### 3.61 $$\int \sin ^3(x) \, dx$$

Optimal. Leaf size=13 $\frac{\cos ^3(x)}{3}-\cos (x)$

[Out]

-Cos[x] + Cos[x]^3/3

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Rubi [A]  time = 0.0054989, antiderivative size = 13, 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} $\frac{\cos ^3(x)}{3}-\cos (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[x]^3,x]

[Out]

-Cos[x] + Cos[x]^3/3

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

Mathematica [A]  time = 0.0017633, size = 15, normalized size = 1.15 $\frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[x]^3,x]

[Out]

(-3*Cos[x])/4 + Cos[3*x]/12

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Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3,x)

[Out]

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

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Maxima [A]  time = 0.928092, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{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)^3,x, algorithm="maxima")

[Out]

1/3*cos(x)^3 - cos(x)

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Fricas [A]  time = 1.97143, size = 31, normalized size = 2.38 \begin{align*} \frac{1}{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)^3,x, algorithm="fricas")

[Out]

1/3*cos(x)^3 - cos(x)

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Sympy [A]  time = 0.06428, size = 8, normalized size = 0.62 \begin{align*} \frac{\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)**3,x)

[Out]

cos(x)**3/3 - cos(x)

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Giac [A]  time = 1.07697, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{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)^3,x, algorithm="giac")

[Out]

1/3*cos(x)^3 - cos(x)