Optimal. Leaf size=67 \[ \frac {14}{45} a \sin (x) \sqrt {a \cos ^3(x)}+\frac {14 a E\left (\left .\frac {x}{2}\right |2\right ) \sqrt {a \cos ^3(x)}}{15 \cos ^{\frac {3}{2}}(x)}+\frac {2}{9} a \sin (x) \cos ^2(x) \sqrt {a \cos ^3(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 67, 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 = {3207, 2635, 2639} \[ \frac {2}{9} a \sin (x) \cos ^2(x) \sqrt {a \cos ^3(x)}+\frac {14}{45} a \sin (x) \sqrt {a \cos ^3(x)}+\frac {14 a E\left (\left .\frac {x}{2}\right |2\right ) \sqrt {a \cos ^3(x)}}{15 \cos ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cos ^3(x)\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {9}{2}}(x) \, dx}{\cos ^{\frac {3}{2}}(x)}\\ &=\frac {2}{9} a \cos ^2(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {\left (7 a \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {5}{2}}(x) \, dx}{9 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {14}{45} a \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{9} a \cos ^2(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {\left (7 a \sqrt {a \cos ^3(x)}\right ) \int \sqrt {\cos (x)} \, dx}{15 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {14 a \sqrt {a \cos ^3(x)} E\left (\left .\frac {x}{2}\right |2\right )}{15 \cos ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{9} a \cos ^2(x) \sqrt {a \cos ^3(x)} \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 50, normalized size = 0.75 \[ \frac {\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac {x}{2}\right |2\right )+(38 \sin (2 x)+5 \sin (4 x)) \sqrt {\cos (x)}\right )}{180 \cos ^{\frac {9}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cos \relax (x)^{3}} a \cos \relax (x)^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 198, normalized size = 2.96 \[ -\frac {2 \left (5 \left (\cos ^{6}\relax (x )\right )-21 i \cos \relax (x ) \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+21 i \cos \relax (x ) \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-21 i \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sin \relax (x )+21 i \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+2 \left (\cos ^{4}\relax (x )\right )+14 \left (\cos ^{2}\relax (x )\right )-21 \cos \relax (x )\right ) \left (a \left (\cos ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}{45 \cos \relax (x )^{5} \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a\,{\cos \relax (x)}^3\right )}^{3/2} \,d x \]
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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