Optimal. Leaf size=89 \[ -\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {16}{3} a \sqrt {a \cosh (x)+a}+\frac {4}{3} a x \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {8}{3} a x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
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Rubi [A] time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3319, 3310, 3296, 2638} \[ -\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {16}{3} a \sqrt {a \cosh (x)+a}+\frac {4}{3} a x \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {8}{3} a x \tanh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 3319
Rubi steps
\begin {align*} \int x (a+a \cosh (x))^{3/2} \, dx &=\left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh ^3\left (\frac {x}{2}\right ) \, dx\\ &=-\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {1}{3} \left (4 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int x \cosh \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {8}{3} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )-\frac {1}{3} \left (8 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \sinh \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {16}{3} a \sqrt {a+a \cosh (x)}-\frac {8}{9} a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}+\frac {4}{3} a x \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )+\frac {8}{3} a x \sqrt {a+a \cosh (x)} \tanh \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 56, normalized size = 0.63 \[ \frac {1}{9} a \text {sech}\left (\frac {x}{2}\right ) \sqrt {a (\cosh (x)+1)} \left (3 x \left (9 \sinh \left (\frac {x}{2}\right )+\sinh \left (\frac {3 x}{2}\right )\right )-54 \cosh \left (\frac {x}{2}\right )-2 \cosh \left (\frac {3 x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 96, normalized size = 1.08 \[ -\frac {1}{18} \, \sqrt {2} {\left (18 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 3 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 36 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (3 \, a^{\frac {3}{2}} x - 2 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 27 \, {\left (a^{\frac {3}{2}} x - 2 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 9 \, {\left (a^{\frac {3}{2}} x + 2 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x \left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 92, normalized size = 1.03 \[ -\frac {1}{18} \, {\left (3 \, \sqrt {2} a^{\frac {3}{2}} x + 2 \, \sqrt {2} a^{\frac {3}{2}} - {\left (3 \, \sqrt {2} a^{\frac {3}{2}} x - 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 27 \, {\left (\sqrt {2} a^{\frac {3}{2}} x - 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 27 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + 2 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \]
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 \[ \int x\,{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a \left (\cosh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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