3.350 \(\int \cos ^6(x) \sin ^6(x) \, dx\)

Optimal. Leaf size=68 \[ \frac{5 x}{1024}-\frac{1}{12} \sin ^5(x) \cos ^7(x)-\frac{1}{24} \sin ^3(x) \cos ^7(x)-\frac{1}{64} \sin (x) \cos ^7(x)+\frac{1}{384} \sin (x) \cos ^5(x)+\frac{5 \sin (x) \cos ^3(x)}{1536}+\frac{5 \sin (x) \cos (x)}{1024} \]

[Out]

(5*x)/1024 + (5*Cos[x]*Sin[x])/1024 + (5*Cos[x]^3*Sin[x])/1536 + (Cos[x]^5*Sin[x])/384 - (Cos[x]^7*Sin[x])/64
- (Cos[x]^7*Sin[x]^3)/24 - (Cos[x]^7*Sin[x]^5)/12

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Rubi [A]  time = 0.084813, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{5 x}{1024}-\frac{1}{12} \sin ^5(x) \cos ^7(x)-\frac{1}{24} \sin ^3(x) \cos ^7(x)-\frac{1}{64} \sin (x) \cos ^7(x)+\frac{1}{384} \sin (x) \cos ^5(x)+\frac{5 \sin (x) \cos ^3(x)}{1536}+\frac{5 \sin (x) \cos (x)}{1024} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6*Sin[x]^6,x]

[Out]

(5*x)/1024 + (5*Cos[x]*Sin[x])/1024 + (5*Cos[x]^3*Sin[x])/1536 + (Cos[x]^5*Sin[x])/384 - (Cos[x]^7*Sin[x])/64
- (Cos[x]^7*Sin[x]^3)/24 - (Cos[x]^7*Sin[x]^5)/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 ^6(x) \sin ^6(x) \, dx &=-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{12} \int \cos ^6(x) \sin ^4(x) \, dx\\ &=-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{1}{8} \int \cos ^6(x) \sin ^2(x) \, dx\\ &=-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{1}{64} \int \cos ^6(x) \, dx\\ &=\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{384} \int \cos ^4(x) \, dx\\ &=\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{512} \int \cos ^2(x) \, dx\\ &=\frac{5 \cos (x) \sin (x)}{1024}+\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5 \int 1 \, dx}{1024}\\ &=\frac{5 x}{1024}+\frac{5 \cos (x) \sin (x)}{1024}+\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)\\ \end{align*}

Mathematica [A]  time = 0.0152357, size = 30, normalized size = 0.44 \[ \frac{5 x}{1024}-\frac{15 \sin (4 x)}{8192}+\frac{3 \sin (8 x)}{8192}-\frac{\sin (12 x)}{24576} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6*Sin[x]^6,x]

[Out]

(5*x)/1024 - (15*Sin[4*x])/8192 + (3*Sin[8*x])/8192 - Sin[12*x]/24576

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Maple [A]  time = 0.008, size = 52, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{5}}{12}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) }{64}}+{\frac{\sin \left ( x \right ) }{384} \left ( \left ( \cos \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( x \right ) }{8}} \right ) }+{\frac{5\,x}{1024}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6*sin(x)^6,x)

[Out]

-1/12*cos(x)^7*sin(x)^5-1/24*cos(x)^7*sin(x)^3-1/64*cos(x)^7*sin(x)+1/384*(cos(x)^5+5/4*cos(x)^3+15/8*cos(x))*
sin(x)+5/1024*x

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Maxima [A]  time = 0.938378, size = 32, normalized size = 0.47 \begin{align*} \frac{1}{6144} \, \sin \left (4 \, x\right )^{3} + \frac{5}{1024} \, x + \frac{3}{8192} \, \sin \left (8 \, x\right ) - \frac{1}{512} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^6,x, algorithm="maxima")

[Out]

1/6144*sin(4*x)^3 + 5/1024*x + 3/8192*sin(8*x) - 1/512*sin(4*x)

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Fricas [A]  time = 2.01354, size = 151, normalized size = 2.22 \begin{align*} -\frac{1}{3072} \,{\left (256 \, \cos \left (x\right )^{11} - 640 \, \cos \left (x\right )^{9} + 432 \, \cos \left (x\right )^{7} - 8 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} - 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{5}{1024} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^6,x, algorithm="fricas")

[Out]

-1/3072*(256*cos(x)^11 - 640*cos(x)^9 + 432*cos(x)^7 - 8*cos(x)^5 - 10*cos(x)^3 - 15*cos(x))*sin(x) + 5/1024*x

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Sympy [A]  time = 0.06577, size = 46, normalized size = 0.68 \begin{align*} \frac{5 x}{1024} - \frac{\sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{768} - \frac{5 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{3072} - \frac{5 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{2048} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**6*sin(x)**6,x)

[Out]

5*x/1024 - sin(2*x)**5*cos(2*x)/768 - 5*sin(2*x)**3*cos(2*x)/3072 - 5*sin(2*x)*cos(2*x)/2048

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Giac [A]  time = 1.05155, size = 30, normalized size = 0.44 \begin{align*} \frac{5}{1024} \, x - \frac{1}{24576} \, \sin \left (12 \, x\right ) + \frac{3}{8192} \, \sin \left (8 \, x\right ) - \frac{15}{8192} \, \sin \left (4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^6,x, algorithm="giac")

[Out]

5/1024*x - 1/24576*sin(12*x) + 3/8192*sin(8*x) - 15/8192*sin(4*x)