Optimal. Leaf size=31 \[ \frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)}+\frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)} \]
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Rubi [A]
time = 0.08, 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 = {4483, 4485,
2678, 2669} \begin {gather*} \frac {2}{3} \sinh (x) \sqrt {\sinh (x) \tanh (x)}+\frac {8}{3} \text {csch}(x) \sqrt {\sinh (x) \tanh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2669
Rule 2678
Rule 4483
Rule 4485
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 = 0.05, size = 23, normalized size = 0.74 \begin {gather*} \frac {2}{3} \left (1+4 \text {csch}^2(x)\right ) \sinh (x) \sqrt {\sinh (x) \tanh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.97, size = 0, normalized size = 0.00 \[\int \left (\sinh \left (x \right ) \tanh \left (x \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (23) = 46\).
time = 0.51, size = 69, normalized size = 2.23 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (23) = 46\).
time = 0.37, size = 95, normalized size = 3.06 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \left (\sinh {\left (x \right )} \tanh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (\mathrm {sinh}\left (x\right )\,\mathrm {tanh}\left (x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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