Optimal. Leaf size=34 \[ \frac {1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac {2}{3} a \coth (x) \sqrt {a \sinh ^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3203, 3207, 2638} \[ \frac {1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac {2}{3} a \coth (x) \sqrt {a \sinh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a \sinh ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac {1}{3} (2 a) \int \sqrt {a \sinh ^2(x)} \, dx\\ &=\frac {1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac {1}{3} \left (2 a \text {csch}(x) \sqrt {a \sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=-\frac {2}{3} a \coth (x) \sqrt {a \sinh ^2(x)}+\frac {1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.76 \[ \frac {1}{12} a (\cosh (3 x)-9 \cosh (x)) \text {csch}(x) \sqrt {a \sinh ^2(x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 226, normalized size = 6.65 \[ \frac {{\left (6 \, a \cosh \relax (x) e^{x} \sinh \relax (x)^{5} + a e^{x} \sinh \relax (x)^{6} + 3 \, {\left (5 \, a \cosh \relax (x)^{2} - 3 \, a\right )} e^{x} \sinh \relax (x)^{4} + 4 \, {\left (5 \, a \cosh \relax (x)^{3} - 9 \, a \cosh \relax (x)\right )} e^{x} \sinh \relax (x)^{3} + 3 \, {\left (5 \, a \cosh \relax (x)^{4} - 18 \, a \cosh \relax (x)^{2} - 3 \, a\right )} e^{x} \sinh \relax (x)^{2} + 6 \, {\left (a \cosh \relax (x)^{5} - 6 \, a \cosh \relax (x)^{3} - 3 \, a \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (a \cosh \relax (x)^{6} - 9 \, a \cosh \relax (x)^{4} - 9 \, a \cosh \relax (x)^{2} + a\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{24 \, {\left (\cosh \relax (x)^{3} e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \relax (x)^{3} - \cosh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} - \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 3 \, {\left (\cosh \relax (x)^{2} e^{\left (2 \, x\right )} - \cosh \relax (x)^{2}\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 70, normalized size = 2.06 \[ -\frac {1}{24} \, {\left ({\left (9 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 9 \, e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 0.71 \[ \frac {a^{2} \sinh \relax (x ) \cosh \relax (x ) \left (\sinh ^{2}\relax (x )-2\right )}{3 \sqrt {a \left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 35, normalized size = 1.03 \[ -\frac {1}{24} \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} + \frac {3}{8} \, a^{\frac {3}{2}} e^{\left (-x\right )} - \frac {1}{24} \, a^{\frac {3}{2}} e^{\left (-3 \, x\right )} + \frac {3}{8} \, a^{\frac {3}{2}} e^{x} \]
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 \[ \int {\left (a\,{\mathrm {sinh}\relax (x)}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (a \sinh ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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