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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 1.00 \[ \frac {1}{36} \cos \left (3 \left (\frac {\pi }{12}-3 x\right )\right )-\frac {1}{4} \cos \left (\frac {\pi }{12}-3 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[-Sin[Pi/12 - 3*x]^3,x]

[Out]

-1/4*Cos[Pi/12 - 3*x] + Cos[3*(Pi/12 - 3*x)]/36

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\sin ^3\left (\frac {\pi }{12}-3 x\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[-Sin[Pi/12 - 3*x]^3,x]

[Out]

Could not integrate

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fricas [A]  time = 0.78, size = 22, normalized size = 0.71 \[ -\frac {1}{9} \, {\left (\cos \left (\frac {5}{12} \, \pi + 3 \, x\right )^{2} + 2\right )} \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]

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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giac [A]  time = 0.74, size = 23, normalized size = 0.74 \[ \frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]

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)

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maple [A]  time = 0.54, size = 22, normalized size = 0.71

method	result	size
risch	\(\frac {\sin \left (\frac {\pi }{4}+9 x \right )}{36}-\frac {\sin \left (\frac {5 \pi }{12}+3 x \right )}{4}\)	\(22\)
derivativedivides	\(-\frac {\left (2+\cos ^{2}\left (\frac {5 \pi }{12}+3 x \right )\right ) \sin \left (\frac {5 \pi }{12}+3 x \right )}{9}\)	\(23\)
default	\(-\frac {\left (2+\cos ^{2}\left (\frac {5 \pi }{12}+3 x \right )\right ) \sin \left (\frac {5 \pi }{12}+3 x \right )}{9}\)	\(23\)
norman	\(\frac {-\frac {4 \left (\tan ^{3}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )}{3}-\frac {2 \tan \left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {5 \pi }{24}+\frac {3 x}{2}\right )\right )^{3}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*sin(1/4*Pi+9*x)-1/4*sin(5/12*Pi+3*x)

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maxima [A]  time = 0.57, size = 23, normalized size = 0.74 \[ \frac {1}{9} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right )^{3} - \frac {1}{3} \, \sin \left (\frac {5}{12} \, \pi + 3 \, x\right ) \]

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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mupad [B]  time = 0.25, size = 22, normalized size = 0.71 \[ \frac {\sin \left (\frac {5\,\Pi }{12}+3\,x\right )\,\left ({\sin \left (\frac {5\,\Pi }{12}+3\,x\right )}^2-3\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cos((5*Pi)/12 + 3*x)^3,x)

[Out]

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

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sympy [A]  time = 0.34, size = 39, normalized size = 1.26 \[ - \frac {2 \sin ^{3}{\left (3 x + \frac {5 \pi }{12} \right )}}{9} - \frac {\sin {\left (3 x + \frac {5 \pi }{12} \right )} \cos ^{2}{\left (3 x + \frac {5 \pi }{12} \right )}}{3} \]

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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