Optimal. Leaf size=53 \[ \frac {8}{15} a^2 \tan (x) \sqrt {a \cos ^2(x)}+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}+\frac {4}{15} a \tan (x) \left (a \cos ^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.04, 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, 2637} \[ \frac {8}{15} a^2 \tan (x) \sqrt {a \cos ^2(x)}+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}+\frac {4}{15} a \tan (x) \left (a \cos ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cos ^2(x)\right )^{5/2} \, dx &=\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{5} (4 a) \int \left (a \cos ^2(x)\right )^{3/2} \, dx\\ &=\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \cos ^2(x)} \, dx\\ &=\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2 \sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\frac {8}{15} a^2 \sqrt {a \cos ^2(x)} \tan (x)+\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.68 \[ \frac {1}{240} a^2 (150 \sin (x)+25 \sin (3 x)+3 \sin (5 x)) \sec (x) \sqrt {a \cos ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 40, normalized size = 0.75 \[ \frac {{\left (3 \, a^{2} \cos \relax (x)^{4} + 4 \, a^{2} \cos \relax (x)^{2} + 8 \, a^{2}\right )} \sqrt {a \cos \relax (x)^{2}} \sin \relax (x)}{15 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 34, normalized size = 0.64 \[ \frac {1}{15} \, {\left (3 \, a^{2} \sin \relax (x)^{5} - 10 \, a^{2} \sin \relax (x)^{3} + 15 \, a^{2} \sin \relax (x)\right )} \sqrt {a} \mathrm {sgn}\left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 32, normalized size = 0.60 \[ \frac {a^{3} \cos \relax (x ) \sin \relax (x ) \left (3 \left (\cos ^{4}\relax (x )\right )+4 \left (\cos ^{2}\relax (x )\right )+8\right )}{15 \sqrt {a \left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 31, normalized size = 0.58 \[ \frac {1}{240} \, {\left (3 \, a^{2} \sin \left (5 \, x\right ) + 25 \, a^{2} \sin \left (3 \, x\right ) + 150 \, a^{2} \sin \relax (x)\right )} \sqrt {a} \]
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 \[ \int {\left (a\,{\cos \relax (x)}^2\right )}^{5/2} \,d x \]
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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