Optimal. Leaf size=132 \[ \frac {21}{128} a^2 \sin (x) \cos (x) \sqrt {a \cos ^4(x)}+\frac {63}{256} a^2 \tan (x) \sqrt {a \cos ^4(x)}+\frac {63}{256} a^2 x \sec ^2(x) \sqrt {a \cos ^4(x)}+\frac {1}{10} a^2 \sin (x) \cos ^7(x) \sqrt {a \cos ^4(x)}+\frac {9}{80} a^2 \sin (x) \cos ^5(x) \sqrt {a \cos ^4(x)}+\frac {21}{160} a^2 \sin (x) \cos ^3(x) \sqrt {a \cos ^4(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 8} \[ \frac {1}{10} a^2 \sin (x) \cos ^7(x) \sqrt {a \cos ^4(x)}+\frac {9}{80} a^2 \sin (x) \cos ^5(x) \sqrt {a \cos ^4(x)}+\frac {21}{160} a^2 \sin (x) \cos ^3(x) \sqrt {a \cos ^4(x)}+\frac {21}{128} a^2 \sin (x) \cos (x) \sqrt {a \cos ^4(x)}+\frac {63}{256} a^2 \tan (x) \sqrt {a \cos ^4(x)}+\frac {63}{256} a^2 x \sec ^2(x) \sqrt {a \cos ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3207
Rubi steps
\begin {align*} \int \left (a \cos ^4(x)\right )^{5/2} \, dx &=\left (a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^{10}(x) \, dx\\ &=\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} \left (9 a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^8(x) \, dx\\ &=\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{80} \left (63 a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^6(x) \, dx\\ &=\frac {21}{160} a^2 \cos ^3(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{32} \left (21 a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^4(x) \, dx\\ &=\frac {21}{128} a^2 \cos (x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {21}{160} a^2 \cos ^3(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{128} \left (63 a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int \cos ^2(x) \, dx\\ &=\frac {21}{128} a^2 \cos (x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {21}{160} a^2 \cos ^3(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {63}{256} a^2 \sqrt {a \cos ^4(x)} \tan (x)+\frac {1}{256} \left (63 a^2 \sqrt {a \cos ^4(x)} \sec ^2(x)\right ) \int 1 \, dx\\ &=\frac {63}{256} a^2 x \sqrt {a \cos ^4(x)} \sec ^2(x)+\frac {21}{128} a^2 \cos (x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {21}{160} a^2 \cos ^3(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {9}{80} a^2 \cos ^5(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {1}{10} a^2 \cos ^7(x) \sqrt {a \cos ^4(x)} \sin (x)+\frac {63}{256} a^2 \sqrt {a \cos ^4(x)} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 53, normalized size = 0.40 \[ \frac {a (2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x)) \sec ^6(x) \left (a \cos ^4(x)\right )^{3/2}}{10240} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 68, normalized size = 0.52 \[ \frac {\sqrt {a \cos \relax (x)^{4}} {\left (315 \, a^{2} x + {\left (128 \, a^{2} \cos \relax (x)^{9} + 144 \, a^{2} \cos \relax (x)^{7} + 168 \, a^{2} \cos \relax (x)^{5} + 210 \, a^{2} \cos \relax (x)^{3} + 315 \, a^{2} \cos \relax (x)\right )} \sin \relax (x)\right )}}{1280 \, \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 57, normalized size = 0.43 \[ \frac {1}{10240} \, {\left (2520 \, a^{2} x + 2 \, a^{2} \sin \left (10 \, x\right ) + 25 \, a^{2} \sin \left (8 \, x\right ) + 150 \, a^{2} \sin \left (6 \, x\right ) + 600 \, a^{2} \sin \left (4 \, x\right ) + 2100 \, a^{2} \sin \left (2 \, x\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 57, normalized size = 0.43 \[ \frac {\left (a \left (\cos ^{4}\relax (x )\right )\right )^{\frac {5}{2}} \left (128 \sin \relax (x ) \left (\cos ^{9}\relax (x )\right )+144 \sin \relax (x ) \left (\cos ^{7}\relax (x )\right )+168 \sin \relax (x ) \left (\cos ^{5}\relax (x )\right )+210 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )+315 \cos \relax (x ) \sin \relax (x )+315 x \right )}{1280 \cos \relax (x )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 85, normalized size = 0.64 \[ \frac {63}{256} \, a^{\frac {5}{2}} x + \frac {315 \, a^{\frac {5}{2}} \tan \relax (x)^{9} + 1470 \, a^{\frac {5}{2}} \tan \relax (x)^{7} + 2688 \, a^{\frac {5}{2}} \tan \relax (x)^{5} + 2370 \, a^{\frac {5}{2}} \tan \relax (x)^{3} + 965 \, a^{\frac {5}{2}} \tan \relax (x)}{1280 \, {\left (\tan \relax (x)^{10} + 5 \, \tan \relax (x)^{8} + 10 \, \tan \relax (x)^{6} + 10 \, \tan \relax (x)^{4} + 5 \, \tan \relax (x)^{2} + 1\right )}} \]
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 \[ \int {\left (a\,{\cos \relax (x)}^4\right )}^{5/2} \,d x \]
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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