Integrand size = 9, antiderivative size = 31 \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)}+\frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4483, 4485, 2678, 2669} \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)}+\frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)} \]
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Rule 2669
Rule 2678
Rule 4483
Rule 4485
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {\sinh (x) \tanh (x)} \int (-\sinh (x) \tanh (x))^{3/2} \, dx}{\sqrt {-\sinh (x) \tanh (x)}} \\ & = -\frac {\sqrt {\sinh (x) \tanh (x)} \int (i \sinh (x))^{3/2} (i \tanh (x))^{3/2} \, dx}{\sqrt {i \sinh (x)} \sqrt {i \tanh (x)}} \\ & = \frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)}-\frac {\left (4 \sqrt {\sinh (x) \tanh (x)}\right ) \int \frac {(i \tanh (x))^{3/2}}{\sqrt {i \sinh (x)}} \, dx}{3 \sqrt {i \sinh (x)} \sqrt {i \tanh (x)}} \\ & = \frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)}+\frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\frac {2}{3} \left (1+4 \text {csch}^2(x)\right ) \sinh (x) \sqrt {\sinh (x) \tanh (x)} \]
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\[\int \left (\sinh \left (x \right ) \tanh \left (x \right )\right )^{\frac {3}{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 7\right )} \sinh \left (x\right )^{2} + 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}}{3 \, \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
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\[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\int \left (\sinh {\left (x \right )} \tanh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (23) = 46\).
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.23 \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=-\frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]
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\[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\int { \left (\sinh \left (x\right ) \tanh \left (x\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (\sinh (x) \tanh (x))^{3/2} \, dx=\int {\left (\mathrm {sinh}\left (x\right )\,\mathrm {tanh}\left (x\right )\right )}^{3/2} \,d x \]
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