Optimal. Leaf size=18 \[ 2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x) \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 8
Rule 12
Rule 2568
Rule 2635
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.56, size = 19, normalized size = 1.06 \[ -2 \, {\left (2 \, \cos \relax (x)^{3} - \cos \relax (x)\right )} \sin \relax (x) + 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 10, normalized size = 0.56 \[ 2 \, x - \frac {1}{2} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 1.06 \[ 2 x +2 \cos \relax (x ) \sin \relax (x )-4 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 10, normalized size = 0.56 \[ 2 \, x - \frac {1}{2} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 18, normalized size = 1.00 \[ 4\,\cos \relax (x)\,{\sin \relax (x)}^3-2\,\cos \relax (x)\,\sin \relax (x)+2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 12, normalized size = 0.67 \[ 2 x - \sin {\left (2 x \right )} \cos {\left (2 x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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