Optimal. Leaf size=117 \[ \frac {26 a^2 \sqrt {a \cos ^3(x)} F\left (\left .\frac {x}{2}\right |2\right )}{77 \cos ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715,
2720} \begin {gather*} \frac {26}{165} a^2 \sin (x) \cos ^3(x) \sqrt {a \cos ^3(x)}+\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \cos ^3(x)}+\frac {26}{77} a^2 \tan (x) \sqrt {a \cos ^3(x)}+\frac {26 a^2 F\left (\left .\frac {x}{2}\right |2\right ) \sqrt {a \cos ^3(x)}}{77 \cos ^{\frac {3}{2}}(x)}+\frac {2}{15} a^2 \sin (x) \cos ^5(x) \sqrt {a \cos ^3(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 3286
Rubi steps
\begin {align*} \int \left (a \cos ^3(x)\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {15}{2}}(x) \, dx}{\cos ^{\frac {3}{2}}(x)}\\ &=\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {\left (13 a^2 \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {11}{2}}(x) \, dx}{15 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {\left (39 a^2 \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {7}{2}}(x) \, dx}{55 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {\left (39 a^2 \sqrt {a \cos ^3(x)}\right ) \int \cos ^{\frac {3}{2}}(x) \, dx}{77 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x)+\frac {\left (13 a^2 \sqrt {a \cos ^3(x)}\right ) \int \frac {1}{\sqrt {\cos (x)}} \, dx}{77 \cos ^{\frac {3}{2}}(x)}\\ &=\frac {26 a^2 \sqrt {a \cos ^3(x)} F\left (\left .\frac {x}{2}\right |2\right )}{77 \cos ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 0.52 \begin {gather*} \frac {a \left (a \cos ^3(x)\right )^{3/2} \left (12480 F\left (\left .\frac {x}{2}\right |2\right )+\sqrt {\cos (x)} (15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x))\right )}{36960 \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 = 0.27, size = 114, normalized size = 0.97
method | result | size |
default | \(\frac {2 \left (-1+\cos \left (x \right )\right ) \left (77 \left (\cos ^{8}\left (x \right )\right )-77 \left (\cos ^{7}\left (x \right )\right )+91 \left (\cos ^{6}\left (x \right )\right )-91 \left (\cos ^{5}\left (x \right )\right )-195 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sin \left (x \right )+117 \left (\cos ^{4}\left (x \right )\right )-117 \left (\cos ^{3}\left (x \right )\right )+195 \left (\cos ^{2}\left (x \right )\right )-195 \cos \left (x \right )\right ) \left (\cos \left (x \right )+1\right )^{2} \left (a \left (\cos ^{3}\left (x \right )\right )\right )^{\frac {5}{2}}}{1155 \cos \left (x \right )^{8} \sin \left (x \right )^{3}}\) | \(114\) |
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 = 0.14, size = 92, normalized size = 0.79 \begin {gather*} \frac {195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{6} + 91 \, a^{2} \cos \left (x\right )^{4} + 117 \, a^{2} \cos \left (x\right )^{2} + 195 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{1155 \, \cos \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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