Optimal. Leaf size=72 \[ \frac {16}{35} a^3 \tanh (x) \sqrt {a \cosh ^2(x)}+\frac {8}{35} a^2 \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{35} a \tanh (x) \left (a \cosh ^2(x)\right )^{5/2} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3203, 3207, 2637} \[ \frac {8}{35} a^2 \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {16}{35} a^3 \tanh (x) \sqrt {a \cosh ^2(x)}+\frac {1}{7} \tanh (x) \left (a \cosh ^2(x)\right )^{7/2}+\frac {6}{35} a \tanh (x) \left (a \cosh ^2(x)\right )^{5/2} \]
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 )^{7/2} \, dx &=\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x)+\frac {1}{7} (6 a) \int \left (a \cosh ^2(x)\right )^{5/2} \, dx\\ &=\frac {6}{35} a \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x)+\frac {1}{35} \left (24 a^2\right ) \int \left (a \cosh ^2(x)\right )^{3/2} \, dx\\ &=\frac {8}{35} a^2 \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {6}{35} a \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x)+\frac {1}{35} \left (16 a^3\right ) \int \sqrt {a \cosh ^2(x)} \, dx\\ &=\frac {8}{35} a^2 \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {6}{35} a \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x)+\frac {1}{35} \left (16 a^3 \sqrt {a \cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\frac {16}{35} a^3 \sqrt {a \cosh ^2(x)} \tanh (x)+\frac {8}{35} a^2 \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {6}{35} a \left (a \cosh ^2(x)\right )^{5/2} \tanh (x)+\frac {1}{7} \left (a \cosh ^2(x)\right )^{7/2} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.58 \[ \frac {a^3 (1225 \sinh (x)+245 \sinh (3 x)+49 \sinh (5 x)+5 \sinh (7 x)) \text {sech}(x) \sqrt {a \cosh ^2(x)}}{2240} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.76, size = 817, normalized size = 11.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 79, normalized size = 1.10 \[ \frac {1}{4480} \, {\left (5 \, a^{3} e^{\left (7 \, x\right )} + 49 \, a^{3} e^{\left (5 \, x\right )} + 245 \, a^{3} e^{\left (3 \, x\right )} + 1225 \, a^{3} e^{x} - {\left (1225 \, a^{3} e^{\left (6 \, x\right )} + 245 \, a^{3} e^{\left (4 \, x\right )} + 49 \, a^{3} e^{\left (2 \, x\right )} + 5 \, a^{3}\right )} e^{\left (-7 \, x\right )}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 38, normalized size = 0.53 \[ \frac {a^{4} \cosh \relax (x ) \sinh \relax (x ) \left (5 \left (\cosh ^{6}\relax (x )\right )+6 \left (\cosh ^{4}\relax (x )\right )+8 \left (\cosh ^{2}\relax (x )\right )+16\right )}{35 \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 0.99 \[ \frac {1}{896} \, a^{\frac {7}{2}} e^{\left (7 \, x\right )} + \frac {7}{640} \, a^{\frac {7}{2}} e^{\left (5 \, x\right )} + \frac {7}{128} \, a^{\frac {7}{2}} e^{\left (3 \, x\right )} - \frac {35}{128} \, a^{\frac {7}{2}} e^{\left (-x\right )} - \frac {7}{128} \, a^{\frac {7}{2}} e^{\left (-3 \, x\right )} - \frac {7}{640} \, a^{\frac {7}{2}} e^{\left (-5 \, x\right )} - \frac {1}{896} \, a^{\frac {7}{2}} e^{\left (-7 \, x\right )} + \frac {35}{128} \, a^{\frac {7}{2}} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^2\right )}^{7/2} \,d x \]
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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