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

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

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Rubi [A]  time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 8

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

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*}

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Mathematica [A]  time = 0.01, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^6(x) \sin ^6(x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [A]  time = 0.79, size = 43, normalized size = 0.63 \[ -\frac {1}{3072} \, {\left (256 \, \cos \relax (x)^{11} - 640 \, \cos \relax (x)^{9} + 432 \, \cos \relax (x)^{7} - 8 \, \cos \relax (x)^{5} - 10 \, \cos \relax (x)^{3} - 15 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {5}{1024} \, x \]

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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giac [A]  time = 0.89, size = 22, normalized size = 0.32 \[ \frac {5}{1024} \, x - \frac {1}{24576} \, \sin \left (12 \, x\right ) + \frac {3}{8192} \, \sin \left (8 \, x\right ) - \frac {15}{8192} \, \sin \left (4 \, x\right ) \]

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)

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maple [A]  time = 0.31, size = 23, normalized size = 0.34

method	result	size
risch	\(\frac {5 x}{1024}-\frac {\sin \left (12 x \right )}{24576}+\frac {3 \sin \left (8 x \right )}{8192}-\frac {15 \sin \left (4 x \right )}{8192}\)	\(23\)
default	\(-\frac {\left (\cos ^{7}\relax (x )\right ) \left (\sin ^{5}\relax (x )\right )}{12}-\frac {\left (\cos ^{7}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{24}-\frac {\left (\cos ^{7}\relax (x )\right ) \sin \relax (x )}{64}+\frac {\left (\cos ^{5}\relax (x )+\frac {5 \left (\cos ^{3}\relax (x )\right )}{4}+\frac {15 \cos \relax (x )}{8}\right ) \sin \relax (x )}{384}+\frac {5 x}{1024}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6*sin(x)^6,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 24, normalized size = 0.35 \[ \frac {1}{6144} \, \sin \left (4 \, x\right )^{3} + \frac {5}{1024} \, x + \frac {3}{8192} \, \sin \left (8 \, x\right ) - \frac {1}{512} \, \sin \left (4 \, x\right ) \]

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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mupad [B]  time = 0.03, size = 44, normalized size = 0.65 \[ \left (\frac {{\cos \relax (x)}^5}{12}+\frac {{\cos \relax (x)}^3}{24}+\frac {\cos \relax (x)}{64}\right )\,{\sin \relax (x)}^7+\frac {5\,x}{1024}-\frac {15\,\sin \left (2\,x\right )}{4096}+\frac {3\,\sin \left (4\,x\right )}{4096}-\frac {\sin \left (6\,x\right )}{12288} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*x)/1024 - (15*sin(2*x))/4096 + (3*sin(4*x))/4096 - sin(6*x)/12288 + sin(x)^7*(cos(x)/64 + cos(x)^3/24 + cos
(x)^5/12)

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sympy [A]  time = 0.07, size = 46, normalized size = 0.68 \[ \frac {5 x}{1024} - \frac {\sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{768} - \frac {5 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{3072} - \frac {5 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{2048} \]

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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