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])/16
0 - (3*Cos[x]^7*Sin[x])/80 - (Cos[x]^7*Sin[x]^3)/10

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Rubi [A]  time = 0.0979612, 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 \[ \frac{3 x}{256}-\frac{1}{10} \sin ^3(x) \cos ^7(x)-\frac{3}{80} \sin (x) \cos ^7(x)+\frac{1}{160} \sin (x) \cos ^5(x)+\frac{1}{128} \sin (x) \cos ^3(x)+\frac{3}{256} \sin (x) \cos (x) \]

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])/16
0 - (3*Cos[x]^7*Sin[x])/80 - (Cos[x]^7*Sin[x]^3)/10

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Rubi in Sympy [A]  time = 3.10782, size = 58, normalized size = 1.04 \[ \frac{3 x}{256} - \frac{\sin ^{3}{\left (x \right )} \cos ^{7}{\left (x \right )}}{10} - \frac{3 \sin{\left (x \right )} \cos ^{7}{\left (x \right )}}{80} + \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{160} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**6*sin(x)**4,x)

[Out]

3*x/256 - sin(x)**3*cos(x)**7/10 - 3*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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Mathematica [A]  time = 0.0234356, 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 = 0.009, size = 42, normalized size = 0.8 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{3}}{10}}-{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) }{80}}+{\frac{\sin \left ( x \right ) }{160} \left ( \left ( \cos \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( x \right ) }{8}} \right ) }+{\frac{3\,x}{256}} \]

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*c
os(x))*sin(x)+3/256*x

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Maxima [A]  time = 1.36461, size = 32, normalized size = 0.57 \[ \frac{1}{320} \, \sin \left (2 \, x\right )^{5} + \frac{3}{256} \, x + \frac{1}{2048} \, \sin \left (8 \, x\right ) - \frac{1}{256} \, \sin \left (4 \, x\right ) \]

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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Fricas [A]  time = 0.307835, size = 50, normalized size = 0.89 \[ \frac{1}{1280} \,{\left (128 \, \cos \left (x\right )^{9} - 176 \, \cos \left (x\right )^{7} + 8 \, \cos \left (x\right )^{5} + 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{256} \, x \]

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 = 0.046293, size = 56, normalized size = 1. \[ \frac{3 x}{256} + \frac{\sin{\left (x \right )} \cos ^{9}{\left (x \right )}}{10} - \frac{11 \sin{\left (x \right )} \cos ^{7}{\left (x \right )}}{80} + \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{160} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{256} \]

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/XCAS [A]  time = 0.199761, size = 46, normalized size = 0.82 \[ \frac{3}{256} \, x + \frac{1}{5120} \, \sin \left (10 \, x\right ) + \frac{1}{2048} \, \sin \left (8 \, x\right ) - \frac{1}{1024} \, \sin \left (6 \, x\right ) - \frac{1}{256} \, \sin \left (4 \, x\right ) + \frac{1}{512} \, \sin \left (2 \, x\right ) \]

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)