3.73 \(\int \cos ^4(x) \sin ^4(x) \, dx\)

Optimal. Leaf size=46 \[ \frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x) \]

[Out]

(3*x)/128 + (3*Cos[x]*Sin[x])/128 + (Cos[x]^3*Sin[x])/64 - (Cos[x]^5*Sin[x])/16
- (Cos[x]^5*Sin[x]^3)/8

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Rubi [A]  time = 0.0794012, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^4*Sin[x]^4,x]

[Out]

(3*x)/128 + (3*Cos[x]*Sin[x])/128 + (Cos[x]^3*Sin[x])/64 - (Cos[x]^5*Sin[x])/16
- (Cos[x]^5*Sin[x]^3)/8

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Rubi in Sympy [A]  time = 2.72374, size = 46, normalized size = 1. \[ \frac{3 x}{128} - \frac{\sin ^{3}{\left (x \right )} \cos ^{5}{\left (x \right )}}{8} - \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{16} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{64} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x/128 - sin(x)**3*cos(x)**5/8 - sin(x)*cos(x)**5/16 + sin(x)*cos(x)**3/64 + 3*
sin(x)*cos(x)/128

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Mathematica [A]  time = 0.0094539, size = 22, normalized size = 0.48 \[ \frac{3 x}{128}-\frac{1}{128} \sin (4 x)+\frac{\sin (8 x)}{1024} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^4*Sin[x]^4,x]

[Out]

(3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024

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Maple [A]  time = 0.011, size = 36, normalized size = 0.8 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{3}}{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{16}}+{\frac{\sin \left ( x \right ) }{64} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{128}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^4*sin(x)^4,x)

[Out]

-1/8*cos(x)^5*sin(x)^3-1/16*cos(x)^5*sin(x)+1/64*(cos(x)^3+3/2*cos(x))*sin(x)+3/
128*x

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Maxima [A]  time = 1.36931, size = 22, normalized size = 0.48 \[ \frac{3}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{128} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^4,x, algorithm="maxima")

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)

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Fricas [A]  time = 0.230924, size = 42, normalized size = 0.91 \[ \frac{1}{128} \,{\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{128} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^4,x, algorithm="fricas")

[Out]

1/128*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/128*x

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Sympy [A]  time = 0.055402, size = 31, normalized size = 0.67 \[ \frac{3 x}{128} - \frac{\sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{128} - \frac{3 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**4*sin(x)**4,x)

[Out]

3*x/128 - sin(2*x)**3*cos(2*x)/128 - 3*sin(2*x)*cos(2*x)/256

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GIAC/XCAS [A]  time = 0.205748, size = 22, normalized size = 0.48 \[ \frac{3}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{128} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^4,x, algorithm="giac")

[Out]

3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)