### 3.72 $$\int \sin ^5(x) \, dx$$

Optimal. Leaf size=21 $-\frac{1}{5} \cos ^5(x)+\frac{2 \cos ^3(x)}{3}-\cos (x)$

[Out]

-Cos[x] + (2*Cos[x]^3)/3 - Cos[x]^5/5

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Rubi [A]  time = 0.0067568, antiderivative size = 21, 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} $-\frac{1}{5} \cos ^5(x)+\frac{2 \cos ^3(x)}{3}-\cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[x]^5,x]

[Out]

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

Mathematica [A]  time = 0.0017605, size = 23, normalized size = 1.1 $-\frac{5 \cos (x)}{8}+\frac{5}{48} \cos (3 x)-\frac{1}{80} \cos (5 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[x]^5,x]

[Out]

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

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Maple [A]  time = 0.004, size = 17, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^5,x)

[Out]

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

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Maxima [A]  time = 0.928642, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Fricas [A]  time = 1.97677, size = 53, normalized size = 2.52 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Sympy [A]  time = 0.06298, size = 17, normalized size = 0.81 \begin{align*} - \frac{\cos ^{5}{\left (x \right )}}{5} + \frac{2 \cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**5,x)

[Out]

-cos(x)**5/5 + 2*cos(x)**3/3 - cos(x)

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Giac [A]  time = 1.04548, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)