3.310 \(\int \left (-3+4 x+x^2\right ) \sin (2 x) \, dx\)

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]

(7*Cos[2*x])/4 - 2*x*Cos[2*x] - (x^2*Cos[2*x])/2 + Sin[2*x] + (x*Sin[2*x])/2

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Rubi [A]  time = 0.100655, 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 \[ -\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\sin (2 x)-2 x \cos (2 x)+\frac{7}{4} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Int[(-3 + 4*x + x^2)*Sin[2*x],x]

[Out]

(7*Cos[2*x])/4 - 2*x*Cos[2*x] - (x^2*Cos[2*x])/2 + Sin[2*x] + (x*Sin[2*x])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x^{2} + 4 x - 3\right ) \sin{\left (2 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+4*x-3)*sin(2*x),x)

[Out]

Integral((x**2 + 4*x - 3)*sin(2*x), x)

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Mathematica [A]  time = 0.0384604, 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]  Integrate[(-3 + 4*x + x^2)*Sin[2*x],x]

[Out]

((7 - 8*x - 2*x^2)*Cos[2*x] + 2*(2 + x)*Sin[2*x])/4

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Maple [A]  time = 0.012, size = 35, normalized size = 0.9 \[{\frac{7\,\cos \left ( 2\,x \right ) }{4}}-2\,x\cos \left ( 2\,x \right ) -{\frac{{x}^{2}\cos \left ( 2\,x \right ) }{2}}+\sin \left ( 2\,x \right ) +{\frac{x\sin \left ( 2\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+4*x-3)*sin(2*x),x)

[Out]

7/4*cos(2*x)-2*x*cos(2*x)-1/2*x^2*cos(2*x)+sin(2*x)+1/2*x*sin(2*x)

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Maxima [A]  time = 1.34438, size = 51, normalized size = 1.27 \[ -\frac{1}{4} \,{\left (2 \, x^{2} - 1\right )} \cos \left (2 \, x\right ) - 2 \, x \cos \left (2 \, x\right ) + \frac{1}{2} \, x \sin \left (2 \, x\right ) + \frac{3}{2} \, \cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 4*x - 3)*sin(2*x),x, algorithm="maxima")

[Out]

-1/4*(2*x^2 - 1)*cos(2*x) - 2*x*cos(2*x) + 1/2*x*sin(2*x) + 3/2*cos(2*x) + sin(2
*x)

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Fricas [A]  time = 0.237115, size = 35, normalized size = 0.88 \[ -\frac{1}{4} \,{\left (2 \, x^{2} + 8 \, x - 7\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \,{\left (x + 2\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 4*x - 3)*sin(2*x),x, algorithm="fricas")

[Out]

-1/4*(2*x^2 + 8*x - 7)*cos(2*x) + 1/2*(x + 2)*sin(2*x)

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Sympy [A]  time = 0.434492, size = 39, normalized size = 0.98 \[ - \frac{x^{2} \cos{\left (2 x \right )}}{2} + \frac{x \sin{\left (2 x \right )}}{2} - 2 x \cos{\left (2 x \right )} + \sin{\left (2 x \right )} + \frac{7 \cos{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+4*x-3)*sin(2*x),x)

[Out]

-x**2*cos(2*x)/2 + x*sin(2*x)/2 - 2*x*cos(2*x) + sin(2*x) + 7*cos(2*x)/4

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GIAC/XCAS [A]  time = 0.215175, size = 35, normalized size = 0.88 \[ -\frac{1}{4} \,{\left (2 \, x^{2} + 8 \, x - 7\right )} \cos \left (2 \, x\right ) + \frac{1}{2} \,{\left (x + 2\right )} \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 4*x - 3)*sin(2*x),x, algorithm="giac")

[Out]

-1/4*(2*x^2 + 8*x - 7)*cos(2*x) + 1/2*(x + 2)*sin(2*x)