3.344 \(\int \cos ^6(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=56 \[ \frac {3 x}{256}-\frac {1}{10} \sin ^3(x) \cos ^7(x)-\frac {3}{80} \sin (x) \cos ^7(x)+\frac {1}{160} \sin (x) \cos ^5(x)+\frac {1}{128} \sin (x) \cos ^3(x)+\frac {3}{256} \sin (x) \cos (x) \]

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Rubi [A]  time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac {3 x}{256}-\frac {1}{10} \sin ^3(x) \cos ^7(x)-\frac {3}{80} \sin (x) \cos ^7(x)+\frac {1}{160} \sin (x) \cos ^5(x)+\frac {1}{128} \sin (x) \cos ^3(x)+\frac {3}{256} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6*Sin[x]^4,x]

[Out]

(3*x)/256 + (3*Cos[x]*Sin[x])/256 + (Cos[x]^3*Sin[x])/128 + (Cos[x]^5*Sin[x])/160 - (3*Cos[x]^7*Sin[x])/80 - (
Cos[x]^7*Sin[x]^3)/10

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 ^4(x) \, dx &=-\frac {1}{10} \cos ^7(x) \sin ^3(x)+\frac {3}{10} \int \cos ^6(x) \sin ^2(x) \, dx\\ &=-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x)+\frac {3}{80} \int \cos ^6(x) \, dx\\ &=\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x)+\frac {1}{32} \int \cos ^4(x) \, dx\\ &=\frac {1}{128} \cos ^3(x) \sin (x)+\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x)+\frac {3}{128} \int \cos ^2(x) \, dx\\ &=\frac {3}{256} \cos (x) \sin (x)+\frac {1}{128} \cos ^3(x) \sin (x)+\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x)+\frac {3 \int 1 \, dx}{256}\\ &=\frac {3 x}{256}+\frac {3}{256} \cos (x) \sin (x)+\frac {1}{128} \cos ^3(x) \sin (x)+\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 46, normalized size = 0.82 \[ \frac {3 x}{256}+\frac {1}{512} \sin (2 x)-\frac {1}{256} \sin (4 x)-\frac {\sin (6 x)}{1024}+\frac {\sin (8 x)}{2048}+\frac {\sin (10 x)}{5120} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6*Sin[x]^4,x]

[Out]

(3*x)/256 + Sin[2*x]/512 - Sin[4*x]/256 - Sin[6*x]/1024 + Sin[8*x]/2048 + Sin[10*x]/5120

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^6(x) \sin ^4(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]^6*Sin[x]^4,x]

[Out]

Could not integrate

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fricas [A]  time = 0.97, size = 37, normalized size = 0.66 \[ \frac {1}{1280} \, {\left (128 \, \cos \relax (x)^{9} - 176 \, \cos \relax (x)^{7} + 8 \, \cos \relax (x)^{5} + 10 \, \cos \relax (x)^{3} + 15 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {3}{256} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^4,x, algorithm="fricas")

[Out]

1/1280*(128*cos(x)^9 - 176*cos(x)^7 + 8*cos(x)^5 + 10*cos(x)^3 + 15*cos(x))*sin(x) + 3/256*x

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giac [A]  time = 0.77, size = 34, normalized size = 0.61 \[ \frac {3}{256} \, x + \frac {1}{5120} \, \sin \left (10 \, x\right ) + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{1024} \, \sin \left (6 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) + \frac {1}{512} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^4,x, algorithm="giac")

[Out]

3/256*x + 1/5120*sin(10*x) + 1/2048*sin(8*x) - 1/1024*sin(6*x) - 1/256*sin(4*x) + 1/512*sin(2*x)

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maple [A]  time = 0.32, size = 35, normalized size = 0.62

method	result	size
risch	\(\frac {3 x}{256}+\frac {\sin \left (10 x \right )}{5120}+\frac {\sin \left (8 x \right )}{2048}-\frac {\sin \left (6 x \right )}{1024}-\frac {\sin \left (4 x \right )}{256}+\frac {\sin \left (2 x \right )}{512}\)	\(35\)
default	\(-\frac {\left (\cos ^{7}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{10}-\frac {3 \left (\cos ^{7}\relax (x )\right ) \sin \relax (x )}{80}+\frac {\left (\cos ^{5}\relax (x )+\frac {5 \left (\cos ^{3}\relax (x )\right )}{4}+\frac {15 \cos \relax (x )}{8}\right ) \sin \relax (x )}{160}+\frac {3 x}{256}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6*sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

3/256*x+1/5120*sin(10*x)+1/2048*sin(8*x)-1/1024*sin(6*x)-1/256*sin(4*x)+1/512*sin(2*x)

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maxima [A]  time = 0.57, size = 24, normalized size = 0.43 \[ \frac {1}{320} \, \sin \left (2 \, x\right )^{5} + \frac {3}{256} \, x + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^4,x, algorithm="maxima")

[Out]

1/320*sin(2*x)^5 + 3/256*x + 1/2048*sin(8*x) - 1/256*sin(4*x)

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mupad [B]  time = 0.04, size = 38, normalized size = 0.68 \[ \left (\frac {{\cos \relax (x)}^5}{10}+\frac {{\cos \relax (x)}^3}{16}+\frac {\cos \relax (x)}{32}\right )\,{\sin \relax (x)}^5+\frac {3\,x}{256}-\frac {\sin \left (2\,x\right )}{128}+\frac {\sin \left (4\,x\right )}{1024} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6*sin(x)^4,x)

[Out]

(3*x)/256 - sin(2*x)/128 + sin(4*x)/1024 + sin(x)^5*(cos(x)/32 + cos(x)^3/16 + cos(x)^5/10)

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sympy [A]  time = 0.07, size = 56, normalized size = 1.00 \[ \frac {3 x}{256} + \frac {\sin {\relax (x )} \cos ^{9}{\relax (x )}}{10} - \frac {11 \sin {\relax (x )} \cos ^{7}{\relax (x )}}{80} + \frac {\sin {\relax (x )} \cos ^{5}{\relax (x )}}{160} + \frac {\sin {\relax (x )} \cos ^{3}{\relax (x )}}{128} + \frac {3 \sin {\relax (x )} \cos {\relax (x )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**6*sin(x)**4,x)

[Out]

3*x/256 + sin(x)*cos(x)**9/10 - 11*sin(x)*cos(x)**7/80 + sin(x)*cos(x)**5/160 + sin(x)*cos(x)**3/128 + 3*sin(x
)*cos(x)/256

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