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 = 0.55239, 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 \[ \frac{1}{5} e^{x/2} x^2 \sin (x)-\frac{3}{37} e^{x/2} x^2 \sin (3 x)+\frac{1}{10} e^{x/2} x^2 \cos (x)-\frac{1}{74} e^{x/2} x^2 \cos (3 x)-\frac{8}{25} e^{x/2} x \sin (x)+\frac{24 e^{x/2} x \sin (3 x)}{1369}-\frac{8}{125} e^{x/2} \sin (x)+\frac{792 e^{x/2} \sin (3 x)}{50653}+\frac{6}{25} e^{x/2} x \cos (x)-\frac{70 e^{x/2} x \cos (3 x)}{1369}-\frac{44}{125} e^{x/2} \cos (x)+\frac{428 e^{x/2} \cos (3 x)}{50653} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{2} e^{\frac{x}{2}} \sin{\left (x \right )}}{5} - \frac{3 x^{2} e^{\frac{x}{2}} \sin{\left (3 x \right )}}{37} + \frac{x^{2} e^{\frac{x}{2}} \cos{\left (x \right )}}{10} - \frac{x^{2} e^{\frac{x}{2}} \cos{\left (3 x \right )}}{74} - \frac{\int x \left (\frac{4 e^{\frac{x}{2}} \sin{\left (x \right )}}{5} + \frac{2 e^{\frac{x}{2}} \cos{\left (x \right )}}{5}\right )\, dx}{2} + \frac{\int x \left (\frac{12 e^{\frac{x}{2}} \sin{\left (3 x \right )}}{37} + \frac{2 e^{\frac{x}{2}} \cos{\left (3 x \right )}}{37}\right )\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(1/2*x)*x**2*cos(x)*sin(x)**2,x)

[Out]

x**2*exp(x/2)*sin(x)/5 - 3*x**2*exp(x/2)*sin(3*x)/37 + x**2*exp(x/2)*cos(x)/10 -
x**2*exp(x/2)*cos(3*x)/74 - Integral(x*(4*exp(x/2)*sin(x)/5 + 2*exp(x/2)*cos(x)
/5), x)/2 + Integral(x*(12*exp(x/2)*sin(3*x)/37 + 2*exp(x/2)*cos(3*x)/37), x)/2

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Mathematica [A]  time = 0.304697, 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 = 0.009, size = 78, normalized size = 0.4 \[{\frac{\cos \left ( x \right ) }{4} \left ({\frac{2\,{x}^{2}}{5}}+{\frac{24\,x}{25}}-{\frac{176}{125}} \right ){{\rm e}^{{\frac{x}{2}}}}}-{\frac{\sin \left ( x \right ) }{4} \left ( -{\frac{4\,{x}^{2}}{5}}+{\frac{32\,x}{25}}+{\frac{32}{125}} \right ){{\rm e}^{{\frac{x}{2}}}}}-{\frac{\cos \left ( 3\,x \right ) }{4} \left ({\frac{2\,{x}^{2}}{37}}+{\frac{280\,x}{1369}}-{\frac{1712}{50653}} \right ){{\rm e}^{{\frac{x}{2}}}}}+{\frac{\sin \left ( 3\,x \right ) }{4} \left ( -{\frac{12\,{x}^{2}}{37}}+{\frac{96\,x}{1369}}+{\frac{3168}{50653}} \right ){{\rm e}^{{\frac{x}{2}}}}} \]

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)*ex
p(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*(-1
2/37*x^2+96/1369*x+3168/50653)*exp(1/2*x)*sin(3*x)

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Maxima [A]  time = 1.38076, size = 104, normalized size = 0.56 \[ -\frac{1}{101306} \,{\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) e^{\left (\frac{1}{2} \, x\right )} + \frac{1}{250} \,{\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) e^{\left (\frac{1}{2} \, x\right )} - \frac{3}{50653} \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (3 \, x\right ) + \frac{1}{125} \,{\left (25 \, x^{2} - 40 \, x - 8\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(x)*e^(1/2*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) + 1/1
25*(25*x^2 - 40*x - 8)*e^(1/2*x)*sin(x)

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Fricas [A]  time = 0.221643, size = 97, normalized size = 0.52 \[ -\frac{4}{6331625} \,{\left (375 \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} \cos \left (x\right )^{2} - 444925 \, x^{2} + 534280 \, x + 126056\right )} e^{\left (\frac{1}{2} \, x\right )} \sin \left (x\right ) - \frac{2}{6331625} \,{\left (125 \,{\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (x\right )^{3} -{\left (444925 \, x^{2} + 1245420 \, x - 1194616\right )} \cos \left (x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(x)*e^(1/2*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 + 1260
56)*e^(1/2*x)*sin(x) - 2/6331625*(125*(1369*x^2 + 5180*x - 856)*cos(x)^3 - (4449
25*x^2 + 1245420*x - 1194616)*cos(x))*e^(1/2*x)

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Sympy [A]  time = 21.8472, size = 202, normalized size = 1.09 \[ \frac{52 x^{2} e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{185} + \frac{26 x^{2} e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{185} - \frac{8 x^{2} e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{185} + \frac{16 x^{2} e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{185} - \frac{11552 x e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{34225} + \frac{13464 x e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{34225} - \frac{9152 x e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{34225} + \frac{6464 x e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{34225} - \frac{504224 e^{\frac{x}{2}} \sin ^{3}{\left (x \right )}}{6331625} - \frac{2389232 e^{\frac{x}{2}} \sin ^{2}{\left (x \right )} \cos{\left (x \right )}}{6331625} - \frac{108224 e^{\frac{x}{2}} \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{6331625} - \frac{2175232 e^{\frac{x}{2}} \cos ^{3}{\left (x \right )}}{6331625} \]

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)*si
n(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/XCAS [A]  time = 0.233029, size = 99, normalized size = 0.54 \[ -\frac{1}{101306} \,{\left ({\left (1369 \, x^{2} + 5180 \, x - 856\right )} \cos \left (3 \, x\right ) + 6 \,{\left (1369 \, x^{2} - 296 \, x - 264\right )} \sin \left (3 \, x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} + \frac{1}{250} \,{\left ({\left (25 \, x^{2} + 60 \, x - 88\right )} \cos \left (x\right ) + 2 \,{\left (25 \, x^{2} - 40 \, x - 8\right )} \sin \left (x\right )\right )} e^{\left (\frac{1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*cos(x)*e^(1/2*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)