3.29 \(\int \sin ^3(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, 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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fricas [A]  time = 0.41, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]

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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giac [A]  time = 0.01, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]

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)

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maple [A]  time = 0.00, size = 11, normalized size = 0.85 \[ -\frac {\left (\sin ^{2}\relax (x )+2\right ) \cos \relax (x )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3,x)

[Out]

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

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maxima [A]  time = 0.58, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]

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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mupad [B]  time = 0.00, size = 10, normalized size = 0.77 \[ \frac {\cos \relax (x)\,\left ({\cos \relax (x)}^2-3\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3,x)

[Out]

(cos(x)*(cos(x)^2 - 3))/3

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sympy [A]  time = 0.07, size = 8, normalized size = 0.62 \[ \frac {\cos ^{3}{\relax (x )}}{3} - \cos {\relax (x )} \]

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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