3.566 \(\int e^{-3 x} x^2 \sin (x) \, dx\)

Optimal. Leaf size=75 \[ -\frac{3}{10} e^{-3 x} x^2 \sin (x)-\frac{1}{10} e^{-3 x} x^2 \cos (x)-\frac{4}{25} e^{-3 x} x \sin (x)-\frac{9}{250} e^{-3 x} \sin (x)-\frac{3}{25} e^{-3 x} x \cos (x)-\frac{13}{250} e^{-3 x} \cos (x) \]

[Out]

(-13*Cos[x])/(250*E^(3*x)) - (3*x*Cos[x])/(25*E^(3*x)) - (x^2*Cos[x])/(10*E^(3*x
)) - (9*Sin[x])/(250*E^(3*x)) - (4*x*Sin[x])/(25*E^(3*x)) - (3*x^2*Sin[x])/(10*E
^(3*x))

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Rubi [A]  time = 0.2228, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{3}{10} e^{-3 x} x^2 \sin (x)-\frac{1}{10} e^{-3 x} x^2 \cos (x)-\frac{4}{25} e^{-3 x} x \sin (x)-\frac{9}{250} e^{-3 x} \sin (x)-\frac{3}{25} e^{-3 x} x \cos (x)-\frac{13}{250} e^{-3 x} \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[(x^2*Sin[x])/E^(3*x),x]

[Out]

(-13*Cos[x])/(250*E^(3*x)) - (3*x*Cos[x])/(25*E^(3*x)) - (x^2*Cos[x])/(10*E^(3*x
)) - (9*Sin[x])/(250*E^(3*x)) - (4*x*Sin[x])/(25*E^(3*x)) - (3*x^2*Sin[x])/(10*E
^(3*x))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 x^{2} e^{- 3 x} \sin{\left (x \right )}}{10} - \frac{x^{2} e^{- 3 x} \cos{\left (x \right )}}{10} - 2 \int x \left (- \frac{3 e^{- 3 x} \sin{\left (x \right )}}{10} - \frac{e^{- 3 x} \cos{\left (x \right )}}{10}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*sin(x)/exp(3*x),x)

[Out]

-3*x**2*exp(-3*x)*sin(x)/10 - x**2*exp(-3*x)*cos(x)/10 - 2*Integral(x*(-3*exp(-3
*x)*sin(x)/10 - exp(-3*x)*cos(x)/10), x)

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Mathematica [A]  time = 0.0406455, size = 38, normalized size = 0.51 \[ \frac{1}{250} e^{-3 x} \left (-\left (75 x^2+40 x+9\right ) \sin (x)-\left (25 x^2+30 x+13\right ) \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*Sin[x])/E^(3*x),x]

[Out]

(-((13 + 30*x + 25*x^2)*Cos[x]) - (9 + 40*x + 75*x^2)*Sin[x])/(250*E^(3*x))

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Maple [A]  time = 0.007, size = 36, normalized size = 0.5 \[ \left ( -{\frac{{x}^{2}}{10}}-{\frac{3\,x}{25}}-{\frac{13}{250}} \right ){{\rm e}^{-3\,x}}\cos \left ( x \right ) + \left ( -{\frac{3\,{x}^{2}}{10}}-{\frac{4\,x}{25}}-{\frac{9}{250}} \right ){{\rm e}^{-3\,x}}\sin \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*sin(x)/exp(3*x),x)

[Out]

(-1/10*x^2-3/25*x-13/250)*exp(-3*x)*cos(x)+(-3/10*x^2-4/25*x-9/250)*exp(-3*x)*si
n(x)

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Maxima [A]  time = 1.38843, size = 45, normalized size = 0.6 \[ -\frac{1}{250} \,{\left ({\left (25 \, x^{2} + 30 \, x + 13\right )} \cos \left (x\right ) +{\left (75 \, x^{2} + 40 \, x + 9\right )} \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/250*((25*x^2 + 30*x + 13)*cos(x) + (75*x^2 + 40*x + 9)*sin(x))*e^(-3*x)

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Fricas [A]  time = 0.214124, size = 50, normalized size = 0.67 \[ -\frac{1}{250} \,{\left (25 \, x^{2} + 30 \, x + 13\right )} \cos \left (x\right ) e^{\left (-3 \, x\right )} - \frac{1}{250} \,{\left (75 \, x^{2} + 40 \, x + 9\right )} e^{\left (-3 \, x\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/250*(25*x^2 + 30*x + 13)*cos(x)*e^(-3*x) - 1/250*(75*x^2 + 40*x + 9)*e^(-3*x)
*sin(x)

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Sympy [A]  time = 3.66555, size = 80, normalized size = 1.07 \[ - \frac{3 x^{2} e^{- 3 x} \sin{\left (x \right )}}{10} - \frac{x^{2} e^{- 3 x} \cos{\left (x \right )}}{10} - \frac{4 x e^{- 3 x} \sin{\left (x \right )}}{25} - \frac{3 x e^{- 3 x} \cos{\left (x \right )}}{25} - \frac{9 e^{- 3 x} \sin{\left (x \right )}}{250} - \frac{13 e^{- 3 x} \cos{\left (x \right )}}{250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*sin(x)/exp(3*x),x)

[Out]

-3*x**2*exp(-3*x)*sin(x)/10 - x**2*exp(-3*x)*cos(x)/10 - 4*x*exp(-3*x)*sin(x)/25
- 3*x*exp(-3*x)*cos(x)/25 - 9*exp(-3*x)*sin(x)/250 - 13*exp(-3*x)*cos(x)/250

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GIAC/XCAS [A]  time = 0.204716, size = 45, normalized size = 0.6 \[ -\frac{1}{250} \,{\left ({\left (25 \, x^{2} + 30 \, x + 13\right )} \cos \left (x\right ) +{\left (75 \, x^{2} + 40 \, x + 9\right )} \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/250*((25*x^2 + 30*x + 13)*cos(x) + (75*x^2 + 40*x + 9)*sin(x))*e^(-3*x)