Optimal. Leaf size=132 \[ \frac{1}{10} a^2 \sin (x) \cos ^7(x) \sqrt{a \cos ^4(x)}+\frac{9}{80} a^2 \sin (x) \cos ^5(x) \sqrt{a \cos ^4(x)}+\frac{21}{160} a^2 \sin (x) \cos ^3(x) \sqrt{a \cos ^4(x)}+\frac{21}{128} a^2 \sin (x) \cos (x) \sqrt{a \cos ^4(x)}+\frac{63}{256} a^2 \tan (x) \sqrt{a \cos ^4(x)}+\frac{63}{256} a^2 x \sec ^2(x) \sqrt{a \cos ^4(x)} \]
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Rubi [A] time = 0.0510577, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3207
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.123115, size = 53, normalized size = 0.4 \[ \frac{a (2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x)) \sec ^6(x) \left (a \cos ^4(x)\right )^{3/2}}{10240} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.295, size = 57, normalized size = 0.4 \begin{align*}{\frac{128\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{9}+144\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{7}+168\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{5}+210\, \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) +315\,\cos \left ( x \right ) \sin \left ( x \right ) +315\,x}{1280\, \left ( \cos \left ( x \right ) \right ) ^{10}} \left ( a \left ( \cos \left ( x \right ) \right ) ^{4} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.47314, size = 115, normalized size = 0.87 \begin{align*} \frac{63}{256} \, a^{\frac{5}{2}} x + \frac{315 \, a^{\frac{5}{2}} \tan \left (x\right )^{9} + 1470 \, a^{\frac{5}{2}} \tan \left (x\right )^{7} + 2688 \, a^{\frac{5}{2}} \tan \left (x\right )^{5} + 2370 \, a^{\frac{5}{2}} \tan \left (x\right )^{3} + 965 \, a^{\frac{5}{2}} \tan \left (x\right )}{1280 \,{\left (\tan \left (x\right )^{10} + 5 \, \tan \left (x\right )^{8} + 10 \, \tan \left (x\right )^{6} + 10 \, \tan \left (x\right )^{4} + 5 \, \tan \left (x\right )^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16141, size = 200, normalized size = 1.52 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{4}}{\left (315 \, a^{2} x +{\left (128 \, a^{2} \cos \left (x\right )^{9} + 144 \, a^{2} \cos \left (x\right )^{7} + 168 \, a^{2} \cos \left (x\right )^{5} + 210 \, a^{2} \cos \left (x\right )^{3} + 315 \, a^{2} \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{1280 \, \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39909, size = 77, normalized size = 0.58 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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