Optimal. Leaf size=18 \[ 2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x) \]
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Rubi [A] time = 0.0282526, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {12, 2568, 2635, 8} \[ 2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int 16 \cos ^2(x) \sin ^2(x) \, dx &=16 \int \cos ^2(x) \sin ^2(x) \, dx\\ &=-4 \cos ^3(x) \sin (x)+4 \int \cos ^2(x) \, dx\\ &=2 \cos (x) \sin (x)-4 \cos ^3(x) \sin (x)+2 \int 1 \, dx\\ &=2 x+2 \cos (x) \sin (x)-4 \cos ^3(x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0079291, size = 16, normalized size = 0.89 \[ 4 \left (\frac{x}{2}-\frac{1}{8} \sin (4 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 19, normalized size = 1.1 \begin{align*} 2\,x+2\,\cos \left ( x \right ) \sin \left ( x \right ) -4\, \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959415, size = 14, normalized size = 0.78 \begin{align*} 2 \, x - \frac{1}{2} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23492, size = 53, normalized size = 2.94 \begin{align*} -2 \,{\left (2 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) + 2 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.066772, size = 12, normalized size = 0.67 \begin{align*} 2 x - \sin{\left (2 x \right )} \cos{\left (2 x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07518, size = 14, normalized size = 0.78 \begin{align*} 2 \, x - \frac{1}{2} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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