Optimal. Leaf size=53 \[ \frac {8}{15} a^2 \coth (x) \sqrt {a \sinh ^2(x)}-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3282, 3286,
2718} \begin {gather*} \frac {8}{15} a^2 \coth (x) \sqrt {a \sinh ^2(x)}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3282
Rule 3286
Rubi steps
\begin {align*} \int \left (a \sinh ^2(x)\right )^{5/2} \, dx &=\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}-\frac {1}{5} (4 a) \int \left (a \sinh ^2(x)\right )^{3/2} \, dx\\ &=-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \sinh ^2(x)} \, dx\\ &=-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}+\frac {1}{15} \left (8 a^2 \text {csch}(x) \sqrt {a \sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=\frac {8}{15} a^2 \coth (x) \sqrt {a \sinh ^2(x)}-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.68 \begin {gather*} \frac {1}{240} a^2 (150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \text {csch}(x) \sqrt {a \sinh ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 32, normalized size = 0.60
method | result | size |
default | \(\frac {a^{3} \sinh \left (x \right ) \cosh \left (x \right ) \left (3 \left (\sinh ^{4}\left (x \right )\right )-4 \left (\sinh ^{2}\left (x \right )\right )+8\right )}{15 \sqrt {a \left (\sinh ^{2}\left (x \right )\right )}}\) | \(32\) |
risch | \(\frac {a^{2} {\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{160 \,{\mathrm e}^{2 x}-160}-\frac {5 a^{2} {\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{96 \left ({\mathrm e}^{2 x}-1\right )}+\frac {5 a^{2} {\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{16 \left ({\mathrm e}^{2 x}-1\right )}+\frac {5 \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}\, a^{2}}{16 \left ({\mathrm e}^{2 x}-1\right )}-\frac {5 a^{2} {\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{96 \left ({\mathrm e}^{2 x}-1\right )}+\frac {a^{2} {\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{160 \,{\mathrm e}^{2 x}-160}\) | \(196\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 53, normalized size = 1.00 \begin {gather*} -\frac {1}{160} \, a^{\frac {5}{2}} e^{\left (5 \, x\right )} + \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} - \frac {5}{16} \, a^{\frac {5}{2}} e^{\left (-x\right )} + \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - \frac {1}{160} \, a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - \frac {5}{16} \, a^{\frac {5}{2}} e^{x} \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. 511 vs.
\(2 (41) = 82\).
time = 0.39, size = 511, normalized size = 9.64 \begin {gather*} \frac {{\left (30 \, a^{2} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{9} + 3 \, a^{2} e^{x} \sinh \left (x\right )^{10} + 5 \, {\left (27 \, a^{2} \cosh \left (x\right )^{2} - 5 \, a^{2}\right )} e^{x} \sinh \left (x\right )^{8} + 40 \, {\left (9 \, a^{2} \cosh \left (x\right )^{3} - 5 \, a^{2} \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right )^{7} + 10 \, {\left (63 \, a^{2} \cosh \left (x\right )^{4} - 70 \, a^{2} \cosh \left (x\right )^{2} + 15 \, a^{2}\right )} e^{x} \sinh \left (x\right )^{6} + 4 \, {\left (189 \, a^{2} \cosh \left (x\right )^{5} - 350 \, a^{2} \cosh \left (x\right )^{3} + 225 \, a^{2} \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right )^{5} + 10 \, {\left (63 \, a^{2} \cosh \left (x\right )^{6} - 175 \, a^{2} \cosh \left (x\right )^{4} + 225 \, a^{2} \cosh \left (x\right )^{2} + 15 \, a^{2}\right )} e^{x} \sinh \left (x\right )^{4} + 40 \, {\left (9 \, a^{2} \cosh \left (x\right )^{7} - 35 \, a^{2} \cosh \left (x\right )^{5} + 75 \, a^{2} \cosh \left (x\right )^{3} + 15 \, a^{2} \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right )^{3} + 5 \, {\left (27 \, a^{2} \cosh \left (x\right )^{8} - 140 \, a^{2} \cosh \left (x\right )^{6} + 450 \, a^{2} \cosh \left (x\right )^{4} + 180 \, a^{2} \cosh \left (x\right )^{2} - 5 \, a^{2}\right )} e^{x} \sinh \left (x\right )^{2} + 10 \, {\left (3 \, a^{2} \cosh \left (x\right )^{9} - 20 \, a^{2} \cosh \left (x\right )^{7} + 90 \, a^{2} \cosh \left (x\right )^{5} + 60 \, a^{2} \cosh \left (x\right )^{3} - 5 \, a^{2} \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (3 \, a^{2} \cosh \left (x\right )^{10} - 25 \, a^{2} \cosh \left (x\right )^{8} + 150 \, a^{2} \cosh \left (x\right )^{6} + 150 \, a^{2} \cosh \left (x\right )^{4} - 25 \, a^{2} \cosh \left (x\right )^{2} + 3 \, a^{2}\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{480 \, {\left (\cosh \left (x\right )^{5} e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{5} - \cosh \left (x\right )^{5} + 5 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + 10 \, {\left (\cosh \left (x\right )^{2} e^{\left (2 \, x\right )} - \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} + 10 \, {\left (\cosh \left (x\right )^{3} e^{\left (2 \, x\right )} - \cosh \left (x\right )^{3}\right )} \sinh \left (x\right )^{2} + 5 \, {\left (\cosh \left (x\right )^{4} e^{\left (2 \, x\right )} - \cosh \left (x\right )^{4}\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 (a \sinh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (41) = 82\).
time = 0.41, size = 120, normalized size = 2.26 \begin {gather*} \frac {1}{480} \, {\left (3 \, a^{2} e^{\left (5 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - 25 \, a^{2} e^{\left (3 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 150 \, a^{2} e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + {\left (150 \, a^{2} e^{\left (4 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - 25 \, a^{2} e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 3 \, a^{2} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} e^{\left (-5 \, x\right )}\right )} \sqrt {a} \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 = -0.02 \begin {gather*} \int {\left (a\,{\mathrm {sinh}\left (x\right )}^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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