Optimal. Leaf size=28 \[ \frac {1}{4} x \sin (3)-\frac {1}{4} \cos (2 x+3)-\frac {1}{16} \cos (4 x+3) \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4574, 2638} \[ \frac {1}{4} x \sin (3)-\frac {1}{4} \cos (2 x+3)-\frac {1}{16} \cos (4 x+3) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 4574
Rubi steps
\begin {align*} \int \cos ^2(x) \sin (3+2 x) \, dx &=\int \left (\frac {\sin (3)}{4}+\frac {1}{2} \sin (3+2 x)+\frac {1}{4} \sin (3+4 x)\right ) \, dx\\ &=\frac {1}{4} x \sin (3)+\frac {1}{4} \int \sin (3+4 x) \, dx+\frac {1}{2} \int \sin (3+2 x) \, dx\\ &=-\frac {1}{4} \cos (3+2 x)-\frac {1}{16} \cos (3+4 x)+\frac {1}{4} x \sin (3)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.00 \[ \frac {1}{4} x \sin (3)-\frac {1}{4} \cos (2 x+3)-\frac {1}{16} \cos (4 x+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 32, normalized size = 1.14 \[ -\frac {1}{2} \, \cos \relax (3) \cos \relax (x)^{4} + \frac {1}{4} \, x \sin \relax (3) + \frac {1}{4} \, {\left (2 \, \cos \relax (x)^{3} \sin \relax (3) + \cos \relax (x) \sin \relax (3)\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 22, normalized size = 0.79 \[ \frac {1}{4} \, x \sin \relax (3) - \frac {1}{16} \, \cos \left (4 \, x + 3\right ) - \frac {1}{4} \, \cos \left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 23, normalized size = 0.82 \[ \frac {\sin \relax (3) x}{4}-\frac {\cos \left (2 x +3\right )}{4}-\frac {\cos \left (4 x +3\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 22, normalized size = 0.79 \[ \frac {1}{4} \, x \sin \relax (3) - \frac {1}{16} \, \cos \left (4 \, x + 3\right ) - \frac {1}{4} \, \cos \left (2 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 22, normalized size = 0.79 \[ \frac {x\,\sin \relax (3)}{4}-\frac {\cos \left (4\,x+3\right )}{16}-\frac {\cos \left (2\,x+3\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.20, size = 75, normalized size = 2.68 \[ - \frac {x \sin ^{2}{\relax (x )} \sin {\left (2 x + 3 \right )}}{4} - \frac {x \sin {\relax (x )} \cos {\relax (x )} \cos {\left (2 x + 3 \right )}}{2} + \frac {x \sin {\left (2 x + 3 \right )} \cos ^{2}{\relax (x )}}{4} - \frac {\sin {\relax (x )} \sin {\left (2 x + 3 \right )} \cos {\relax (x )}}{4} - \frac {\cos ^{2}{\relax (x )} \cos {\left (2 x + 3 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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