Optimal. Leaf size=31 \[ \frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)}+\frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)} \]
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Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4398, 4400, 2598, 2589} \[ \frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)}+\frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2598
Rule 4398
Rule 4400
Rubi steps
\begin {align*} \int (\sinh (x) \tanh (x))^{3/2} \, dx &=-\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*}
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Mathematica [A] time = 0.07, size = 23, normalized size = 0.74 \[ \frac {2}{3} \sinh (x) \left (4 \text {csch}^2(x)+1\right ) \sqrt {\sinh (x) \tanh (x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 95, normalized size = 3.06 \[ \frac {\sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 7\right )} \sinh \relax (x)^{2} + 14 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}}{3 \, \sqrt {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sinh \relax (x) \tanh \relax (x)\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (\sinh \relax (x ) \tanh \relax (x )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 69, normalized size = 2.23 \[ -\frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (\mathrm {sinh}\relax (x)\,\mathrm {tanh}\relax (x)\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\sinh {\relax (x )} \tanh {\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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