3.71 \(\int \cos ^4(2 x) \sin ^2(2 x) \, dx\)

Optimal. Leaf size=46 \[ \frac{x}{16}-\frac{1}{12} \sin (2 x) \cos ^5(2 x)+\frac{1}{48} \sin (2 x) \cos ^3(2 x)+\frac{1}{32} \sin (2 x) \cos (2 x) \]

[Out]

x/16 + (Cos[2*x]*Sin[2*x])/32 + (Cos[2*x]^3*Sin[2*x])/48 - (Cos[2*x]^5*Sin[2*x])
/12

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Rubi [A]  time = 0.0575614, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x}{16}-\frac{1}{12} \sin (2 x) \cos ^5(2 x)+\frac{1}{48} \sin (2 x) \cos ^3(2 x)+\frac{1}{32} \sin (2 x) \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[2*x]^4*Sin[2*x]^2,x]

[Out]

x/16 + (Cos[2*x]*Sin[2*x])/32 + (Cos[2*x]^3*Sin[2*x])/48 - (Cos[2*x]^5*Sin[2*x])
/12

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Rubi in Sympy [A]  time = 1.8985, size = 41, normalized size = 0.89 \[ \frac{x}{16} - \frac{\sin{\left (2 x \right )} \cos ^{5}{\left (2 x \right )}}{12} + \frac{\sin{\left (2 x \right )} \cos ^{3}{\left (2 x \right )}}{48} + \frac{\sin{\left (2 x \right )} \cos{\left (2 x \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(2*x)**4*sin(2*x)**2,x)

[Out]

x/16 - sin(2*x)*cos(2*x)**5/12 + sin(2*x)*cos(2*x)**3/48 + sin(2*x)*cos(2*x)/32

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Mathematica [A]  time = 0.0206472, size = 30, normalized size = 0.65 \[ \frac{x}{16}+\frac{1}{128} \sin (4 x)-\frac{1}{128} \sin (8 x)-\frac{1}{384} \sin (12 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[2*x]^4*Sin[2*x]^2,x]

[Out]

x/16 + Sin[4*x]/128 - Sin[8*x]/128 - Sin[12*x]/384

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Maple [A]  time = 0.019, size = 36, normalized size = 0.8 \[ -{\frac{ \left ( \cos \left ( 2\,x \right ) \right ) ^{5}\sin \left ( 2\,x \right ) }{12}}+{\frac{\sin \left ( 2\,x \right ) }{48} \left ( \left ( \cos \left ( 2\,x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( 2\,x \right ) }{2}} \right ) }+{\frac{x}{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(2*x)^4*sin(2*x)^2,x)

[Out]

-1/12*cos(2*x)^5*sin(2*x)+1/48*(cos(2*x)^3+3/2*cos(2*x))*sin(2*x)+1/16*x

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Maxima [A]  time = 1.33205, size = 24, normalized size = 0.52 \[ \frac{1}{96} \, \sin \left (4 \, x\right )^{3} + \frac{1}{16} \, x - \frac{1}{128} \, \sin \left (8 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="maxima")

[Out]

1/96*sin(4*x)^3 + 1/16*x - 1/128*sin(8*x)

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Fricas [A]  time = 0.234865, size = 45, normalized size = 0.98 \[ -\frac{1}{96} \,{\left (8 \, \cos \left (2 \, x\right )^{5} - 2 \, \cos \left (2 \, x\right )^{3} - 3 \, \cos \left (2 \, x\right )\right )} \sin \left (2 \, x\right ) + \frac{1}{16} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="fricas")

[Out]

-1/96*(8*cos(2*x)^5 - 2*cos(2*x)^3 - 3*cos(2*x))*sin(2*x) + 1/16*x

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Sympy [A]  time = 0.046342, size = 41, normalized size = 0.89 \[ \frac{x}{16} - \frac{\sin{\left (2 x \right )} \cos ^{5}{\left (2 x \right )}}{12} + \frac{\sin{\left (2 x \right )} \cos ^{3}{\left (2 x \right )}}{48} + \frac{\sin{\left (2 x \right )} \cos{\left (2 x \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)**4*sin(2*x)**2,x)

[Out]

x/16 - sin(2*x)*cos(2*x)**5/12 + sin(2*x)*cos(2*x)**3/48 + sin(2*x)*cos(2*x)/32

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GIAC/XCAS [A]  time = 0.199566, size = 30, normalized size = 0.65 \[ \frac{1}{16} \, x - \frac{1}{384} \, \sin \left (12 \, x\right ) - \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{1}{128} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(2*x)^4*sin(2*x)^2,x, algorithm="giac")

[Out]

1/16*x - 1/384*sin(12*x) - 1/128*sin(8*x) + 1/128*sin(4*x)