\(\int \cos ^5(x) \, dx\) [74]

Optimal result

Integrand size = 4, antiderivative size = 19 \[ \int \cos ^5(x) \, dx=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \cos ^5(x) \, dx=\frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]

[In]

Int[Cos[x]^5,x]

[Out]

Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5

Rule 2713

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*} \text {integral}& = -\text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \cos ^5(x) \, dx=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5} \]

[In]

Integrate[Cos[x]^5,x]

[Out]

Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {\left (\frac {8}{3}+\cos ^{4}\left (x \right )+\frac {4 \left (\cos ^{2}\left (x \right )\right )}{3}\right ) \sin \left (x \right )}{5}\)	\(17\)
risch	\(\frac {5 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\)	\(18\)
parallelrisch	\(\frac {5 \sin \left (x \right )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\)	\(18\)

[In]

int(cos(x)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \cos ^5(x) \, dx=\frac {1}{15} \, {\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \cos ^5(x) \, dx=\frac {\sin ^{5}{\left (x \right )}}{5} - \frac {2 \sin ^{3}{\left (x \right )}}{3} + \sin {\left (x \right )} \]

[In]

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

[Out]

sin(x)**5/5 - 2*sin(x)**3/3 + sin(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \cos ^5(x) \, dx=\frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

[In]

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

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \cos ^5(x) \, dx=\frac {1}{5} \, \sin \left (x\right )^{5} - \frac {2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

[In]

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

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \cos ^5(x) \, dx=\frac {\sin \left (x\right )\,{\cos \left (x\right )}^4}{5}+\frac {4\,\sin \left (x\right )\,{\cos \left (x\right )}^2}{15}+\frac {8\,\sin \left (x\right )}{15} \]

[In]

int(cos(x)^5,x)

[Out]

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