Optimal. Leaf size=57 \[ \frac{1}{4} x^2 \sin (2 x)+\frac{1}{16} x^2 \sin (8 x)-\frac{1}{8} \sin (2 x)-\frac{1}{512} \sin (8 x)+\frac{1}{4} x \cos (2 x)+\frac{1}{64} x \cos (8 x) \]
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Rubi [A] time = 0.0734621, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4429, 3296, 2637} \[ \frac{1}{4} x^2 \sin (2 x)+\frac{1}{16} x^2 \sin (8 x)-\frac{1}{8} \sin (2 x)-\frac{1}{512} \sin (8 x)+\frac{1}{4} x \cos (2 x)+\frac{1}{64} x \cos (8 x) \]
Antiderivative was successfully verified.
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Rule 4429
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \cos (3 x) \cos (5 x) \, dx &=\int \left (\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x^2 \cos (8 x)\right ) \, dx\\ &=\frac{1}{2} \int x^2 \cos (2 x) \, dx+\frac{1}{2} \int x^2 \cos (8 x) \, dx\\ &=\frac{1}{4} x^2 \sin (2 x)+\frac{1}{16} x^2 \sin (8 x)-\frac{1}{8} \int x \sin (8 x) \, dx-\frac{1}{2} \int x \sin (2 x) \, dx\\ &=\frac{1}{4} x \cos (2 x)+\frac{1}{64} x \cos (8 x)+\frac{1}{4} x^2 \sin (2 x)+\frac{1}{16} x^2 \sin (8 x)-\frac{1}{64} \int \cos (8 x) \, dx-\frac{1}{4} \int \cos (2 x) \, dx\\ &=\frac{1}{4} x \cos (2 x)+\frac{1}{64} x \cos (8 x)-\frac{1}{8} \sin (2 x)+\frac{1}{4} x^2 \sin (2 x)-\frac{1}{512} \sin (8 x)+\frac{1}{16} x^2 \sin (8 x)\\ \end{align*}
Mathematica [A] time = 0.0914518, size = 49, normalized size = 0.86 \[ \frac{1}{512} \left (128 x^2 \sin (2 x)+32 x^2 \sin (8 x)-64 \sin (2 x)-\sin (8 x)+128 x \cos (2 x)+8 x \cos (8 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 46, normalized size = 0.8 \begin{align*}{\frac{x\cos \left ( 2\,x \right ) }{4}}+{\frac{x\cos \left ( 8\,x \right ) }{64}}-{\frac{\sin \left ( 2\,x \right ) }{8}}+{\frac{{x}^{2}\sin \left ( 2\,x \right ) }{4}}-{\frac{\sin \left ( 8\,x \right ) }{512}}+{\frac{{x}^{2}\sin \left ( 8\,x \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985726, size = 55, normalized size = 0.96 \begin{align*} \frac{1}{64} \, x \cos \left (8 \, x\right ) + \frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{512} \,{\left (32 \, x^{2} - 1\right )} \sin \left (8 \, x\right ) + \frac{1}{8} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25457, size = 220, normalized size = 3.86 \begin{align*} 2 \, x \cos \left (x\right )^{8} - 4 \, x \cos \left (x\right )^{6} + \frac{5}{2} \, x \cos \left (x\right )^{4} + \frac{1}{64} \,{\left (16 \,{\left (32 \, x^{2} - 1\right )} \cos \left (x\right )^{7} - 24 \,{\left (32 \, x^{2} - 1\right )} \cos \left (x\right )^{5} + 10 \,{\left (32 \, x^{2} - 1\right )} \cos \left (x\right )^{3} - 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) - \frac{15}{64} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.72247, size = 90, normalized size = 1.58 \begin{align*} - \frac{3 x^{2} \sin{\left (3 x \right )} \cos{\left (5 x \right )}}{16} + \frac{5 x^{2} \sin{\left (5 x \right )} \cos{\left (3 x \right )}}{16} + \frac{15 x \sin{\left (3 x \right )} \sin{\left (5 x \right )}}{64} + \frac{17 x \cos{\left (3 x \right )} \cos{\left (5 x \right )}}{64} + \frac{63 \sin{\left (3 x \right )} \cos{\left (5 x \right )}}{512} - \frac{65 \sin{\left (5 x \right )} \cos{\left (3 x \right )}}{512} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09981, size = 55, normalized size = 0.96 \begin{align*} \frac{1}{64} \, x \cos \left (8 \, x\right ) + \frac{1}{4} \, x \cos \left (2 \, x\right ) + \frac{1}{512} \,{\left (32 \, x^{2} - 1\right )} \sin \left (8 \, x\right ) + \frac{1}{8} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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