3.65 \(\int \cos ^2(x) \sin ^2(x) \, dx\)

Optimal. Leaf size=24 \[ \frac{x}{8}-\frac{1}{4} \sin (x) \cos ^3(x)+\frac{1}{8} \sin (x) \cos (x) \]

[Out]

x/8 + (Cos[x]*Sin[x])/8 - (Cos[x]^3*Sin[x])/4

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Rubi [A]  time = 0.0387615, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x}{8}-\frac{1}{4} \sin (x) \cos ^3(x)+\frac{1}{8} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^2*Sin[x]^2,x]

[Out]

x/8 + (Cos[x]*Sin[x])/8 - (Cos[x]^3*Sin[x])/4

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Rubi in Sympy [A]  time = 1.62506, size = 20, normalized size = 0.83 \[ \frac{x}{8} - \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**2*sin(x)**2,x)

[Out]

x/8 - sin(x)*cos(x)**3/4 + sin(x)*cos(x)/8

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Mathematica [A]  time = 0.00647966, size = 14, normalized size = 0.58 \[ \frac{x}{8}-\frac{1}{32} \sin (4 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^2*Sin[x]^2,x]

[Out]

x/8 - Sin[4*x]/32

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Maple [A]  time = 0.002, size = 19, normalized size = 0.8 \[{\frac{x}{8}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2*sin(x)^2,x)

[Out]

1/8*x+1/8*cos(x)*sin(x)-1/4*cos(x)^3*sin(x)

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Maxima [A]  time = 1.34772, size = 14, normalized size = 0.58 \[ \frac{1}{8} \, x - \frac{1}{32} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^2,x, algorithm="maxima")

[Out]

1/8*x - 1/32*sin(4*x)

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Fricas [A]  time = 0.221972, size = 26, normalized size = 1.08 \[ -\frac{1}{8} \,{\left (2 \, \cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{8} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^2,x, algorithm="fricas")

[Out]

-1/8*(2*cos(x)^3 - cos(x))*sin(x) + 1/8*x

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Sympy [A]  time = 0.052749, size = 14, normalized size = 0.58 \[ \frac{x}{8} - \frac{\sin{\left (2 x \right )} \cos{\left (2 x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**2*sin(x)**2,x)

[Out]

x/8 - sin(2*x)*cos(2*x)/16

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GIAC/XCAS [A]  time = 0.219669, size = 14, normalized size = 0.58 \[ \frac{1}{8} \, x - \frac{1}{32} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^2,x, algorithm="giac")

[Out]

1/8*x - 1/32*sin(4*x)