\(\int \cos ^6(x) \sin ^4(x) \, dx\) [344]

Optimal result

Integrand size = 9, antiderivative size = 56 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256}+\frac {3}{256} \cos (x) \sin (x)+\frac {1}{128} \cos ^3(x) \sin (x)+\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x) \]

[Out]

3/256*x+3/256*cos(x)*sin(x)+1/128*cos(x)^3*sin(x)+1/160*cos(x)^5*sin(x)-3/80*cos(x)^7*sin(x)-1/10*cos(x)^7*sin
(x)^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2648, 2715, 8} \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256}-\frac {1}{10} \sin ^3(x) \cos ^7(x)-\frac {3}{80} \sin (x) \cos ^7(x)+\frac {1}{160} \sin (x) \cos ^5(x)+\frac {1}{128} \sin (x) \cos ^3(x)+\frac {3}{256} \sin (x) \cos (x) \]

[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 2648

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 2715

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] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256}+\frac {1}{512} \sin (2 x)-\frac {1}{256} \sin (4 x)-\frac {\sin (6 x)}{1024}+\frac {\sin (8 x)}{2048}+\frac {\sin (10 x)}{5120} \]

[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

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {1}{1280} \, {\left (128 \, \cos \left (x\right )^{9} - 176 \, \cos \left (x\right )^{7} + 8 \, \cos \left (x\right )^{5} + 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{256} \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256} + \frac {\sin {\left (x \right )} \cos ^{9}{\left (x \right )}}{10} - \frac {11 \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{80} + \frac {\sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{160} + \frac {\sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + \frac {3 \sin {\left (x \right )} \cos {\left (x \right )}}{256} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.43 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {1}{320} \, \sin \left (2 \, x\right )^{5} + \frac {3}{256} \, x + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3}{256} \, x + \frac {1}{5120} \, \sin \left (10 \, x\right ) + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{1024} \, \sin \left (6 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) + \frac {1}{512} \, \sin \left (2 \, x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\left (\frac {{\cos \left (x\right )}^5}{10}+\frac {{\cos \left (x\right )}^3}{16}+\frac {\cos \left (x\right )}{32}\right )\,{\sin \left (x\right )}^5+\frac {3\,x}{256}-\frac {\sin \left (2\,x\right )}{128}+\frac {\sin \left (4\,x\right )}{1024} \]

[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)