Integrand size = 10, antiderivative size = 117 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {26 a^2 \sqrt {a \cos ^3(x)} \operatorname {EllipticF}\left (\frac {x}{2},2\right )}{77 \cos ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x) \]
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Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2720} \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {26}{165} a^2 \sin (x) \cos ^3(x) \sqrt {a \cos ^3(x)}+\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \cos ^3(x)}+\frac {26}{77} a^2 \tan (x) \sqrt {a \cos ^3(x)}+\frac {26 a^2 \operatorname {EllipticF}\left (\frac {x}{2},2\right ) \sqrt {a \cos ^3(x)}}{77 \cos ^{\frac {3}{2}}(x)}+\frac {2}{15} a^2 \sin (x) \cos ^5(x) \sqrt {a \cos ^3(x)} \]
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Rule 2715
Rule 2720
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \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)} \operatorname {EllipticF}\left (\frac {x}{2},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*}
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.52 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {a \left (a \cos ^3(x)\right )^{3/2} \left (12480 \operatorname {EllipticF}\left (\frac {x}{2},2\right )+\sqrt {\cos (x)} (15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x))\right )}{36960 \cos ^{\frac {9}{2}}(x)} \]
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Result contains complex when optimal does not.
Time = 12.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {2 \sqrt {a \left (\cos ^{3}\left (x \right )\right )}\, a^{2} \left (-77 \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )-91 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )+195 i \sec \left (x \right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+195 i \left (\sec ^{2}\left (x \right )\right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-117 \cos \left (x \right ) \sin \left (x \right )-195 \tan \left (x \right )\right )}{1155}\) | \(117\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{6} + 91 \, a^{2} \cos \left (x\right )^{4} + 117 \, a^{2} \cos \left (x\right )^{2} + 195 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{1155 \, \cos \left (x\right )} \]
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Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\text {Timed out} \]
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\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^3\right )}^{5/2} \,d x \]
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