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 = 0.0437049, 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, Rules used = {4469, 2194, 4433} \[ \frac{e^{2 x}}{16}-\frac{1}{40} e^{2 x} \sin (4 x)-\frac{1}{80} e^{2 x} \cos (4 x) \]

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

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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]

Rubi steps

\begin{align*} \int e^{2 x} \cos ^2(x) \sin ^2(x) \, dx &=\int \left (\frac{e^{2 x}}{8}-\frac{1}{8} e^{2 x} \cos (4 x)\right ) \, dx\\ &=\frac{1}{8} \int e^{2 x} \, dx-\frac{1}{8} \int e^{2 x} \cos (4 x) \, dx\\ &=\frac{e^{2 x}}{16}-\frac{1}{80} e^{2 x} \cos (4 x)-\frac{1}{40} e^{2 x} \sin (4 x)\\ \end{align*}

Mathematica [A]  time = 0.0324941, 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 = 0.007, size = 28, normalized size = 0.8 \begin{align*} -{\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}} \end{align*}

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 = 0.961003, size = 36, normalized size = 1. \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(x)^2*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 = 1.86622, size = 120, normalized size = 3.33 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*cos(x)^2*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 [B]  time = 5.17987, size = 70, normalized size = 1.94 \begin{align*} \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} \end{align*}

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 [A]  time = 1.12767, size = 32, normalized size = 0.89 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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