\(\int \sin ^3(x) \, dx\) [85]

Optimal result

Integrand size = 4, antiderivative size = 13 \[ \int \sin ^3(x) \, dx=-\cos (x)+\frac {\cos ^3(x)}{3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, 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} \[ \int \sin ^3(x) \, dx=\frac {\cos ^3(x)}{3}-\cos (x) \]

[In]

Int[Sin[x]^3,x]

[Out]

-Cos[x] + Cos[x]^3/3

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \sin ^3(x) \, dx=-\frac {3 \cos (x)}{4}+\frac {1}{12} \cos (3 x) \]

[In]

Integrate[Sin[x]^3,x]

[Out]

(-3*Cos[x])/4 + Cos[3*x]/12

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85

method	result	size
default	\(-\frac {\left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}\)	\(11\)
risch	\(-\frac {3 \cos \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{12}\)	\(12\)
parallelrisch	\(-\frac {2}{3}-\frac {3 \cos \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{12}\)	\(13\)
norman	\(\frac {-4 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {4}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\)	\(22\)

[In]

int(sin(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

integrate(sin(x)^3,x, algorithm="fricas")

[Out]

1/3*cos(x)^3 - cos(x)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \sin ^3(x) \, dx=\frac {\cos ^{3}{\left (x \right )}}{3} - \cos {\left (x \right )} \]

[In]

integrate(sin(x)**3,x)

[Out]

cos(x)**3/3 - cos(x)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

integrate(sin(x)^3,x, algorithm="maxima")

[Out]

1/3*cos(x)^3 - cos(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sin ^3(x) \, dx=\frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

[In]

integrate(sin(x)^3,x, algorithm="giac")

[Out]

1/3*cos(x)^3 - cos(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \sin ^3(x) \, dx=\frac {\cos \left (x\right )\,\left ({\cos \left (x\right )}^2-3\right )}{3} \]

[In]

int(sin(x)^3,x)

[Out]

(cos(x)*(cos(x)^2 - 3))/3