Integrand size = 10, antiderivative size = 36 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x) \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2686, 30, 5316, 12, 464, 212} \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x)) \]
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Rule 12
Rule 30
Rule 212
Rule 464
Rule 2686
Rule 5316
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\int \frac {2 \text {csch}^2(x)}{3 (-3-\cosh (2 x))} \, dx \\ & = -\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {2}{3} \int \frac {\text {csch}^2(x)}{-3-\cosh (2 x)} \, dx \\ & = -\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {2}{3} \text {Subst}\left (\int \frac {1-x^2}{2 x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = -\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {1}{3} \text {Subst}\left (\int \frac {1-x^2}{x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = \frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {1}{6} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right ) \\ & = \frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {1}{48} \text {csch}^3(x) \left (-16 \cot ^{-1}(\cosh (x))-2 \cosh (x)+2 \cosh (3 x)-3 i \sqrt {2} \arctan \left (1-i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+3 i \sqrt {2} \arctan \left (1+i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (x)+i \sqrt {2} \arctan \left (1-i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)-i \sqrt {2} \arctan \left (1+i \sqrt {2} \tanh \left (\frac {x}{2}\right )\right ) \sinh (3 x)\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 850, normalized size of antiderivative = 23.61
\[\text {Expression too large to display}\]
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Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 423, normalized size of antiderivative = 11.75 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {8 \, \cosh \left (x\right )^{4} + 32 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 8 \, \sinh \left (x\right )^{4} + 16 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 64 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )} \arctan \left (\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right ) - 16 \, \cosh \left (x\right )^{2} + {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{4} - 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} - 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} - 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) - \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) + 32 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 8}{24 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (34) = 68\).
Time = 78.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 5.94 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=- \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} + \frac {\sqrt {2} \log {\left (4 \tanh ^{2}{\left (\frac {x}{2} \right )} + 4 \sqrt {2} \tanh {\left (\frac {x}{2} \right )} + 4 \right )}}{24} - \frac {\tanh ^{3}{\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24} + \frac {\tanh {\left (\frac {x}{2} \right )} \operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8} + \frac {\tanh {\left (\frac {x}{2} \right )}}{12} - \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{8 \tanh {\left (\frac {x}{2} \right )}} + \frac {1}{12 \tanh {\left (\frac {x}{2} \right )}} + \frac {\operatorname {acot}{\left (\frac {\tanh ^{2}{\left (\frac {x}{2} \right )}}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {1}{\tanh ^{2}{\left (\frac {x}{2} \right )} - 1} \right )}}{24 \tanh ^{3}{\left (\frac {x}{2} \right )}} \]
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=-\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac {1}{3 \, {\left (e^{\left (-2 \, x\right )} - 1\right )}} - \frac {\operatorname {arccot}\left (\cosh \left (x\right )\right )}{3 \, \sinh \left (x\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (27) = 54\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.94 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac {1}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \frac {8 \, \arctan \left (\frac {2}{e^{\left (-x\right )} + e^{x}}\right )}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \]
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Time = 0.63 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.86 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx=\frac {\sqrt {2}\,\ln \left (-\frac {2\,{\mathrm {e}}^{2\,x}}{3}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}\right )}{24}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}-\frac {2\,{\mathrm {e}}^{2\,x}}{3}\right )}{24}+\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{3\,x}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )} \]
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