Optimal. Leaf size=33 \[ \frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x) \]
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Rubi [A] time = 0.0256267, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3175, 2635, 8} \[ \frac{3 a^2 x}{8}+\frac{1}{4} a^2 \sin (x) \cos ^3(x)+\frac{3}{8} a^2 \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a-a \sin ^2(x)\right )^2 \, dx &=a^2 \int \cos ^4(x) \, dx\\ &=\frac{1}{4} a^2 \cos ^3(x) \sin (x)+\frac{1}{4} \left (3 a^2\right ) \int \cos ^2(x) \, dx\\ &=\frac{3}{8} a^2 \cos (x) \sin (x)+\frac{1}{4} a^2 \cos ^3(x) \sin (x)+\frac{1}{8} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{8}+\frac{3}{8} a^2 \cos (x) \sin (x)+\frac{1}{4} a^2 \cos ^3(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0027889, size = 26, normalized size = 0.79 \[ a^2 \left (\frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 43, normalized size = 1.3 \begin{align*}{a}^{2} \left ( -{\frac{\cos \left ( x \right ) }{4} \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{3\,\sin \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{8}} \right ) -2\,{a}^{2} \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948838, size = 54, normalized size = 1.64 \begin{align*} \frac{1}{32} \, a^{2}{\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} - \frac{1}{2} \, a^{2}{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64587, size = 76, normalized size = 2.3 \begin{align*} \frac{3}{8} \, a^{2} x + \frac{1}{8} \,{\left (2 \, a^{2} \cos \left (x\right )^{3} + 3 \, a^{2} \cos \left (x\right )\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.06638, size = 110, normalized size = 3.33 \begin{align*} \frac{3 a^{2} x \sin ^{4}{\left (x \right )}}{8} + \frac{3 a^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} - a^{2} x \sin ^{2}{\left (x \right )} + \frac{3 a^{2} x \cos ^{4}{\left (x \right )}}{8} - a^{2} x \cos ^{2}{\left (x \right )} + a^{2} x - \frac{5 a^{2} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{8} - \frac{3 a^{2} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + a^{2} \sin{\left (x \right )} \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12513, size = 34, normalized size = 1.03 \begin{align*} \frac{3}{8} \, a^{2} x + \frac{1}{32} \, a^{2} \sin \left (4 \, x\right ) + \frac{1}{4} \, a^{2} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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