \begin{align*} -\frac{e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 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{\log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac{\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 = 2.5358, size = 2060, normalized size = 44.78 \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{\operatorname{csch}^{3}{\left (x \right )}}{\operatorname{sech}{\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 = 1.1941, size = 122, normalized size = 2.65 \begin{align*} \frac{\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac{\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac{e^{\left (-x\right )} + e^{x} - 6}{16 \, a{\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac{3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, e^{\left (-x\right )} + 12 \, e^{x} - 4}{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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