3.70 \(\int \cos ^6(x) \, dx\)

Optimal. Leaf size=34 \[ \frac{5 x}{16}+\frac{1}{6} \sin (x) \cos ^5(x)+\frac{5}{24} \sin (x) \cos ^3(x)+\frac{5}{16} \sin (x) \cos (x) \]

[Out]

(5*x)/16 + (5*Cos[x]*Sin[x])/16 + (5*Cos[x]^3*Sin[x])/24 + (Cos[x]^5*Sin[x])/6

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Rubi [A]  time = 0.0168791, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{5 x}{16}+\frac{1}{6} \sin (x) \cos ^5(x)+\frac{5}{24} \sin (x) \cos ^3(x)+\frac{5}{16} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6,x]

[Out]

(5*x)/16 + (5*Cos[x]*Sin[x])/16 + (5*Cos[x]^3*Sin[x])/24 + (Cos[x]^5*Sin[x])/6

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 ^6(x) \, dx &=\frac{1}{6} \cos ^5(x) \sin (x)+\frac{5}{6} \int \cos ^4(x) \, dx\\ &=\frac{5}{24} \cos ^3(x) \sin (x)+\frac{1}{6} \cos ^5(x) \sin (x)+\frac{5}{8} \int \cos ^2(x) \, dx\\ &=\frac{5}{16} \cos (x) \sin (x)+\frac{5}{24} \cos ^3(x) \sin (x)+\frac{1}{6} \cos ^5(x) \sin (x)+\frac{5 \int 1 \, dx}{16}\\ &=\frac{5 x}{16}+\frac{5}{16} \cos (x) \sin (x)+\frac{5}{24} \cos ^3(x) \sin (x)+\frac{1}{6} \cos ^5(x) \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0018655, size = 30, normalized size = 0.88 \[ \frac{5 x}{16}+\frac{15}{64} \sin (2 x)+\frac{3}{64} \sin (4 x)+\frac{1}{192} \sin (6 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6,x]

[Out]

(5*x)/16 + (15*Sin[2*x])/64 + (3*Sin[4*x])/64 + Sin[6*x]/192

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Maple [A]  time = 0.029, size = 24, normalized size = 0.7 \begin{align*}{\frac{\sin \left ( x \right ) }{6} \left ( \left ( \cos \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6,x)

[Out]

1/6*(cos(x)^5+5/4*cos(x)^3+15/8*cos(x))*sin(x)+5/16*x

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Maxima [A]  time = 0.927446, size = 32, normalized size = 0.94 \begin{align*} -\frac{1}{48} \, \sin \left (2 \, x\right )^{3} + \frac{5}{16} \, x + \frac{3}{64} \, \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)^6,x, algorithm="maxima")

[Out]

-1/48*sin(2*x)^3 + 5/16*x + 3/64*sin(4*x) + 1/4*sin(2*x)

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Fricas [A]  time = 2.05129, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{48} \,{\left (8 \, \cos \left (x\right )^{5} + 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{5}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6,x, algorithm="fricas")

[Out]

1/48*(8*cos(x)^5 + 10*cos(x)^3 + 15*cos(x))*sin(x) + 5/16*x

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Sympy [A]  time = 0.057272, size = 36, normalized size = 1.06 \begin{align*} \frac{5 x}{16} + \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{6} + \frac{5 \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{24} + \frac{5 \sin{\left (x \right )} \cos{\left (x \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**6,x)

[Out]

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

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Giac [A]  time = 1.04698, size = 30, normalized size = 0.88 \begin{align*} \frac{5}{16} \, x + \frac{1}{192} \, \sin \left (6 \, x\right ) + \frac{3}{64} \, \sin \left (4 \, x\right ) + \frac{15}{64} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6,x, algorithm="giac")

[Out]

5/16*x + 1/192*sin(6*x) + 3/64*sin(4*x) + 15/64*sin(2*x)