3.361 \(\int \cos ^5(x) \, dx\)

Optimal. Leaf size=19 \[ \frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]

[Out]

sin(x)-2/3*sin(x)^3+1/5*sin(x)^5

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2633} \[ \frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^5,x]

[Out]

Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \cos ^5(x) \, dx &=-\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 23, normalized size = 1.21 \[ \frac {5 \sin (x)}{8}+\frac {5}{48} \sin (3 x)+\frac {1}{80} \sin (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^5,x]

[Out]

(5*Sin[x])/8 + (5*Sin[3*x])/48 + Sin[5*x]/80

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 18, normalized size = 0.95 \[ \frac {1}{15} \, {\left (3 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{2} + 8\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5,x, algorithm="fricas")

[Out]

1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)

________________________________________________________________________________________

giac [A]  time = 1.05, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} - \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5,x, algorithm="giac")

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 17, normalized size = 0.89 \[ \frac {\left (\cos ^{4}\relax (x )+\frac {4 \left (\cos ^{2}\relax (x )\right )}{3}+\frac {8}{3}\right ) \sin \relax (x )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5,x)

[Out]

1/5*(8/3+cos(x)^4+4/3*cos(x)^2)*sin(x)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} - \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5,x, algorithm="maxima")

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 21, normalized size = 1.11 \[ \frac {\sin \relax (x)\,{\cos \relax (x)}^4}{5}+\frac {4\,\sin \relax (x)\,{\cos \relax (x)}^2}{15}+\frac {8\,\sin \relax (x)}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5,x)

[Out]

(8*sin(x))/15 + (4*cos(x)^2*sin(x))/15 + (cos(x)^4*sin(x))/5

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 17, normalized size = 0.89 \[ \frac {\sin ^{5}{\relax (x )}}{5} - \frac {2 \sin ^{3}{\relax (x )}}{3} + \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**5,x)

[Out]

sin(x)**5/5 - 2*sin(x)**3/3 + sin(x)

________________________________________________________________________________________