3.569 \(\int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx\)

Optimal. Leaf size=185 \[ \frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)+\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{8}{125} e^{x/2} \sin (x)+\frac{792 e^{x/2} \sin (3 x)}{50653}+\frac{6}{25} e^{x/2} x \cos (x)-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{44}{125} e^{x/2} \cos (x)+\frac{428 e^{x/2} \cos (3 x)}{50653} \]

[Out]

(-44*E^(x/2)*Cos[x])/125 + (6*E^(x/2)*x*Cos[x])/25 + (E^(x/2)*x^2*Cos[x])/10 + (428*E^(x/2)*Cos[3*x])/50653 -
(70*E^(x/2)*x*Cos[3*x])/1369 - (E^(x/2)*x^2*Cos[3*x])/74 - (8*E^(x/2)*Sin[x])/125 - (8*E^(x/2)*x*Sin[x])/25 +
(E^(x/2)*x^2*Sin[x])/5 + (792*E^(x/2)*Sin[3*x])/50653 + (24*E^(x/2)*x*Sin[3*x])/1369 - (3*E^(x/2)*x^2*Sin[3*x]
)/37

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Rubi [A]  time = 0.356214, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {4470, 4433, 4466, 14, 4432, 4465} \[ \frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)+\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{8}{125} e^{x/2} \sin (x)+\frac{792 e^{x/2} \sin (3 x)}{50653}+\frac{6}{25} e^{x/2} x \cos (x)-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{44}{125} e^{x/2} \cos (x)+\frac{428 e^{x/2} \cos (3 x)}{50653} \]

Antiderivative was successfully verified.

[In]

Int[E^(x/2)*x^2*Cos[x]*Sin[x]^2,x]

[Out]

(-44*E^(x/2)*Cos[x])/125 + (6*E^(x/2)*x*Cos[x])/25 + (E^(x/2)*x^2*Cos[x])/10 + (428*E^(x/2)*Cos[3*x])/50653 -
(70*E^(x/2)*x*Cos[3*x])/1369 - (E^(x/2)*x^2*Cos[3*x])/74 - (8*E^(x/2)*Sin[x])/125 - (8*E^(x/2)*x*Sin[x])/25 +
(E^(x/2)*x^2*Sin[x])/5 + (792*E^(x/2)*Sin[3*x])/50653 + (24*E^(x/2)*x*Sin[3*x])/1369 - (3*E^(x/2)*x^2*Sin[3*x]
)/37

Rule 4470

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(x_)^(p_.)*Sin[(d_.) + (e_.)*(x_)]^(m_.),
x_Symbol] :> Int[ExpandTrigReduce[x^p*F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b
, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

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]

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 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]

Rubi steps

\begin{align*} \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx &=\int \left (\frac{1}{4} e^{x/2} x^2 \cos (x)-\frac{1}{4} e^{x/2} x^2 \cos (3 x)\right ) \, dx\\ &=\frac{1}{4} \int e^{x/2} x^2 \cos (x) \, dx-\frac{1}{4} \int e^{x/2} x^2 \cos (3 x) \, dx\\ &=\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)+\frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)-\frac{1}{2} \int x \left (\frac{2}{5} e^{x/2} \cos (x)+\frac{4}{5} e^{x/2} \sin (x)\right ) \, dx+\frac{1}{2} \int x \left (\frac{2}{37} e^{x/2} \cos (3 x)+\frac{12}{37} e^{x/2} \sin (3 x)\right ) \, dx\\ &=\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)+\frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)-\frac{1}{2} \int \left (\frac{2}{5} e^{x/2} x \cos (x)+\frac{4}{5} e^{x/2} x \sin (x)\right ) \, dx+\frac{1}{2} \int \left (\frac{2}{37} e^{x/2} x \cos (3 x)+\frac{12}{37} e^{x/2} x \sin (3 x)\right ) \, dx\\ &=\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)+\frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)+\frac{1}{37} \int e^{x/2} x \cos (3 x) \, dx+\frac{6}{37} \int e^{x/2} x \sin (3 x) \, dx-\frac{1}{5} \int e^{x/2} x \cos (x) \, dx-\frac{2}{5} \int e^{x/2} x \sin (x) \, dx\\ &=\frac{6}{25} e^{x/2} x \cos (x)+\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{1}{5} e^{x/2} x^2 \sin (x)+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{3}{37} e^{x/2} x^2 \sin (3 x)-\frac{1}{37} \int \left (\frac{2}{37} e^{x/2} \cos (3 x)+\frac{12}{37} e^{x/2} \sin (3 x)\right ) \, dx-\frac{6}{37} \int \left (-\frac{12}{37} e^{x/2} \cos (3 x)+\frac{2}{37} e^{x/2} \sin (3 x)\right ) \, dx+\frac{1}{5} \int \left (\frac{2}{5} e^{x/2} \cos (x)+\frac{4}{5} e^{x/2} \sin (x)\right ) \, dx+\frac{2}{5} \int \left (-\frac{4}{5} e^{x/2} \cos (x)+\frac{2}{5} e^{x/2} \sin (x)\right ) \, dx\\ &=\frac{6}{25} e^{x/2} x \cos (x)+\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{1}{5} e^{x/2} x^2 \sin (x)+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{3}{37} e^{x/2} x^2 \sin (3 x)-\frac{2 \int e^{x/2} \cos (3 x) \, dx}{1369}-2 \frac{12 \int e^{x/2} \sin (3 x) \, dx}{1369}+\frac{72 \int e^{x/2} \cos (3 x) \, dx}{1369}+\frac{2}{25} \int e^{x/2} \cos (x) \, dx+2 \left (\frac{4}{25} \int e^{x/2} \sin (x) \, dx\right )-\frac{8}{25} \int e^{x/2} \cos (x) \, dx\\ &=-\frac{12}{125} e^{x/2} \cos (x)+\frac{6}{25} e^{x/2} x \cos (x)+\frac{1}{10} e^{x/2} x^2 \cos (x)+\frac{140 e^{x/2} \cos (3 x)}{50653}-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{24}{125} e^{x/2} \sin (x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{1}{5} e^{x/2} x^2 \sin (x)+2 \left (-\frac{16}{125} e^{x/2} \cos (x)+\frac{8}{125} e^{x/2} \sin (x)\right )+\frac{840 e^{x/2} \sin (3 x)}{50653}+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{3}{37} e^{x/2} x^2 \sin (3 x)-2 \left (-\frac{144 e^{x/2} \cos (3 x)}{50653}+\frac{24 e^{x/2} \sin (3 x)}{50653}\right )\\ \end{align*}

Mathematica [A]  time = 0.210563, size = 76, normalized size = 0.41 \[ \frac{e^{x/2} \left (50653 \left (2 \left (25 x^2-40 x-8\right ) \sin (x)+\left (25 x^2+60 x-88\right ) \cos (x)\right )-125 \left (6 \left (1369 x^2-296 x-264\right ) \sin (3 x)+\left (1369 x^2+5180 x-856\right ) \cos (3 x)\right )\right )}{12663250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(x/2)*(50653*((-88 + 60*x + 25*x^2)*Cos[x] + 2*(-8 - 40*x + 25*x^2)*Sin[x]) - 125*((-856 + 5180*x + 1369*x^
2)*Cos[3*x] + 6*(-264 - 296*x + 1369*x^2)*Sin[3*x])))/12663250

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Maple [A]  time = 0.024, size = 78, normalized size = 0.4 \begin{align*}{\frac{\cos \left ( x \right ) }{4} \left ({\frac{2\,{x}^{2}}{5}}+{\frac{24\,x}{25}}-{\frac{176}{125}} \right ){{\rm e}^{{\frac{x}{2}}}}}-{\frac{\sin \left ( x \right ) }{4} \left ( -{\frac{4\,{x}^{2}}{5}}+{\frac{32\,x}{25}}+{\frac{32}{125}} \right ){{\rm e}^{{\frac{x}{2}}}}}-{\frac{\cos \left ( 3\,x \right ) }{4} \left ({\frac{2\,{x}^{2}}{37}}+{\frac{280\,x}{1369}}-{\frac{1712}{50653}} \right ){{\rm e}^{{\frac{x}{2}}}}}+{\frac{\sin \left ( 3\,x \right ) }{4} \left ( -{\frac{12\,{x}^{2}}{37}}+{\frac{96\,x}{1369}}+{\frac{3168}{50653}} \right ){{\rm e}^{{\frac{x}{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x)

[Out]

1/4*(2/5*x^2+24/25*x-176/125)*exp(1/2*x)*cos(x)-1/4*(-4/5*x^2+32/25*x+32/125)*exp(1/2*x)*sin(x)-1/4*(2/37*x^2+
280/1369*x-1712/50653)*exp(1/2*x)*cos(3*x)+1/4*(-12/37*x^2+96/1369*x+3168/50653)*exp(1/2*x)*sin(3*x)

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Maxima [A]  time = 0.98504, size = 104, normalized size = 0.56 \begin{align*} -\frac{1}{101306} \,{\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac{1}{2} \, x\right )} + \frac{1}{250} \,{\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac{1}{2} \, x\right )} - \frac{3}{50653} \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac{1}{125} \,{\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="maxima")

[Out]

-1/101306*(1369*x^2 + 5180*x - 856)*cos(3*x)*e^(1/2*x) + 1/250*(25*x^2 + 60*x - 88)*cos(x)*e^(1/2*x) - 3/50653
*(1369*x^2 - 296*x - 264)*e^(1/2*x)*sin(3*x) + 1/125*(25*x^2 - 40*x - 8)*e^(1/2*x)*sin(x)

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Fricas [A]  time = 2.25934, size = 282, normalized size = 1.52 \begin{align*} -\frac{4}{6331625} \,{\left (375 \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (x\right ) - \frac{2}{6331625} \,{\left (125 \,{\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} -{\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \left (x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="fricas")

[Out]

-4/6331625*(375*(1369*x^2 - 296*x - 264)*cos(x)^2 - 444925*x^2 + 534280*x + 126056)*e^(1/2*x)*sin(x) - 2/63316
25*(125*(1369*x^2 + 5180*x - 856)*cos(x)^3 - (444925*x^2 + 1245420*x - 1194616)*cos(x))*e^(1/2*x)

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Sympy [A]  time = 12.7763, size = 202, normalized size = 1.09 \begin{align*} \frac{52 x^{2} e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac{26 x^{2} e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{185} - \frac{8 x^{2} e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac{16 x^{2} e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac{11552 x e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac{13464 x e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{34225} - \frac{9152 x e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac{6464 x e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac{504224 e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac{2389232 e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{6331625} - \frac{108224 e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac{2175232 e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/2*x)*x**2*cos(x)*sin(x)**2,x)

[Out]

52*x**2*exp(x/2)*sin(x)**3/185 + 26*x**2*exp(x/2)*sin(x)**2*cos(x)/185 - 8*x**2*exp(x/2)*sin(x)*cos(x)**2/185
+ 16*x**2*exp(x/2)*cos(x)**3/185 - 11552*x*exp(x/2)*sin(x)**3/34225 + 13464*x*exp(x/2)*sin(x)**2*cos(x)/34225
- 9152*x*exp(x/2)*sin(x)*cos(x)**2/34225 + 6464*x*exp(x/2)*cos(x)**3/34225 - 504224*exp(x/2)*sin(x)**3/6331625
- 2389232*exp(x/2)*sin(x)**2*cos(x)/6331625 - 108224*exp(x/2)*sin(x)*cos(x)**2/6331625 - 2175232*exp(x/2)*cos
(x)**3/6331625

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Giac [A]  time = 1.14673, size = 99, normalized size = 0.54 \begin{align*} -\frac{1}{101306} \,{\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} + \frac{1}{250} \,{\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \,{\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/2*x)*x^2*cos(x)*sin(x)^2,x, algorithm="giac")

[Out]

-1/101306*((1369*x^2 + 5180*x - 856)*cos(3*x) + 6*(1369*x^2 - 296*x - 264)*sin(3*x))*e^(1/2*x) + 1/250*((25*x^
2 + 60*x - 88)*cos(x) + 2*(25*x^2 - 40*x - 8)*sin(x))*e^(1/2*x)