Optimal. Leaf size=40 \[ -\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\sin (2 x)-2 x \cos (2 x)+\frac{7}{4} \cos (2 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0667267, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6742, 2638, 3296, 2637} \[ -\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\sin (2 x)-2 x \cos (2 x)+\frac{7}{4} \cos (2 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx &=\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*}
Mathematica [A] time = 0.0374874, size = 29, normalized size = 0.72 \[ \frac{1}{4} \left (\left (-2 x^2-8 x+7\right ) \cos (2 x)+2 (x+2) \sin (2 x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 35, normalized size = 0.9 \begin{align*}{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.95162, size = 51, normalized size = 1.27 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.26788, size = 76, normalized size = 1.9 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.31514, size = 39, normalized size = 0.98 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.04826, size = 35, normalized size = 0.88 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]