Integrand size = 13, antiderivative size = 40 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=\frac {7}{4} \cos (2 x)-2 x \cos (2 x)-\frac {1}{2} x^2 \cos (2 x)+\sin (2 x)+\frac {1}{2} x \sin (2 x) \]
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Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6874, 2718, 3377, 2717} \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=-\frac {1}{2} x^2 \cos (2 x)+\frac {1}{2} x \sin (2 x)+\sin (2 x)-2 x \cos (2 x)+\frac {7}{4} \cos (2 x) \]
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Rule 2717
Rule 2718
Rule 3377
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-3 \sin (2 x)+4 x \sin (2 x)+x^2 \sin (2 x)\right ) \, dx \\ & = -(3 \int \sin (2 x) \, dx)+4 \int x \sin (2 x) \, dx+\int x^2 \sin (2 x) \, dx \\ & = \frac {3}{2} \cos (2 x)-2 x \cos (2 x)-\frac {1}{2} x^2 \cos (2 x)+2 \int \cos (2 x) \, dx+\int x \cos (2 x) \, dx \\ & = \frac {3}{2} \cos (2 x)-2 x \cos (2 x)-\frac {1}{2} x^2 \cos (2 x)+\sin (2 x)+\frac {1}{2} x \sin (2 x)-\frac {1}{2} \int \sin (2 x) \, dx \\ & = \frac {7}{4} \cos (2 x)-2 x \cos (2 x)-\frac {1}{2} x^2 \cos (2 x)+\sin (2 x)+\frac {1}{2} x \sin (2 x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=\frac {1}{4} \left (\left (7-8 x-2 x^2\right ) \cos (2 x)+2 (2+x) \sin (2 x)\right ) \]
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Time = 0.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\left (-\frac {1}{2} x^{2}-2 x +\frac {7}{4}\right ) \cos \left (2 x \right )+\frac {\left (2+x \right ) \sin \left (2 x \right )}{2}\) | \(26\) |
derivativedivides | \(\frac {7 \cos \left (2 x \right )}{4}-2 x \cos \left (2 x \right )-\frac {x^{2} \cos \left (2 x \right )}{2}+\sin \left (2 x \right )+\frac {x \sin \left (2 x \right )}{2}\) | \(35\) |
default | \(\frac {7 \cos \left (2 x \right )}{4}-2 x \cos \left (2 x \right )-\frac {x^{2} \cos \left (2 x \right )}{2}+\sin \left (2 x \right )+\frac {x \sin \left (2 x \right )}{2}\) | \(35\) |
parts | \(\frac {7 \cos \left (2 x \right )}{4}-2 x \cos \left (2 x \right )-\frac {x^{2} \cos \left (2 x \right )}{2}+\sin \left (2 x \right )+\frac {x \sin \left (2 x \right )}{2}\) | \(35\) |
parallelrisch | \(\frac {\left (x^{2}+4 x \right ) \tan \left (x \right )^{2}+\left (2 x +4\right ) \tan \left (x \right )-x^{2}-4 x +7}{2+2 \tan \left (x \right )^{2}}\) | \(42\) |
norman | \(\frac {x \tan \left (x \right )-2 x -\frac {x^{2}}{2}+2 x \tan \left (x \right )^{2}+\frac {x^{2} \tan \left (x \right )^{2}}{2}+2 \tan \left (x \right )+\frac {7}{2}}{1+\tan \left (x \right )^{2}}\) | \(44\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-2 x^{2}+1\right ) \cos \left (2 x \right )}{2 \sqrt {\pi }}+\frac {x \sin \left (2 x \right )}{\sqrt {\pi }}\right )}{2}+2 \sqrt {\pi }\, \left (-\frac {x \cos \left (2 x \right )}{\sqrt {\pi }}+\frac {\sin \left (2 x \right )}{2 \sqrt {\pi }}\right )-\frac {3 \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{2}\) | \(81\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} + 8 \, x - 7\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, {\left (x + 2\right )} \sin \left (2 \, x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=- \frac {x^{2} \cos {\left (2 x \right )}}{2} + \frac {x \sin {\left (2 x \right )}}{2} - 2 x \cos {\left (2 x \right )} + \sin {\left (2 x \right )} + \frac {7 \cos {\left (2 x \right )}}{4} \]
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Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) - 2 \, x \cos \left (2 \, x\right ) + \frac {1}{2} \, x \sin \left (2 \, x\right ) + \frac {3}{2} \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=-\frac {1}{4} \, {\left (2 \, x^{2} + 8 \, x - 7\right )} \cos \left (2 \, x\right ) + \frac {1}{2} \, {\left (x + 2\right )} \sin \left (2 \, x\right ) \]
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Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx=\frac {7\,\cos \left (2\,x\right )}{4}+\sin \left (2\,x\right )-2\,x\,\cos \left (2\,x\right )+\frac {x\,\sin \left (2\,x\right )}{2}-\frac {x^2\,\cos \left (2\,x\right )}{2} \]
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