Optimal. Leaf size=185 \[ \frac {4}{3} a x^3 \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x^3 \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+16 a x^2 \sqrt {a \cos (x)+a}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {1280}{9} a \sqrt {a \cos (x)+a}-\frac {32}{9} a x \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {640}{9} a x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]
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Rubi [A] time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3319, 3311, 3296, 2638, 3310} \[ \frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+16 a x^2 \sqrt {a \cos (x)+a}+\frac {4}{3} a x^3 \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x^3 \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {1280}{9} a \sqrt {a \cos (x)+a}-\frac {32}{9} a x \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}-\frac {640}{9} a x \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3310
Rule 3311
Rule 3319
Rubi steps
\begin {align*} \int x^3 (a+a \cos (x))^{3/2} \, dx &=\left (2 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x^3 \cos ^3\left (\frac {x}{2}\right ) \, dx\\ &=\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {1}{3} \left (4 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x^3 \cos \left (\frac {x}{2}\right ) \, dx-\frac {1}{3} \left (16 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos ^3\left (\frac {x}{2}\right ) \, dx\\ &=-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}-\frac {32}{9} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )-\frac {1}{9} \left (32 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx-\left (8 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x^2 \sin \left (\frac {x}{2}\right ) \, dx\\ &=16 a x^2 \sqrt {a+a \cos (x)}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}-\frac {32}{9} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {64}{9} a x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {1}{9} \left (64 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, dx-\left (32 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {128}{9} a \sqrt {a+a \cos (x)}+16 a x^2 \sqrt {a+a \cos (x)}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}-\frac {32}{9} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {640}{9} a x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\left (64 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, dx\\ &=-\frac {1280}{9} a \sqrt {a+a \cos (x)}+16 a x^2 \sqrt {a+a \cos (x)}-\frac {64}{27} a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {8}{3} a x^2 \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}-\frac {32}{9} a x \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {4}{3} a x^3 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {640}{9} a x \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^3 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 67, normalized size = 0.36 \[ \frac {2}{27} a \sqrt {a (\cos (x)+1)} \left (234 x^2+3 \left (15 x^2-328\right ) x \tan \left (\frac {x}{2}\right )+\cos (x) \left (2 \left (9 x^2-8\right )+3 x \left (3 x^2-8\right ) \tan \left (\frac {x}{2}\right )\right )-1936\right ) \]
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 = 0.48, size = 113, normalized size = 0.61 \[ \frac {1}{27} \, \sqrt {2} {\left (2 \, {\left (9 \, a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {3}{2} \, x\right ) + 486 \, {\left (a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {1}{2} \, x\right ) + 3 \, {\left (3 \, a x^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {3}{2} \, x\right ) + 81 \, {\left (a x^{3} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 24 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +a \cos \relax (x )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 98, normalized size = 0.53 \[ \frac {1}{27} \, {\left (81 \, \sqrt {2} a x^{3} \sin \left (\frac {1}{2} \, x\right ) + 486 \, \sqrt {2} a x^{2} \cos \left (\frac {1}{2} \, x\right ) - 1944 \, \sqrt {2} a x \sin \left (\frac {1}{2} \, x\right ) - 3888 \, \sqrt {2} a \cos \left (\frac {1}{2} \, x\right ) + 2 \, {\left (9 \, \sqrt {2} a x^{2} - 8 \, \sqrt {2} a\right )} \cos \left (\frac {3}{2} \, x\right ) + 3 \, {\left (3 \, \sqrt {2} a x^{3} - 8 \, \sqrt {2} a x\right )} \sin \left (\frac {3}{2} \, x\right )\right )} \sqrt {a} \]
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 \[ \int x^3\,{\left (a+a\,\cos \relax (x)\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \left (a \left (\cos {\relax (x )} + 1\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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