### 3.73 $$\int \cos ^4(x) \sin ^4(x) \, dx$$

Optimal. Leaf size=46 $\frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x)$

[Out]

(3*x)/128 + (3*Cos[x]*Sin[x])/128 + (Cos[x]^3*Sin[x])/64 - (Cos[x]^5*Sin[x])/16 - (Cos[x]^5*Sin[x]^3)/8

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Rubi [A]  time = 0.0516621, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {2568, 2635, 8} $\frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*x)/128 + (3*Cos[x]*Sin[x])/128 + (Cos[x]^3*Sin[x])/64 - (Cos[x]^5*Sin[x])/16 - (Cos[x]^5*Sin[x]^3)/8

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(x) \sin ^4(x) \, dx &=-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3}{8} \int \cos ^4(x) \sin ^2(x) \, dx\\ &=-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{1}{16} \int \cos ^4(x) \, dx\\ &=\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3}{64} \int \cos ^2(x) \, dx\\ &=\frac{3}{128} \cos (x) \sin (x)+\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3}{128} \cos (x) \sin (x)+\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0066726, size = 22, normalized size = 0.48 $\frac{3 x}{128}-\frac{1}{128} \sin (4 x)+\frac{\sin (8 x)}{1024}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{3}}{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{16}}+{\frac{\sin \left ( x \right ) }{64} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{128}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*cos(x)^5*sin(x)^3-1/16*cos(x)^5*sin(x)+1/64*(cos(x)^3+3/2*cos(x))*sin(x)+3/128*x

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Maxima [A]  time = 0.929238, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{1}{1024} \, \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(x)^4*sin(x)^4,x, algorithm="maxima")

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)

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Fricas [A]  time = 2.10534, size = 103, normalized size = 2.24 \begin{align*} \frac{1}{128} \,{\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}{128} \, x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/128*x

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Sympy [A]  time = 0.067331, size = 31, normalized size = 0.67 \begin{align*} \frac{3 x}{128} - \frac{\sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{128} - \frac{3 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{256} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/128 - sin(2*x)**3*cos(2*x)/128 - 3*sin(2*x)*cos(2*x)/256

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Giac [A]  time = 1.04786, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{1}{1024} \, \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(x)^4*sin(x)^4,x, algorithm="giac")

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)