Optimal. Leaf size=77 \[ -\frac{a \coth (x)}{a^2-b^2}+\frac{b \text{csch}(x)}{a^2-b^2}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]
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Rubi [A] time = 0.0935763, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2727, 3767, 8, 2606, 2659, 208} \[ -\frac{a \coth (x)}{a^2-b^2}+\frac{b \text{csch}(x)}{a^2-b^2}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2727
Rule 3767
Rule 8
Rule 2606
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{a+b \cosh (x)} \, dx &=\frac{a \int \text{csch}^2(x) \, dx}{a^2-b^2}+\frac{a^2 \int \frac{1}{a+b \cosh (x)} \, dx}{a^2-b^2}-\frac{b \int \coth (x) \text{csch}(x) \, dx}{a^2-b^2}\\ &=-\frac{(i a) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))}{a^2-b^2}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2-b^2}+\frac{(i b) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(x))}{a^2-b^2}\\ &=\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}-\frac{a \coth (x)}{a^2-b^2}+\frac{b \text{csch}(x)}{a^2-b^2}\\ \end{align*}
Mathematica [A] time = 0.20125, size = 77, normalized size = 1. \[ \frac{2 a^2 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}-\frac{\tanh \left (\frac{x}{2}\right )}{2 (a-b)}-\frac{\coth \left (\frac{x}{2}\right )}{2 (a+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 78, normalized size = 1. \begin{align*} -{\frac{1}{2\,a-2\,b}\tanh \left ({\frac{x}{2}} \right ) }+2\,{\frac{{a}^{2}}{ \left ( a+b \right ) \left ( a-b \right ) \sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{2\,a+2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96197, size = 1172, normalized size = 15.22 \begin{align*} \left [\frac{2 \, a^{3} - 2 \, a b^{2} +{\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt{a^{2} - b^{2}} \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) - 2 \,{\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) - 2 \,{\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}, \frac{2 \,{\left (a^{3} - a b^{2} +{\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}\right )} \sqrt{-a^{2} + b^{2}} \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) -{\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) -{\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}}\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 ^{2}{\left (x \right )}}{a + b \cosh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19427, size = 103, normalized size = 1.34 \begin{align*} \frac{2 \, a^{2} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}}} + \frac{2 \,{\left (b e^{x} - a\right )}}{{\left (a^{2} - b^{2}\right )}{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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