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

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 = 0.0129631, 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, Rules used = {2635, 8} \[ \frac{3 x}{8}+\frac{1}{4} \sin (x) \cos ^3(x)+\frac{3}{8} \sin (x) \cos (x) \]

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^4(x) \, dx &=\frac{1}{4} \cos ^3(x) \sin (x)+\frac{3}{4} \int \cos ^2(x) \, dx\\ &=\frac{3}{8} \cos (x) \sin (x)+\frac{1}{4} \cos ^3(x) \sin (x)+\frac{3 \int 1 \, dx}{8}\\ &=\frac{3 x}{8}+\frac{3}{8} \cos (x) \sin (x)+\frac{1}{4} \cos ^3(x) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0018096, 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 = 0.001, size = 18, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

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 = 0.952741, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

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 = 2.08891, size = 59, normalized size = 2.46 \begin{align*} \frac{1}{8} \,{\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{8} \, x \end{align*}

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 = 0.058508, size = 24, normalized size = 1. \begin{align*} \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} \end{align*}

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 [A]  time = 1.05795, size = 22, normalized size = 0.92 \begin{align*} \frac{3}{8} \, x + \frac{1}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

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)