3.565 \(\int e^x x^2 \sin (x) \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{2} e^x x^2 \sin (x)-\frac{1}{2} e^x x^2 \cos (x)-\frac{1}{2} e^x \sin (x)+e^x x \cos (x)-\frac{1}{2} e^x \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.107696, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4432, 4465, 14, 4433, 4466} \[ \frac{1}{2} e^x x^2 \sin (x)-\frac{1}{2} e^x x^2 \cos (x)-\frac{1}{2} e^x \sin (x)+e^x x \cos (x)-\frac{1}{2} e^x \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[E^x*x^2*Sin[x],x]

[Out]

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

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4465

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.)*Sin[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Sin[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4466

Int[Cos[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rubi steps

\begin{align*} \int e^x x^2 \sin (x) \, dx &=-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)-2 \int x \left (-\frac{1}{2} e^x \cos (x)+\frac{1}{2} e^x \sin (x)\right ) \, dx\\ &=-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)-2 \int \left (-\frac{1}{2} e^x x \cos (x)+\frac{1}{2} e^x x \sin (x)\right ) \, dx\\ &=-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)+\int e^x x \cos (x) \, dx-\int e^x x \sin (x) \, dx\\ &=e^x x \cos (x)-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)+\int \left (-\frac{1}{2} e^x \cos (x)+\frac{1}{2} e^x \sin (x)\right ) \, dx-\int \left (\frac{1}{2} e^x \cos (x)+\frac{1}{2} e^x \sin (x)\right ) \, dx\\ &=e^x x \cos (x)-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)-2 \left (\frac{1}{2} \int e^x \cos (x) \, dx\right )\\ &=e^x x \cos (x)-\frac{1}{2} e^x x^2 \cos (x)+\frac{1}{2} e^x x^2 \sin (x)-2 \left (\frac{1}{4} e^x \cos (x)+\frac{1}{4} e^x \sin (x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0357919, size = 25, normalized size = 0.5 \[ \frac{1}{2} e^x \left (\left (x^2-1\right ) \sin (x)-(x-1)^2 \cos (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x*x^2*Sin[x],x]

[Out]

(E^x*(-((-1 + x)^2*Cos[x]) + (-1 + x^2)*Sin[x]))/2

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Maple [A]  time = 0.003, size = 27, normalized size = 0.5 \begin{align*} \left ( -{\frac{{x}^{2}}{2}}+x-{\frac{1}{2}} \right ){{\rm e}^{x}}\cos \left ( x \right ) + \left ({\frac{{x}^{2}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{x}}\sin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*x^2*sin(x),x)

[Out]

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

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Maxima [A]  time = 0.979202, size = 35, normalized size = 0.7 \begin{align*} -\frac{1}{2} \,{\left (x^{2} - 2 \, x + 1\right )} \cos \left (x\right ) e^{x} + \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{x} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^2*sin(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.74131, size = 81, normalized size = 1.62 \begin{align*} -\frac{1}{2} \,{\left (x^{2} - 2 \, x + 1\right )} \cos \left (x\right ) e^{x} + \frac{1}{2} \,{\left (x^{2} - 1\right )} e^{x} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^2*sin(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.10396, size = 48, normalized size = 0.96 \begin{align*} \frac{x^{2} e^{x} \sin{\left (x \right )}}{2} - \frac{x^{2} e^{x} \cos{\left (x \right )}}{2} + x e^{x} \cos{\left (x \right )} - \frac{e^{x} \sin{\left (x \right )}}{2} - \frac{e^{x} \cos{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x**2*sin(x),x)

[Out]

x**2*exp(x)*sin(x)/2 - x**2*exp(x)*cos(x)/2 + x*exp(x)*cos(x) - exp(x)*sin(x)/2 - exp(x)*cos(x)/2

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Giac [A]  time = 1.09025, size = 34, normalized size = 0.68 \begin{align*} -\frac{1}{2} \,{\left ({\left (x^{2} - 2 \, x + 1\right )} \cos \left (x\right ) -{\left (x^{2} - 1\right )} \sin \left (x\right )\right )} e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*x^2*sin(x),x, algorithm="giac")

[Out]

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