Optimal. Leaf size=67 \[ \frac {14 a \sqrt {a \cos ^3(x)} E\left (\left .\frac {x}{2}\right |2\right )}{15 \cos ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{9} a \cos ^2(x) \sqrt {a \cos ^3(x)} \sin (x) \]
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Rubi [A]
time = 0.02, 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 = {3286, 2715,
2719} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2719
Rule 3286
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 = 0.05, size = 50, normalized size = 0.75 \begin {gather*} \frac {\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac {x}{2}\right |2\right )+\sqrt {\cos (x)} (38 \sin (2 x)+5 \sin (4 x))\right )}{180 \cos ^{\frac {9}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.08, size = 198, normalized size = 2.96
method | result | size |
default | \(-\frac {2 \left (5 \left (\cos ^{6}\left (x \right )\right )+21 i \cos \left (x \right ) \sin \left (x \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-21 i \cos \left (x \right ) \sin \left (x \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+21 i \sin \left (x \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-21 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sin \left (x \right )+2 \left (\cos ^{4}\left (x \right )\right )+14 \left (\cos ^{2}\left (x \right )\right )-21 \cos \left (x \right )\right ) \left (a \left (\cos ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}{45 \cos \left (x \right )^{5} \sin \left (x \right )}\) | \(198\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 66, normalized size = 0.99 \begin {gather*} -\frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + \frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, \sqrt {a \cos \left (x\right )^{3}} {\left (5 \, a \cos \left (x\right )^{2} + 7 \, a\right )} \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a\,{\cos \left (x\right )}^3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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