3.835 \(\int 4 \cos ^4(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=46 \[ \frac{3 x}{32}-\frac{1}{2} \sin ^3(x) \cos ^5(x)-\frac{1}{4} \sin (x) \cos ^5(x)+\frac{1}{16} \sin (x) \cos ^3(x)+\frac{3}{32} \sin (x) \cos (x) \]

[Out]

(3*x)/32 + (3*Cos[x]*Sin[x])/32 + (Cos[x]^3*Sin[x])/16 - (Cos[x]^5*Sin[x])/4 - (Cos[x]^5*Sin[x]^3)/2

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Rubi [A]  time = 0.056109, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {12, 2568, 2635, 8} \[ \frac{3 x}{32}-\frac{1}{2} \sin ^3(x) \cos ^5(x)-\frac{1}{4} \sin (x) \cos ^5(x)+\frac{1}{16} \sin (x) \cos ^3(x)+\frac{3}{32} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/32 + (3*Cos[x]*Sin[x])/32 + (Cos[x]^3*Sin[x])/16 - (Cos[x]^5*Sin[x])/4 - (Cos[x]^5*Sin[x]^3)/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0071436, size = 24, normalized size = 0.52 \[ 4 \left (\frac{3 x}{128}-\frac{1}{128} \sin (4 x)+\frac{\sin (8 x)}{1024}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*((3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{3}}{2}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{4}}+{\frac{\sin \left ( x \right ) }{16} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cos(x)^5*sin(x)^3-1/4*cos(x)^5*sin(x)+1/16*(cos(x)^3+3/2*cos(x))*sin(x)+3/32*x

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Maxima [A]  time = 0.959757, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{32} \, x + \frac{1}{256} \, \sin \left (8 \, x\right ) - \frac{1}{32} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*x + 1/256*sin(8*x) - 1/32*sin(4*x)

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Fricas [A]  time = 2.29601, size = 100, normalized size = 2.17 \begin{align*} \frac{1}{32} \,{\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{32} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/32*x

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Sympy [A]  time = 0.069102, size = 31, normalized size = 0.67 \begin{align*} \frac{3 x}{32} - \frac{\sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{32} - \frac{3 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{64} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/32 - sin(2*x)**3*cos(2*x)/32 - 3*sin(2*x)*cos(2*x)/64

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Giac [A]  time = 1.07825, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{32} \, x + \frac{1}{256} \, \sin \left (8 \, x\right ) - \frac{1}{32} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*x + 1/256*sin(8*x) - 1/32*sin(4*x)