Optimal. Leaf size=53 \[ -\frac {8}{15} a^2 \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}-\frac {4}{15} a \cot (x) \left (a \sin ^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 53, 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 = {3203, 3207, 2638} \[ -\frac {8}{15} a^2 \cot (x) \sqrt {a \sin ^2(x)}-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}-\frac {4}{15} a \cot (x) \left (a \sin ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a \sin ^2(x)\right )^{5/2} \, dx &=-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}+\frac {1}{5} (4 a) \int \left (a \sin ^2(x)\right )^{3/2} \, dx\\ &=-\frac {4}{15} a \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \sin ^2(x)} \, dx\\ &=-\frac {4}{15} a \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}+\frac {1}{15} \left (8 a^2 \csc (x) \sqrt {a \sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\frac {8}{15} a^2 \cot (x) \sqrt {a \sin ^2(x)}-\frac {4}{15} a \cot (x) \left (a \sin ^2(x)\right )^{3/2}-\frac {1}{5} \cot (x) \left (a \sin ^2(x)\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.68 \[ -\frac {1}{240} a^2 (150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x) \sqrt {a \sin ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 43, normalized size = 0.81 \[ -\frac {{\left (3 \, a^{2} \cos \relax (x)^{5} - 10 \, a^{2} \cos \relax (x)^{3} + 15 \, a^{2} \cos \relax (x)\right )} \sqrt {-a \cos \relax (x)^{2} + a}}{15 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.74, size = 45, normalized size = 0.85 \[ \frac {1}{15} \, {\left (8 \, a^{2} \mathrm {sgn}\left (\sin \relax (x)\right ) - {\left (3 \, a^{2} \cos \relax (x)^{5} - 10 \, a^{2} \cos \relax (x)^{3} + 15 \, a^{2} \cos \relax (x)\right )} \mathrm {sgn}\left (\sin \relax (x)\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 32, normalized size = 0.60 \[ -\frac {a^{3} \sin \relax (x ) \cos \relax (x ) \left (3 \left (\sin ^{4}\relax (x )\right )+4 \left (\sin ^{2}\relax (x )\right )+8\right )}{15 \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]
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 \[ \int \left (a \sin \relax (x)^{2}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a\,{\sin \relax (x)}^2\right )}^{5/2} \,d x \]
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 \[ \int \left (a \sin ^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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