Integrand size = 10, antiderivative size = 132 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {63}{256} a^2 x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {21}{128} a^2 \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {63}{256} a^2 \sqrt {a \cosh ^4(x)} \tanh (x) \]
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Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8} \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {21}{128} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 x \text {sech}^2(x) \sqrt {a \cosh ^4(x)}+\frac {1}{10} a^2 \sinh (x) \cosh ^7(x) \sqrt {a \cosh ^4(x)}+\frac {9}{80} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^4(x)}+\frac {21}{160} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^4(x)} \]
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Rule 8
Rule 2715
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \left (a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^{10}(x) \, dx \\ & = \frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} \left (9 a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^8(x) \, dx \\ & = \frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{80} \left (63 a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^6(x) \, dx \\ & = \frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{32} \left (21 a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^4(x) \, dx \\ & = \frac {21}{128} a^2 \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{128} \left (63 a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^2(x) \, dx \\ & = \frac {21}{128} a^2 \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {63}{256} a^2 \sqrt {a \cosh ^4(x)} \tanh (x)+\frac {1}{256} \left (63 a^2 \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int 1 \, dx \\ & = \frac {63}{256} a^2 x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {21}{128} a^2 \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {21}{160} a^2 \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {9}{80} a^2 \cosh ^5(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{10} a^2 \cosh ^7(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {63}{256} a^2 \sqrt {a \cosh ^4(x)} \tanh (x) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.40 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {a \left (a \cosh ^4(x)\right )^{3/2} \text {sech}^6(x) (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x))}{10240} \]
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Time = 8.67 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.34
method | result | size |
default | \(\frac {\sqrt {8}\, \sqrt {2}\, a^{\frac {3}{2}} \left (1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \left (8 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \sinh \left (2 x \right )^{4}+50 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \cosh \left (2 x \right ) \sinh \left (2 x \right )^{2}+160 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}\, \sinh \left (2 x \right )^{2}+325 \cosh \left (2 x \right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+640 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+315 \ln \left (\sqrt {a}\, \cosh \left (2 x \right )+\sqrt {a \sinh \left (2 x \right )^{2}}\right ) a \right )}{10240 \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(177\) |
risch | \(\frac {63 a^{2} {\mathrm e}^{2 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}\, x}{256 \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {a^{2} {\mathrm e}^{12 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {5 a^{2} {\mathrm e}^{10 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{8 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{6 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{512 \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {105 a^{2} {\mathrm e}^{4 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{1024 \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {105 \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}\, a^{2}}{1024 \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{-2 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{512 \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{-4 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {5 a^{2} {\mathrm e}^{-6 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {a^{2} {\mathrm e}^{-8 x} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left (1+{\mathrm e}^{2 x}\right )^{2}}\) | \(362\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1597 vs. \(2 (108) = 216\).
Time = 0.32 (sec) , antiderivative size = 1597, normalized size of antiderivative = 12.10 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {63}{256} \, a^{\frac {5}{2}} x + \frac {1}{20480} \, {\left (25 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 150 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 2100 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} - 2100 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} - 150 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac {5}{2}} e^{\left (-18 \, x\right )} - 2 \, a^{\frac {5}{2}} e^{\left (-20 \, x\right )} + 2 \, a^{\frac {5}{2}}\right )} e^{\left (10 \, x\right )} \]
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Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\frac {1}{20480} \, {\left (5040 \, a^{2} x + 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} + 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} + 2100 \, a^{2} e^{\left (2 \, x\right )} - {\left (5754 \, a^{2} e^{\left (10 \, x\right )} + 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} + 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} + 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt {a} \]
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Timed out. \[ \int \left (a \cosh ^4(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {cosh}\left (x\right )}^4\right )}^{5/2} \,d x \]
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