3.38 \(\int \cos ^4(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0185558, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{3 x}{8}+\frac{1}{4} \sin (x) \cos ^3(x)+\frac{3}{8} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 0.59117, size = 24, normalized size = 1. \[ \frac{3 x}{8} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**4,x)

[Out]

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

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Mathematica [A]  time = 0.00288593, size = 22, normalized size = 0.92 \[ \frac{3 x}{8}+\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^4,x]

[Out]

(3*x)/8 + Sin[2*x]/4 + Sin[4*x]/32

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Maple [A]  time = 0.062, size = 18, normalized size = 0.8 \[{\frac{\sin \left ( x \right ) }{4} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^4,x)

[Out]

1/4*(cos(x)^3+3/2*cos(x))*sin(x)+3/8*x

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Maxima [A]  time = 1.35768, size = 22, normalized size = 0.92 \[ \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*x + 1/32*sin(4*x) + 1/4*sin(2*x)

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Fricas [A]  time = 0.246385, size = 26, normalized size = 1.08 \[ \frac{1}{8} \,{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{8} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.03856, size = 24, normalized size = 1. \[ \frac{3 x}{8} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.236757, size = 22, normalized size = 0.92 \[ \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*x + 1/32*sin(4*x) + 1/4*sin(2*x)