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 = 0.0387325, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3203
Rule 3207
Rule 2637
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*}
Mathematica [A] time = 0.019473, 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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Maple [A] time = 0.628, size = 32, normalized size = 0.6 \begin{align*}{\frac{{a}^{3}\cos \left ( x \right ) \sin \left ( x \right ) \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{4}+4\, \left ( \cos \left ( x \right ) \right ) ^{2}+8 \right ) }{15}{\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.34722, size = 42, normalized size = 0.79 \begin{align*} \frac{1}{240} \,{\left (3 \, a^{2} \sin \left (5 \, x\right ) + 25 \, a^{2} \sin \left (3 \, x\right ) + 150 \, a^{2} \sin \left (x\right )\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12342, size = 107, normalized size = 2.02 \begin{align*} \frac{{\left (3 \, a^{2} \cos \left (x\right )^{4} + 4 \, a^{2} \cos \left (x\right )^{2} + 8 \, a^{2}\right )} \sqrt{a \cos \left (x\right )^{2}} \sin \left (x\right )}{15 \, \cos \left (x\right )} \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 = 1.40348, size = 46, normalized size = 0.87 \begin{align*} \frac{1}{15} \,{\left (3 \, a^{2} \sin \left (x\right )^{5} - 10 \, a^{2} \sin \left (x\right )^{3} + 15 \, a^{2} \sin \left (x\right )\right )} \sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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