3.336 \(\int -\sin ^3(\frac{\pi }{12}-3 x) \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{9} \cos ^3\left (\frac{\pi }{12}-3 x\right )-\frac{1}{3} \cos \left (\frac{\pi }{12}-3 x\right ) \]

[Out]

-Cos[Pi/12 - 3*x]/3 + Cos[Pi/12 - 3*x]^3/9

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Rubi [A]  time = 0.0107794, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2633} \[ \frac{1}{9} \cos ^3\left (\frac{\pi }{12}-3 x\right )-\frac{1}{3} \cos \left (\frac{\pi }{12}-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Int[-Sin[Pi/12 - 3*x]^3,x]

[Out]

-Cos[Pi/12 - 3*x]/3 + Cos[Pi/12 - 3*x]^3/9

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\left (\frac{\pi }{12}-3 x\right ) \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\frac{\pi }{12}-3 x\right )\right )\right )\\ &=-\frac{1}{3} \cos \left (\frac{\pi }{12}-3 x\right )+\frac{1}{9} \cos ^3\left (\frac{\pi }{12}-3 x\right )\\ \end{align*}

Mathematica [A]  time = 0.0163376, size = 31, normalized size = 1. \[ \frac{1}{36} \cos \left (3 \left (\frac{\pi }{12}-3 x\right )\right )-\frac{1}{4} \cos \left (\frac{\pi }{12}-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[-Sin[Pi/12 - 3*x]^3,x]

[Out]

-Cos[Pi/12 - 3*x]/4 + Cos[3*(Pi/12 - 3*x)]/36

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Maple [A]  time = 0.007, size = 23, normalized size = 0.7 \begin{align*} -{\frac{1}{9} \left ( 2+ \left ( \cos \left ({\frac{5\,\pi }{12}}+3\,x \right ) \right ) ^{2} \right ) \sin \left ({\frac{5\,\pi }{12}}+3\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos(5/12*Pi+3*x)^3,x)

[Out]

-1/9*(2+cos(5/12*Pi+3*x)^2)*sin(5/12*Pi+3*x)

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Maxima [A]  time = 0.936318, size = 31, normalized size = 1. \begin{align*} \frac{1}{9} \, \sin \left (\frac{5}{12} \, \pi + 3 \, x\right )^{3} - \frac{1}{3} \, \sin \left (\frac{5}{12} \, \pi + 3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(5/12*pi+3*x)^3,x, algorithm="maxima")

[Out]

1/9*sin(5/12*pi + 3*x)^3 - 1/3*sin(5/12*pi + 3*x)

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Fricas [A]  time = 1.5673, size = 70, normalized size = 2.26 \begin{align*} -\frac{1}{9} \,{\left (\cos \left (\frac{5}{12} \, \pi + 3 \, x\right )^{2} + 2\right )} \sin \left (\frac{5}{12} \, \pi + 3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(5/12*pi+3*x)^3,x, algorithm="fricas")

[Out]

-1/9*(cos(5/12*pi + 3*x)^2 + 2)*sin(5/12*pi + 3*x)

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Sympy [A]  time = 0.309147, size = 39, normalized size = 1.26 \begin{align*} - \frac{2 \sin ^{3}{\left (3 x + \frac{5 \pi }{12} \right )}}{9} - \frac{\sin{\left (3 x + \frac{5 \pi }{12} \right )} \cos ^{2}{\left (3 x + \frac{5 \pi }{12} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(5/12*pi+3*x)**3,x)

[Out]

-2*sin(3*x + 5*pi/12)**3/9 - sin(3*x + 5*pi/12)*cos(3*x + 5*pi/12)**2/3

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Giac [A]  time = 1.06113, size = 31, normalized size = 1. \begin{align*} \frac{1}{9} \, \sin \left (\frac{5}{12} \, \pi + 3 \, x\right )^{3} - \frac{1}{3} \, \sin \left (\frac{5}{12} \, \pi + 3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cos(5/12*pi+3*x)^3,x, algorithm="giac")

[Out]

1/9*sin(5/12*pi + 3*x)^3 - 1/3*sin(5/12*pi + 3*x)