Optimal. Leaf size=132 \[ \frac {21}{128} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 x \text {sech}^2(x) \sqrt {a \cosh ^4(x)}+\frac {1}{10} a^2 \sinh (x) \cosh ^7(x) \sqrt {a \cosh ^4(x)}+\frac {9}{80} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^4(x)}+\frac {21}{160} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^4(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 8} \[ \frac {1}{10} a^2 \sinh (x) \cosh ^7(x) \sqrt {a \cosh ^4(x)}+\frac {9}{80} a^2 \sinh (x) \cosh ^5(x) \sqrt {a \cosh ^4(x)}+\frac {21}{160} a^2 \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^4(x)}+\frac {21}{128} a^2 \sinh (x) \cosh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {63}{256} a^2 x \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cosh ^4(x)\right )^{5/2} \, dx &=\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*}
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Mathematica [A] time = 0.12, size = 53, normalized size = 0.40 \[ \frac {a (2520 x+2100 \sinh (2 x)+600 \sinh (4 x)+150 \sinh (6 x)+25 \sinh (8 x)+2 \sinh (10 x)) \text {sech}^6(x) \left (a \cosh ^4(x)\right )^{3/2}}{10240} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 1597, normalized size = 12.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 114, normalized size = 0.86 \[ \frac {1}{20480} \, {\left (5040 \, a^{2} x + 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} + 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} + 2100 \, a^{2} e^{\left (2 \, x\right )} - {\left (5754 \, a^{2} e^{\left (10 \, x\right )} + 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} + 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} + 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 177, normalized size = 1.34 \[ \frac {\sqrt {8}\, \left (\cosh \left (2 x \right )+1\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, \sqrt {2}\, a^{\frac {3}{2}} \left (8 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \left (\sinh ^{4}\left (2 x \right )\right )+50 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \cosh \left (2 x \right ) \left (\sinh ^{2}\left (2 x \right )\right )+160 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \left (\sinh ^{2}\left (2 x \right )\right )+325 \cosh \left (2 x \right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+640 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+315 \ln \left (\sqrt {a}\, \cosh \left (2 x \right )+\sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\right ) a \right )}{10240 \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 100, normalized size = 0.76 \[ \frac {63}{256} \, a^{\frac {5}{2}} x + \frac {1}{20480} \, {\left (25 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 150 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 2100 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} - 2100 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} - 150 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac {5}{2}} e^{\left (-18 \, x\right )} - 2 \, a^{\frac {5}{2}} e^{\left (-20 \, x\right )} + 2 \, a^{\frac {5}{2}}\right )} e^{\left (10 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^4\right )}^{5/2} \,d x \]
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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