Optimal. Leaf size=31 \[ \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \text {sech}(x) \sqrt {\cosh (x) \coth (x)} \]
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Rubi [A] time = 0.12, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4397, 4398, 4400, 2598, 2589} \[ \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \text {sech}(x) \sqrt {\cosh (x) \coth (x)} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2598
Rule 4397
Rule 4398
Rule 4400
Rubi steps
\begin {align*} \int (\text {csch}(x)+\sinh (x))^{3/2} \, dx &=\int (\cosh (x) \coth (x))^{3/2} \, dx\\ &=\frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int (-i \cosh (x) \coth (x))^{3/2} \, dx}{\sqrt {-i \cosh (x) \coth (x)}}\\ &=\frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int \cosh ^{\frac {3}{2}}(x) (-i \coth (x))^{3/2} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}+\frac {\left (4 i \sqrt {\cosh (x) \coth (x)}\right ) \int \frac {(-i \coth (x))^{3/2}}{\sqrt {\cosh (x)}} \, dx}{3 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 21, normalized size = 0.68 \[ \frac {2}{3} \left (\cosh ^2(x)-4\right ) \text {sech}(x) \sqrt {\cosh (x) \coth (x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 97, normalized size = 3.13 \[ \frac {\sqrt {\frac {1}{2}} {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 7\right )} \sinh \relax (x)^{2} - 14 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 7 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}}{3 \, \sqrt {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
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 \[ \int {\left (\operatorname {csch}\relax (x) + \sinh \relax (x)\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \left (\mathrm {csch}\relax (x )+\sinh \relax (x )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 109, normalized size = 3.52 \[ \frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} \]
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 \[ \int {\left (\mathrm {sinh}\relax (x)+\frac {1}{\mathrm {sinh}\relax (x)}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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