Integrand size = 9, antiderivative size = 50 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \sqrt {\cosh (x) \coth (x)} \tanh (x) \]
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Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4483, 4485, 2678, 2674, 2669} \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \tanh (x) \sqrt {\cosh (x) \coth (x)} \]
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Rule 2669
Rule 2674
Rule 2678
Rule 4483
Rule 4485
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {\cosh (x) \coth (x)} \int (-i \cosh (x) \coth (x))^{5/2} \, dx}{\sqrt {-i \cosh (x) \coth (x)}} \\ & = -\frac {\sqrt {\cosh (x) \coth (x)} \int \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{5/2} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = \frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}-\frac {\left (8 \sqrt {\cosh (x) \coth (x)}\right ) \int \sqrt {\cosh (x)} (-i \coth (x))^{5/2} \, dx}{5 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = -\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {\left (32 \sqrt {\cosh (x) \coth (x)}\right ) \int \sqrt {\cosh (x)} \sqrt {-i \coth (x)} \, dx}{15 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = -\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \sqrt {\cosh (x) \coth (x)} \tanh (x) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {1}{15} \sqrt {\cosh (x) \coth (x)} \left (-10 \coth (x)+6 \cosh (x) \sinh (x)+57 \text {csch}(x) \text {sech}(x) \left (-\sinh ^2(x)\right )^{3/4}+64 \tanh (x)\right ) \]
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\[\int \left (\coth \left (x \right ) \cosh \left (x \right )\right )^{\frac {5}{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 5.18 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (x\right )^{8} + 24 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, \sinh \left (x\right )^{8} + 12 \, {\left (7 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{6} + 108 \, \cosh \left (x\right )^{6} + 24 \, {\left (7 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} + 810 \, \cosh \left (x\right )^{2} - 151\right )} \sinh \left (x\right )^{4} - 302 \, \cosh \left (x\right )^{4} + 8 \, {\left (21 \, \cosh \left (x\right )^{5} + 270 \, \cosh \left (x\right )^{3} - 151 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 12 \, {\left (7 \, \cosh \left (x\right )^{6} + 135 \, \cosh \left (x\right )^{4} - 151 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{2} + 108 \, \cosh \left (x\right )^{2} + 8 \, {\left (3 \, \cosh \left (x\right )^{7} + 81 \, \cosh \left (x\right )^{5} - 151 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{30 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}} \]
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Timed out. \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (38) = 76\).
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.26 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {\sqrt {2} e^{\left (\frac {5}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {7 \, \sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {41 \, \sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {41 \, \sqrt {2} e^{\left (-\frac {7}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {7 \, \sqrt {2} e^{\left (-\frac {11}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {15}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} \]
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\[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\int { \left (\cosh \left (x\right ) \coth \left (x\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\int {\left (\mathrm {cosh}\left (x\right )\,\mathrm {coth}\left (x\right )\right )}^{5/2} \,d x \]
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