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

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

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Rubi [A]  time = 0.36, antiderivative size = 185, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 (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*}

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Mathematica [A]  time = 0.22, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x/2} x^2 \cos (x) \sin ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 1.15, size = 72, normalized size = 0.39 \[ -\frac {4}{6331625} \, {\left (375 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \relax (x)^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) - \frac {2}{6331625} \, {\left (125 \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \relax (x)^{3} - {\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]

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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giac [A]  time = 0.60, size = 73, normalized size = 0.39 \[ -\frac {1}{101306} \, {\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) + 2 \, {\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \relax (x)\right )} e^{\left (\frac {1}{2} \, x\right )} \]

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)

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maple [C]  time = 0.09, size = 106, normalized size = 0.57

method	result	size
risch	\(\left (-\frac {1}{202612}+\frac {3 i}{101306}\right ) \left (1369 x^{2}+888 i x -148 x -96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}+3 i\right ) x}+\left (\frac {1}{500}-\frac {i}{250}\right ) \left (25 x^{2}+40 i x -20 x -32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}+i\right ) x}+\left (\frac {1}{500}+\frac {i}{250}\right ) \left (25 x^{2}-40 i x -20 x +32 i-24\right ) {\mathrm e}^{\left (\frac {1}{2}-i\right ) x}+\left (-\frac {1}{202612}-\frac {3 i}{101306}\right ) \left (1369 x^{2}-888 i x -148 x +96 i-280\right ) {\mathrm e}^{\left (\frac {1}{2}-3 i\right ) x}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/202612+3/101306*I)*(888*I*x+1369*x^2-280-96*I-148*x)*exp((1/2+3*I)*x)+(1/500-1/250*I)*(40*I*x+25*x^2-24-32
*I-20*x)*exp((1/2+I)*x)+(1/500+1/250*I)*(-40*I*x+25*x^2-24+32*I-20*x)*exp((1/2-I)*x)-(1/202612+3/101306*I)*(-8
88*I*x+1369*x^2-280+96*I-148*x)*exp((1/2-3*I)*x)

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maxima [A]  time = 0.49, size = 77, normalized size = 0.42 \[ -\frac {1}{101306} \, {\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac {1}{2} \, x\right )} + \frac {1}{250} \, {\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \relax (x) e^{\left (\frac {1}{2} \, x\right )} - \frac {3}{50653} \, {\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac {1}{125} \, {\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac {1}{2} \, x\right )} \sin \relax (x) \]

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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mupad [B]  time = 0.53, size = 83, normalized size = 0.45 \[ \frac {{\mathrm {e}}^{x/2}\,\left (107000\,\cos \left (3\,x\right )+198000\,\sin \left (3\,x\right )-4457464\,\cos \relax (x)-810448\,\sin \relax (x)-647500\,x\,\cos \left (3\,x\right )+1266325\,x^2\,\cos \relax (x)+222000\,x\,\sin \left (3\,x\right )+2532650\,x^2\,\sin \relax (x)-171125\,x^2\,\cos \left (3\,x\right )-1026750\,x^2\,\sin \left (3\,x\right )+3039180\,x\,\cos \relax (x)-4052240\,x\,\sin \relax (x)\right )}{12663250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(x/2)*(107000*cos(3*x) + 198000*sin(3*x) - 4457464*cos(x) - 810448*sin(x) - 647500*x*cos(3*x) + 1266325*x^
2*cos(x) + 222000*x*sin(3*x) + 2532650*x^2*sin(x) - 171125*x^2*cos(3*x) - 1026750*x^2*sin(3*x) + 3039180*x*cos
(x) - 4052240*x*sin(x)))/12663250

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sympy [A]  time = 11.81, size = 202, normalized size = 1.09 \[ \frac {52 x^{2} e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{185} + \frac {26 x^{2} e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{185} - \frac {8 x^{2} e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{185} + \frac {16 x^{2} e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{185} - \frac {11552 x e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{34225} + \frac {13464 x e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{34225} - \frac {9152 x e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{34225} + \frac {6464 x e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{34225} - \frac {504224 e^{\frac {x}{2}} \sin ^{3}{\relax (x )}}{6331625} - \frac {2389232 e^{\frac {x}{2}} \sin ^{2}{\relax (x )} \cos {\relax (x )}}{6331625} - \frac {108224 e^{\frac {x}{2}} \sin {\relax (x )} \cos ^{2}{\relax (x )}}{6331625} - \frac {2175232 e^{\frac {x}{2}} \cos ^{3}{\relax (x )}}{6331625} \]

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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