3.71 \(\int \cos ^4(2 x) \sin ^2(2 x) \, dx\)

Optimal. Leaf size=46 \[ \frac{x}{16}-\frac{1}{12} \sin (2 x) \cos ^5(2 x)+\frac{1}{48} \sin (2 x) \cos ^3(2 x)+\frac{1}{32} \sin (2 x) \cos (2 x) \]

[Out]

x/16 + (Cos[2*x]*Sin[2*x])/32 + (Cos[2*x]^3*Sin[2*x])/48 - (Cos[2*x]^5*Sin[2*x])/12

________________________________________________________________________________________

Rubi [A]  time = 0.0360118, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2568, 2635, 8} \[ \frac{x}{16}-\frac{1}{12} \sin (2 x) \cos ^5(2 x)+\frac{1}{48} \sin (2 x) \cos ^3(2 x)+\frac{1}{32} \sin (2 x) \cos (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*x]^4*Sin[2*x]^2,x]

[Out]

x/16 + (Cos[2*x]*Sin[2*x])/32 + (Cos[2*x]^3*Sin[2*x])/48 - (Cos[2*x]^5*Sin[2*x])/12

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^4(2 x) \sin ^2(2 x) \, dx &=-\frac{1}{12} \cos ^5(2 x) \sin (2 x)+\frac{1}{6} \int \cos ^4(2 x) \, dx\\ &=\frac{1}{48} \cos ^3(2 x) \sin (2 x)-\frac{1}{12} \cos ^5(2 x) \sin (2 x)+\frac{1}{8} \int \cos ^2(2 x) \, dx\\ &=\frac{1}{32} \cos (2 x) \sin (2 x)+\frac{1}{48} \cos ^3(2 x) \sin (2 x)-\frac{1}{12} \cos ^5(2 x) \sin (2 x)+\frac{\int 1 \, dx}{16}\\ &=\frac{x}{16}+\frac{1}{32} \cos (2 x) \sin (2 x)+\frac{1}{48} \cos ^3(2 x) \sin (2 x)-\frac{1}{12} \cos ^5(2 x) \sin (2 x)\\ \end{align*}

Mathematica [A]  time = 0.0138663, size = 30, normalized size = 0.65 \[ \frac{x}{16}+\frac{1}{128} \sin (4 x)-\frac{1}{128} \sin (8 x)-\frac{1}{384} \sin (12 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*x]^4*Sin[2*x]^2,x]

[Out]

x/16 + Sin[4*x]/128 - Sin[8*x]/128 - Sin[12*x]/384

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{5}\sin \left ( 2\,x \right ) }{12}}+{\frac{\sin \left ( 2\,x \right ) }{48} \left ( \left ( \cos \left ( 2\,x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) }+{\frac{x}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(2*x)^4*sin(2*x)^2,x)

[Out]

-1/12*cos(2*x)^5*sin(2*x)+1/48*(cos(2*x)^3+3/2*cos(2*x))*sin(2*x)+1/16*x

________________________________________________________________________________________

Maxima [A]  time = 0.938778, size = 24, normalized size = 0.52 \begin{align*} \frac{1}{96} \, \sin \left (4 \, x\right )^{3} + \frac{1}{16} \, x - \frac{1}{128} \, \sin \left (8 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="maxima")

[Out]

1/96*sin(4*x)^3 + 1/16*x - 1/128*sin(8*x)

________________________________________________________________________________________

Fricas [A]  time = 1.99528, size = 92, normalized size = 2. \begin{align*} -\frac{1}{96} \,{\left (8 \, \cos \left (2 \, x\right )^{5} - 2 \, \cos \left (2 \, x\right )^{3} - 3 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac{1}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="fricas")

[Out]

-1/96*(8*cos(2*x)^5 - 2*cos(2*x)^3 - 3*cos(2*x))*sin(2*x) + 1/16*x

________________________________________________________________________________________

Sympy [A]  time = 0.061795, size = 41, normalized size = 0.89 \begin{align*} \frac{x}{16} - \frac{\sin{\left (2 x \right )} \cos ^{5}{\left (2 x \right )}}{12} + \frac{\sin{\left (2 x \right )} \cos ^{3}{\left (2 x \right )}}{48} + \frac{\sin{\left (2 x \right )} \cos{\left (2 x \right )}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)**4*sin(2*x)**2,x)

[Out]

x/16 - sin(2*x)*cos(2*x)**5/12 + sin(2*x)*cos(2*x)**3/48 + sin(2*x)*cos(2*x)/32

________________________________________________________________________________________

Giac [A]  time = 1.05166, size = 30, normalized size = 0.65 \begin{align*} \frac{1}{16} \, x - \frac{1}{384} \, \sin \left (12 \, x\right ) - \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{1}{128} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="giac")

[Out]

1/16*x - 1/384*sin(12*x) - 1/128*sin(8*x) + 1/128*sin(4*x)