Optimal. Leaf size=78 \[ \frac {5}{24} a \sinh (x) \cosh (x) \sqrt {a \cosh ^4(x)}+\frac {5}{16} a \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {5}{16} a x \text {sech}^2(x) \sqrt {a \cosh ^4(x)}+\frac {1}{6} a \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^4(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 78, 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 = {3207, 2635, 8} \[ \frac {1}{6} a \sinh (x) \cosh ^3(x) \sqrt {a \cosh ^4(x)}+\frac {5}{24} a \sinh (x) \cosh (x) \sqrt {a \cosh ^4(x)}+\frac {5}{16} a \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {5}{16} a x \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \]
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 )^{3/2} \, dx &=\left (a \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^6(x) \, dx\\ &=\frac {1}{6} a \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{6} \left (5 a \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^4(x) \, dx\\ &=\frac {5}{24} a \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{6} a \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{8} \left (5 a \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^2(x) \, dx\\ &=\frac {5}{24} a \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{6} a \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {5}{16} a \sqrt {a \cosh ^4(x)} \tanh (x)+\frac {1}{16} \left (5 a \sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int 1 \, dx\\ &=\frac {5}{16} a x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {5}{24} a \cosh (x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {1}{6} a \cosh ^3(x) \sqrt {a \cosh ^4(x)} \sinh (x)+\frac {5}{16} a \sqrt {a \cosh ^4(x)} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 0.49 \[ \frac {1}{192} (60 x+45 \sinh (2 x)+9 \sinh (4 x)+\sinh (6 x)) \text {sech}^6(x) \left (a \cosh ^4(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 659, normalized size = 8.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 52, normalized size = 0.67 \[ -\frac {1}{384} \, {\left ({\left (110 \, e^{\left (6 \, x\right )} + 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-6 \, x\right )} - 120 \, x - e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 45 \, e^{\left (2 \, x\right )}\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 131, normalized size = 1.68 \[ \frac {\sqrt {8}\, \left (\cosh \left (2 x \right )+1\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, \sqrt {2}\, \sqrt {a}\, \left (2 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \left (\sinh ^{2}\left (2 x \right )\right )+9 \cosh \left (2 x \right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+24 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+15 \ln \left (\sqrt {a}\, \cosh \left (2 x \right )+\sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\right ) a \right )}{384 \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 62, normalized size = 0.79 \[ \frac {5}{16} \, a^{\frac {3}{2}} x + \frac {1}{384} \, {\left (9 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 45 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} - 45 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} - 9 \, a^{\frac {3}{2}} e^{\left (-10 \, x\right )} - a^{\frac {3}{2}} e^{\left (-12 \, x\right )} + a^{\frac {3}{2}}\right )} e^{\left (6 \, x\right )} \]
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 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^4\right )}^{3/2} \,d x \]
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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