### 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*Sin[x]^3)/3 + Sin[x]^5/5

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Rubi [A]  time = 0.0086871, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, 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 = 0.0019552, 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

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Maple [A]  time = 0.027, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

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)

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Maxima [A]  time = 0.942157, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}

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)

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Fricas [A]  time = 1.9634, size = 58, normalized size = 3.05 \begin{align*} \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \end{align*}

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)

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Sympy [A]  time = 0.058083, size = 17, normalized size = 0.89 \begin{align*} \frac{\sin ^{5}{\left (x \right )}}{5} - \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \end{align*}

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)

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Giac [A]  time = 1.07823, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}

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)