Optimal. Leaf size=24 \[ \frac{\text{csch}(x)}{3 (a \cosh (x)+a)}-\frac{2 \coth (x)}{3 a} \]
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Rubi [A] time = 0.0483311, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac{\text{csch}(x)}{3 (a \cosh (x)+a)}-\frac{2 \coth (x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{a+a \cosh (x)} \, dx &=\frac{\text{csch}(x)}{3 (a+a \cosh (x))}+\frac{2 \int \text{csch}^2(x) \, dx}{3 a}\\ &=\frac{\text{csch}(x)}{3 (a+a \cosh (x))}-\frac{(2 i) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))}{3 a}\\ &=-\frac{2 \coth (x)}{3 a}+\frac{\text{csch}(x)}{3 (a+a \cosh (x))}\\ \end{align*}
Mathematica [A] time = 0.0460004, size = 30, normalized size = 1.25 \[ -\frac{(2 \cosh (x)+\cosh (2 x)) \text{csch}\left (\frac{x}{2}\right ) \text{sech}^3\left (\frac{x}{2}\right )}{12 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 29, normalized size = 1.2 \begin{align*}{\frac{1}{4\,a} \left ({\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-2\,\tanh \left ( x/2 \right ) - \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00997, size = 80, normalized size = 3.33 \begin{align*} -\frac{8 \, e^{\left (-x\right )}}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac{4}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.85497, size = 292, normalized size = 12.17 \begin{align*} -\frac{4 \,{\left (2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 1\right )}}{3 \,{\left (a \cosh \left (x\right )^{4} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{3} + 2 \,{\left (2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + 6 \,{\left (a \cosh \left (x\right )^{2} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) + 2 \,{\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} - a\right )} \sinh \left (x\right ) - a\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*} \frac{\int \frac{\operatorname{csch}^{2}{\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 [A] time = 1.17293, size = 47, normalized size = 1.96 \begin{align*} -\frac{1}{2 \, a{\left (e^{x} - 1\right )}} + \frac{3 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5}{6 \, a{\left (e^{x} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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