3.567 \(\int e^{x/2} x^2 \cos ^3(x) \, dx\)

Optimal. Leaf size=187 \[ \frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac{24}{125} e^{x/2} \sin (x)-\frac{24}{25} e^{x/2} x \sin (x)-\frac{792 e^{x/2} \sin (3 x)}{50653}-\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{132}{125} e^{x/2} \cos (x)+\frac{18}{25} e^{x/2} x \cos (x)-\frac{428 e^{x/2} \cos (3 x)}{50653}+\frac{70 e^{x/2} x \cos (3 x)}{1369} \]

[Out]

(-132*E^(x/2)*Cos[x])/125 + (18*E^(x/2)*x*Cos[x])/25 + (48*E^(x/2)*x^2*Cos[x])/185 + (2*E^(x/2)*x^2*Cos[x]^3)/
37 - (428*E^(x/2)*Cos[3*x])/50653 + (70*E^(x/2)*x*Cos[3*x])/1369 - (24*E^(x/2)*Sin[x])/125 - (24*E^(x/2)*x*Sin
[x])/25 + (96*E^(x/2)*x^2*Sin[x])/185 + (12*E^(x/2)*x^2*Cos[x]^2*Sin[x])/37 - (792*E^(x/2)*Sin[3*x])/50653 - (
24*E^(x/2)*x*Sin[3*x])/1369

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Rubi [A]  time = 0.478478, antiderivative size = 253, normalized size of antiderivative = 1.35, number of steps used = 31, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {4435, 4433, 4466, 14, 4432, 4469, 4465, 4470} \[ \frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{12}{37} e^{x/2} x^2 \sin (x) \cos ^2(x)-\frac{1218672 e^{x/2} \sin (x)}{6331625}-\frac{32556 e^{x/2} x \sin (x)}{34225}-\frac{816 e^{x/2} \sin (3 x)}{50653}-\frac{12 e^{x/2} x \sin (3 x)}{1369}+\frac{16 e^{x/2} \cos ^3(x)}{50653}-\frac{8 e^{x/2} x \cos ^3(x)}{1369}-\frac{6687696 e^{x/2} \cos (x)}{6331625}+\frac{24792 e^{x/2} x \cos (x)}{34225}-\frac{432 e^{x/2} \cos (3 x)}{50653}+\frac{72 e^{x/2} x \cos (3 x)}{1369}+\frac{96 e^{x/2} \sin (x) \cos ^2(x)}{50653}-\frac{48 e^{x/2} x \sin (x) \cos ^2(x)}{1369} \]

Antiderivative was successfully verified.

[In]

Int[E^(x/2)*x^2*Cos[x]^3,x]

[Out]

(-6687696*E^(x/2)*Cos[x])/6331625 + (24792*E^(x/2)*x*Cos[x])/34225 + (48*E^(x/2)*x^2*Cos[x])/185 + (16*E^(x/2)
*Cos[x]^3)/50653 - (8*E^(x/2)*x*Cos[x]^3)/1369 + (2*E^(x/2)*x^2*Cos[x]^3)/37 - (432*E^(x/2)*Cos[3*x])/50653 +
(72*E^(x/2)*x*Cos[3*x])/1369 - (1218672*E^(x/2)*Sin[x])/6331625 - (32556*E^(x/2)*x*Sin[x])/34225 + (96*E^(x/2)
*x^2*Sin[x])/185 + (96*E^(x/2)*Cos[x]^2*Sin[x])/50653 - (48*E^(x/2)*x*Cos[x]^2*Sin[x])/1369 + (12*E^(x/2)*x^2*
Cos[x]^2*Sin[x])/37 - (816*E^(x/2)*Sin[3*x])/50653 - (12*E^(x/2)*x*Sin[3*x])/1369

Rule 4435

Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*
x))*Cos[d + e*x]^m)/(e^2*m^2 + b^2*c^2*Log[F]^2), x] + (Dist[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^2*Log[F]^2), Int
[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x] + Simp[(e*m*F^(c*(a + b*x))*Sin[d + e*x]*Cos[d + e*x]^(m - 1))/(
e^2*m^2 + b^2*c^2*Log[F]^2), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[
m, 1]

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 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[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]

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 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]

Rubi steps

\begin{align*} \int e^{x/2} x^2 \cos ^3(x) \, dx &=\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int x \left (\frac{48}{185} e^{x/2} \cos (x)+\frac{2}{37} e^{x/2} \cos ^3(x)+\frac{96}{185} e^{x/2} \sin (x)+\frac{12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx\\ &=\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-2 \int \left (\frac{48}{185} e^{x/2} x \cos (x)+\frac{2}{37} e^{x/2} x \cos ^3(x)+\frac{96}{185} e^{x/2} x \sin (x)+\frac{12}{37} e^{x/2} x \cos ^2(x) \sin (x)\right ) \, dx\\ &=\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)-\frac{4}{37} \int e^{x/2} x \cos ^3(x) \, dx-\frac{96}{185} \int e^{x/2} x \cos (x) \, dx-\frac{24}{37} \int e^{x/2} x \cos ^2(x) \sin (x) \, dx-\frac{192}{185} \int e^{x/2} x \sin (x) \, dx\\ &=\frac{20352 e^{x/2} x \cos (x)}{34225}+\frac{48}{185} e^{x/2} x^2 \cos (x)-\frac{8 e^{x/2} x \cos ^3(x)}{1369}+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)-\frac{30336 e^{x/2} x \sin (x)}{34225}+\frac{96}{185} e^{x/2} x^2 \sin (x)-\frac{48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac{4}{37} \int \left (\frac{48}{185} e^{x/2} \cos (x)+\frac{2}{37} e^{x/2} \cos ^3(x)+\frac{96}{185} e^{x/2} \sin (x)+\frac{12}{37} e^{x/2} \cos ^2(x) \sin (x)\right ) \, dx+\frac{96}{185} \int \left (\frac{2}{5} e^{x/2} \cos (x)+\frac{4}{5} e^{x/2} \sin (x)\right ) \, dx-\frac{24}{37} \int \left (\frac{1}{4} e^{x/2} x \sin (x)+\frac{1}{4} e^{x/2} x \sin (3 x)\right ) \, dx+\frac{192}{185} \int \left (-\frac{4}{5} e^{x/2} \cos (x)+\frac{2}{5} e^{x/2} \sin (x)\right ) \, dx\\ &=\frac{20352 e^{x/2} x \cos (x)}{34225}+\frac{48}{185} e^{x/2} x^2 \cos (x)-\frac{8 e^{x/2} x \cos ^3(x)}{1369}+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)-\frac{30336 e^{x/2} x \sin (x)}{34225}+\frac{96}{185} e^{x/2} x^2 \sin (x)-\frac{48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+\frac{8 \int e^{x/2} \cos ^3(x) \, dx}{1369}+\frac{192 \int e^{x/2} \cos (x) \, dx}{6845}+\frac{48 \int e^{x/2} \cos ^2(x) \sin (x) \, dx}{1369}+\frac{384 \int e^{x/2} \sin (x) \, dx}{6845}-\frac{6}{37} \int e^{x/2} x \sin (x) \, dx-\frac{6}{37} \int e^{x/2} x \sin (3 x) \, dx+\frac{192}{925} \int e^{x/2} \cos (x) \, dx+2 \left (\frac{384}{925} \int e^{x/2} \sin (x) \, dx\right )-\frac{768}{925} \int e^{x/2} \cos (x) \, dx\\ &=-\frac{48384 e^{x/2} \cos (x)}{171125}+\frac{24792 e^{x/2} x \cos (x)}{34225}+\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{16 e^{x/2} \cos ^3(x)}{50653}-\frac{8 e^{x/2} x \cos ^3(x)}{1369}+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{72 e^{x/2} x \cos (3 x)}{1369}-\frac{77568 e^{x/2} \sin (x)}{171125}-\frac{32556 e^{x/2} x \sin (x)}{34225}+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac{48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac{1536 e^{x/2} \cos (x)}{4625}+\frac{768 e^{x/2} \sin (x)}{4625}\right )-\frac{12 e^{x/2} x \sin (3 x)}{1369}+\frac{192 \int e^{x/2} \cos (x) \, dx}{50653}+\frac{48 \int \left (\frac{1}{4} e^{x/2} \sin (x)+\frac{1}{4} e^{x/2} \sin (3 x)\right ) \, dx}{1369}+\frac{6}{37} \int \left (-\frac{4}{5} e^{x/2} \cos (x)+\frac{2}{5} e^{x/2} \sin (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{1780608 e^{x/2} \cos (x)}{6331625}+\frac{24792 e^{x/2} x \cos (x)}{34225}+\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{16 e^{x/2} \cos ^3(x)}{50653}-\frac{8 e^{x/2} x \cos ^3(x)}{1369}+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)+\frac{72 e^{x/2} x \cos (3 x)}{1369}-\frac{2850816 e^{x/2} \sin (x)}{6331625}-\frac{32556 e^{x/2} x \sin (x)}{34225}+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac{48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac{1536 e^{x/2} \cos (x)}{4625}+\frac{768 e^{x/2} \sin (x)}{4625}\right )-\frac{12 e^{x/2} x \sin (3 x)}{1369}+\frac{12 \int e^{x/2} \sin (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{12}{185} \int e^{x/2} \sin (x) \, dx-\frac{24}{185} \int e^{x/2} \cos (x) \, dx\\ &=-\frac{2482128 e^{x/2} \cos (x)}{6331625}+\frac{24792 e^{x/2} x \cos (x)}{34225}+\frac{48}{185} e^{x/2} x^2 \cos (x)+\frac{16 e^{x/2} \cos ^3(x)}{50653}-\frac{8 e^{x/2} x \cos ^3(x)}{1369}+\frac{2}{37} e^{x/2} x^2 \cos ^3(x)-\frac{144 e^{x/2} \cos (3 x)}{50653}+\frac{72 e^{x/2} x \cos (3 x)}{1369}-\frac{3321456 e^{x/2} \sin (x)}{6331625}-\frac{32556 e^{x/2} x \sin (x)}{34225}+\frac{96}{185} e^{x/2} x^2 \sin (x)+\frac{96 e^{x/2} \cos ^2(x) \sin (x)}{50653}-\frac{48 e^{x/2} x \cos ^2(x) \sin (x)}{1369}+\frac{12}{37} e^{x/2} x^2 \cos ^2(x) \sin (x)+2 \left (-\frac{1536 e^{x/2} \cos (x)}{4625}+\frac{768 e^{x/2} \sin (x)}{4625}\right )-\frac{864 e^{x/2} \sin (3 x)}{50653}-\frac{12 e^{x/2} x \sin (3 x)}{1369}+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 = 0.148817, size = 72, normalized size = 0.39 \[ \frac{e^{x/2} \left (303918 \left (25 x^2-40 x-8\right ) \sin (x)+750 \left (1369 x^2-296 x-264\right ) \sin (3 x)+151959 \left (25 x^2+60 x-88\right ) \cos (x)+125 \left (1369 x^2+5180 x-856\right ) \cos (3 x)\right )}{12663250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(x/2)*(151959*(-88 + 60*x + 25*x^2)*Cos[x] + 125*(-856 + 5180*x + 1369*x^2)*Cos[3*x] + 303918*(-8 - 40*x +
25*x^2)*Sin[x] + 750*(-264 - 296*x + 1369*x^2)*Sin[3*x]))/12663250

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Maple [A]  time = 0.039, size = 78, normalized size = 0.4 \begin{align*}{\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}}}}}+{\frac{3\,\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{3\,\sin \left ( x \right ) }{4} \left ( -{\frac{4\,{x}^{2}}{5}}+{\frac{32\,x}{25}}+{\frac{32}{125}} \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)^3,x)

[Out]

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)+3/4*(2/5*x^2+24/25*x-176/125)*exp(1/2*x)*cos(x)-3/4*(-4/5*x^2+32/25*x+32/125)*exp(1/2*x)*sin(x)

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Maxima [A]  time = 0.990361, 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{3}{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{3}{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.01906, size = 279, normalized size = 1.49 \begin{align*} \frac{12}{6331625} \,{\left (125 \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} + 273800 \, x^{2} - 497280 \, x - 93056\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} + 24 \,{\left (34225 \, x^{2} + 74740 \, x - 135952\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)^3,x, algorithm="fricas")

[Out]

12/6331625*(125*(1369*x^2 - 296*x - 264)*cos(x)^2 + 273800*x^2 - 497280*x - 93056)*e^(1/2*x)*sin(x) + 2/633162
5*(125*(1369*x^2 + 5180*x - 856)*cos(x)^3 + 24*(34225*x^2 + 74740*x - 135952)*cos(x))*e^(1/2*x)

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Sympy [A]  time = 12.812, size = 202, normalized size = 1.08 \begin{align*} \frac{96 x^{2} e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac{48 x^{2} e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{185} + \frac{156 x^{2} e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac{58 x^{2} e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac{32256 x e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac{19392 x e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{34225} - \frac{34656 x e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac{26392 x e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac{1116672 e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac{6525696 e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{6331625} - \frac{1512672 e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac{6739696 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)**3,x)

[Out]

96*x**2*exp(x/2)*sin(x)**3/185 + 48*x**2*exp(x/2)*sin(x)**2*cos(x)/185 + 156*x**2*exp(x/2)*sin(x)*cos(x)**2/18
5 + 58*x**2*exp(x/2)*cos(x)**3/185 - 32256*x*exp(x/2)*sin(x)**3/34225 + 19392*x*exp(x/2)*sin(x)**2*cos(x)/3422
5 - 34656*x*exp(x/2)*sin(x)*cos(x)**2/34225 + 26392*x*exp(x/2)*cos(x)**3/34225 - 1116672*exp(x/2)*sin(x)**3/63
31625 - 6525696*exp(x/2)*sin(x)**2*cos(x)/6331625 - 1512672*exp(x/2)*sin(x)*cos(x)**2/6331625 - 6739696*exp(x/
2)*cos(x)**3/6331625

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Giac [A]  time = 1.08749, size = 99, normalized size = 0.53 \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{3}{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)^3,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) + 3/250*((25*x^2
+ 60*x - 88)*cos(x) + 2*(25*x^2 - 40*x - 8)*sin(x))*e^(1/2*x)