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

Optimal. Leaf size=36 \[ \frac{e^{2 x}}{16}-\frac{1}{40} e^{2 x} \sin (4 x)-\frac{1}{80} e^{2 x} \cos (4 x) \]

[Out]

E^(2*x)/16 - (E^(2*x)*Cos[4*x])/80 - (E^(2*x)*Sin[4*x])/40

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Rubi [A]  time = 0.0705463, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{e^{2 x}}{16}-\frac{1}{40} e^{2 x} \sin (4 x)-\frac{1}{80} e^{2 x} \cos (4 x) \]

Antiderivative was successfully verified.

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

[Out]

E^(2*x)/16 - (E^(2*x)*Cos[4*x])/80 - (E^(2*x)*Sin[4*x])/40

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Rubi in Sympy [A]  time = 5.31334, size = 29, normalized size = 0.81 \[ - \frac{e^{2 x} \sin{\left (4 x \right )}}{40} - \frac{e^{2 x} \cos{\left (4 x \right )}}{80} + \frac{e^{2 x}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-exp(2*x)*sin(4*x)/40 - exp(2*x)*cos(4*x)/80 + exp(2*x)/16

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Mathematica [A]  time = 0.0298775, size = 21, normalized size = 0.58 \[ -\frac{1}{80} e^{2 x} (2 \sin (4 x)+\cos (4 x)-5) \]

Antiderivative was successfully verified.

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

[Out]

-(E^(2*x)*(-5 + Cos[4*x] + 2*Sin[4*x]))/80

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Maple [A]  time = 0.008, size = 28, normalized size = 0.8 \[ -{\frac{{{\rm e}^{2\,x}}\cos \left ( 4\,x \right ) }{80}}-{\frac{{{\rm e}^{2\,x}}\sin \left ( 4\,x \right ) }{40}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)*cos(x)^2*sin(x)^2,x)

[Out]

-1/80*exp(2*x)*cos(4*x)-1/40*exp(2*x)*sin(4*x)+1/16*exp(x)^2

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Maxima [A]  time = 1.34544, size = 36, normalized size = 1. \[ -\frac{1}{80} \, \cos \left (4 \, x\right ) e^{\left (2 \, x\right )} - \frac{1}{40} \, e^{\left (2 \, x\right )} \sin \left (4 \, x\right ) + \frac{1}{16} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*e^(2*x)*sin(x)^2,x, algorithm="maxima")

[Out]

-1/80*cos(4*x)*e^(2*x) - 1/40*e^(2*x)*sin(4*x) + 1/16*e^(2*x)

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Fricas [A]  time = 0.218727, size = 54, normalized size = 1.5 \[ -\frac{1}{10} \,{\left (2 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} e^{\left (2 \, x\right )} \sin \left (x\right ) - \frac{1}{20} \,{\left (2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*e^(2*x)*sin(x)^2,x, algorithm="fricas")

[Out]

-1/10*(2*cos(x)^3 - cos(x))*e^(2*x)*sin(x) - 1/20*(2*cos(x)^4 - 2*cos(x)^2 - 1)*
e^(2*x)

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Sympy [A]  time = 8.3612, size = 70, normalized size = 1.94 \[ \frac{e^{2 x} \sin ^{4}{\left (x \right )}}{20} + \frac{e^{2 x} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{10} + \frac{e^{2 x} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{5} - \frac{e^{2 x} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{10} + \frac{e^{2 x} \cos ^{4}{\left (x \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)*cos(x)**2*sin(x)**2,x)

[Out]

exp(2*x)*sin(x)**4/20 + exp(2*x)*sin(x)**3*cos(x)/10 + exp(2*x)*sin(x)**2*cos(x)
**2/5 - exp(2*x)*sin(x)*cos(x)**3/10 + exp(2*x)*cos(x)**4/20

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GIAC/XCAS [A]  time = 0.209015, size = 32, normalized size = 0.89 \[ -\frac{1}{80} \,{\left (\cos \left (4 \, x\right ) + 2 \, \sin \left (4 \, x\right )\right )} e^{\left (2 \, x\right )} + \frac{1}{16} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*e^(2*x)*sin(x)^2,x, algorithm="giac")

[Out]

-1/80*(cos(4*x) + 2*sin(4*x))*e^(2*x) + 1/16*e^(2*x)