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

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

[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

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Rubi [A]  time = 0.0623641, antiderivative size = 56, normalized size of antiderivative = 1., 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 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 ^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*}

Mathematica [A]  time = 0.0154152, 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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Maple [A]  time = 0.008, size = 42, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{3}}{10}}-{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) }{80}}+{\frac{\sin \left ( x \right ) }{160} \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{3\,x}{256}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*cos(x)^7*sin(x)^3-3/80*cos(x)^7*sin(x)+1/160*(cos(x)^5+5/4*cos(x)^3+15/8*cos(x))*sin(x)+3/256*x

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Maxima [A]  time = 0.930434, size = 32, normalized size = 0.57 \begin{align*} \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 ) \end{align*}

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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Fricas [A]  time = 1.98124, size = 127, normalized size = 2.27 \begin{align*} \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 \end{align*}

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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Sympy [A]  time = 0.06141, size = 56, normalized size = 1. \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.05848, size = 46, normalized size = 0.82 \begin{align*} \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 ) \end{align*}

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)