Optimal. Leaf size=53 \[ \frac {8}{15} a^2 \tanh (x) \sqrt {a \cosh ^2(x)}+\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{15} a \tanh (x) \left (a \cosh ^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.04, 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 = {3203, 3207, 2637} \[ \frac {8}{15} a^2 \tanh (x) \sqrt {a \cosh ^2(x)}+\frac {1}{5} \tanh (x) \left (a \cosh ^2(x)\right )^{5/2}+\frac {4}{15} a \tanh (x) \left (a \cosh ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cosh ^2(x)\right )^{5/2} \, dx &=\frac {1}{5} \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{5} (4 a) \int \left (a \cosh ^2(x)\right )^{3/2} \, dx\\ &=\frac {4}{15} a \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {1}{5} \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \cosh ^2(x)} \, dx\\ &=\frac {4}{15} a \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {1}{5} \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{15} \left (8 a^2 \sqrt {a \cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\frac {8}{15} a^2 \sqrt {a \cosh ^2(x)} \tanh (x)+\frac {4}{15} a \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {1}{5} \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.68 \[ \frac {1}{240} a^2 (150 \sinh (x)+25 \sinh (3 x)+3 \sinh (5 x)) \text {sech}(x) \sqrt {a \cosh ^2(x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 501, normalized size = 9.45 \[ \frac {{\left (30 \, a^{2} \cosh \relax (x) e^{x} \sinh \relax (x)^{9} + 3 \, a^{2} e^{x} \sinh \relax (x)^{10} + 5 \, {\left (27 \, a^{2} \cosh \relax (x)^{2} + 5 \, a^{2}\right )} e^{x} \sinh \relax (x)^{8} + 40 \, {\left (9 \, a^{2} \cosh \relax (x)^{3} + 5 \, a^{2} \cosh \relax (x)\right )} e^{x} \sinh \relax (x)^{7} + 10 \, {\left (63 \, a^{2} \cosh \relax (x)^{4} + 70 \, a^{2} \cosh \relax (x)^{2} + 15 \, a^{2}\right )} e^{x} \sinh \relax (x)^{6} + 4 \, {\left (189 \, a^{2} \cosh \relax (x)^{5} + 350 \, a^{2} \cosh \relax (x)^{3} + 225 \, a^{2} \cosh \relax (x)\right )} e^{x} \sinh \relax (x)^{5} + 10 \, {\left (63 \, a^{2} \cosh \relax (x)^{6} + 175 \, a^{2} \cosh \relax (x)^{4} + 225 \, a^{2} \cosh \relax (x)^{2} - 15 \, a^{2}\right )} e^{x} \sinh \relax (x)^{4} + 40 \, {\left (9 \, a^{2} \cosh \relax (x)^{7} + 35 \, a^{2} \cosh \relax (x)^{5} + 75 \, a^{2} \cosh \relax (x)^{3} - 15 \, a^{2} \cosh \relax (x)\right )} e^{x} \sinh \relax (x)^{3} + 5 \, {\left (27 \, a^{2} \cosh \relax (x)^{8} + 140 \, a^{2} \cosh \relax (x)^{6} + 450 \, a^{2} \cosh \relax (x)^{4} - 180 \, a^{2} \cosh \relax (x)^{2} - 5 \, a^{2}\right )} e^{x} \sinh \relax (x)^{2} + 10 \, {\left (3 \, a^{2} \cosh \relax (x)^{9} + 20 \, a^{2} \cosh \relax (x)^{7} + 90 \, a^{2} \cosh \relax (x)^{5} - 60 \, a^{2} \cosh \relax (x)^{3} - 5 \, a^{2} \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (3 \, a^{2} \cosh \relax (x)^{10} + 25 \, a^{2} \cosh \relax (x)^{8} + 150 \, a^{2} \cosh \relax (x)^{6} - 150 \, a^{2} \cosh \relax (x)^{4} - 25 \, a^{2} \cosh \relax (x)^{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 \relax (x)^{5} e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{5} + \cosh \relax (x)^{5} + 5 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x)^{4} + 10 \, {\left (\cosh \relax (x)^{2} e^{\left (2 \, x\right )} + \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{3} + 10 \, {\left (\cosh \relax (x)^{3} e^{\left (2 \, x\right )} + \cosh \relax (x)^{3}\right )} \sinh \relax (x)^{2} + 5 \, {\left (\cosh \relax (x)^{4} e^{\left (2 \, x\right )} + \cosh \relax (x)^{4}\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 61, normalized size = 1.15 \[ \frac {1}{480} \, {\left (3 \, a^{2} e^{\left (5 \, x\right )} + 25 \, a^{2} e^{\left (3 \, x\right )} + 150 \, a^{2} e^{x} - {\left (150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2}\right )} e^{\left (-5 \, x\right )}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 32, normalized size = 0.60 \[ \frac {a^{3} \cosh \relax (x ) \sinh \relax (x ) \left (3 \left (\cosh ^{4}\relax (x )\right )+4 \left (\cosh ^{2}\relax (x )\right )+8\right )}{15 \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 53, normalized size = 1.00 \[ \frac {1}{160} \, a^{\frac {5}{2}} e^{\left (5 \, x\right )} + \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} - \frac {5}{16} \, a^{\frac {5}{2}} e^{\left (-x\right )} - \frac {5}{96} \, a^{\frac {5}{2}} e^{\left (-3 \, x\right )} - \frac {1}{160} \, a^{\frac {5}{2}} e^{\left (-5 \, x\right )} + \frac {5}{16} \, a^{\frac {5}{2}} e^{x} \]
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 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^2\right )}^{5/2} \,d x \]
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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