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

Optimal. Leaf size=36 \[ \frac{x}{16}-\frac{1}{6} \sin ^3(x) \cos ^3(x)-\frac{1}{8} \sin (x) \cos ^3(x)+\frac{1}{16} \sin (x) \cos (x) \]

[Out]

x/16 + (Cos[x]*Sin[x])/16 - (Cos[x]^3*Sin[x])/8 - (Cos[x]^3*Sin[x]^3)/6

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Rubi [A]  time = 0.0698417, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x}{16}-\frac{1}{6} \sin ^3(x) \cos ^3(x)-\frac{1}{8} \sin (x) \cos ^3(x)+\frac{1}{16} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^2*Sin[x]^4,x]

[Out]

x/16 + (Cos[x]*Sin[x])/16 - (Cos[x]^3*Sin[x])/8 - (Cos[x]^3*Sin[x]^3)/6

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Rubi in Sympy [A]  time = 2.65918, size = 32, normalized size = 0.89 \[ \frac{x}{16} - \frac{\sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/16 - sin(x)**3*cos(x)**3/6 - sin(x)*cos(x)**3/8 + sin(x)*cos(x)/16

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Mathematica [A]  time = 0.0125056, size = 30, normalized size = 0.83 \[ \frac{x}{16}-\frac{1}{64} \sin (2 x)-\frac{1}{64} \sin (4 x)+\frac{1}{192} \sin (6 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^2*Sin[x]^4,x]

[Out]

x/16 - Sin[2*x]/64 - Sin[4*x]/64 + Sin[6*x]/192

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Maple [A]  time = 0.01, size = 29, normalized size = 0.8 \[{\frac{x}{16}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{16}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{3}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2*sin(x)^4,x)

[Out]

1/16*x+1/16*cos(x)*sin(x)-1/8*cos(x)^3*sin(x)-1/6*cos(x)^3*sin(x)^3

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Maxima [A]  time = 1.34965, size = 24, normalized size = 0.67 \[ -\frac{1}{48} \, \sin \left (2 \, x\right )^{3} + \frac{1}{16} \, x - \frac{1}{64} \, \sin \left (4 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*sin(2*x)^3 + 1/16*x - 1/64*sin(4*x)

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Fricas [A]  time = 0.259057, size = 34, normalized size = 0.94 \[ \frac{1}{48} \,{\left (8 \, \cos \left (x\right )^{5} - 14 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{16} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(8*cos(x)^5 - 14*cos(x)^3 + 3*cos(x))*sin(x) + 1/16*x

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Sympy [A]  time = 0.041135, size = 31, normalized size = 0.86 \[ \frac{x}{16} + \frac{\sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{6} - \frac{\sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{24} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/16 + sin(x)**5*cos(x)/6 - sin(x)**3*cos(x)/24 - sin(x)*cos(x)/16

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GIAC/XCAS [A]  time = 0.20482, size = 30, normalized size = 0.83 \[ \frac{1}{16} \, x + \frac{1}{192} \, \sin \left (6 \, x\right ) - \frac{1}{64} \, \sin \left (4 \, x\right ) - \frac{1}{64} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*x + 1/192*sin(6*x) - 1/64*sin(4*x) - 1/64*sin(2*x)