Integrand size = 10, antiderivative size = 53 \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\frac {8}{15} a^2 \coth (x) \sqrt {a \sinh ^2(x)}-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3282, 3286, 2718} \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\frac {8}{15} a^2 \coth (x) \sqrt {a \sinh ^2(x)}+\frac {1}{5} \coth (x) \left (a \sinh ^2(x)\right )^{5/2}-\frac {4}{15} a \coth (x) \left (a \sinh ^2(x)\right )^{3/2} \]
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Rule 2718
Rule 3282
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.68 \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\frac {1}{240} a^2 (150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \text {csch}(x) \sqrt {a \sinh ^2(x)} \]
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Time = 1.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {a^{3} \sinh \left (x \right ) \cosh \left (x \right ) \left (3 \sinh \left (x \right )^{4}-4 \sinh \left (x \right )^{2}+8\right )}{15 \sqrt {a \sinh \left (x \right )^{2}}}\) | \(32\) |
risch | \(\frac {a^{2} {\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{160 \,{\mathrm e}^{2 x}-160}-\frac {5 a^{2} {\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{96 \left ({\mathrm e}^{2 x}-1\right )}+\frac {5 a^{2} {\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{16 \left ({\mathrm e}^{2 x}-1\right )}+\frac {5 \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}\, a^{2}}{16 \left ({\mathrm e}^{2 x}-1\right )}-\frac {5 a^{2} {\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{96 \left ({\mathrm e}^{2 x}-1\right )}+\frac {a^{2} {\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{160 \,{\mathrm e}^{2 x}-160}\) | \(196\) |
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Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 511, normalized size of antiderivative = 9.64 \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\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 )}} \]
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\[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\int \left (a \sinh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=-\frac {1}{160} \, a^{\frac {5}{2}} e^{\left (5 \, x\right )} + \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} - \frac {5}{16} \, a^{\frac {5}{2}} e^{\left (-x\right )} + \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - \frac {1}{160} \, a^{\frac {5}{2}} e^{\left (-5 \, x\right )} - \frac {5}{16} \, a^{\frac {5}{2}} e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.26 \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\frac {1}{480} \, {\left (3 \, a^{2} e^{\left (5 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - 25 \, a^{2} e^{\left (3 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 150 \, a^{2} e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + {\left (150 \, a^{2} e^{\left (4 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - 25 \, a^{2} e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 3 \, a^{2} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} e^{\left (-5 \, x\right )}\right )} \sqrt {a} \]
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Timed out. \[ \int \left (a \sinh ^2(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^2\right )}^{5/2} \,d x \]
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