3.1.58 \(\int \sin ^2(3 x) \, dx\) [58]

Optimal. Leaf size=18 \[ \frac {x}{2}-\frac {1}{6} \cos (3 x) \sin (3 x) \]

[Out]

1/2*x-1/6*cos(3*x)*sin(3*x)

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2715, 8} \begin {gather*} \frac {x}{2}-\frac {1}{6} \sin (3 x) \cos (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[3*x]^2,x]

[Out]

x/2 - (Cos[3*x]*Sin[3*x])/6

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps

\begin {align*} \int \sin ^2(3 x) \, dx &=-\frac {1}{6} \cos (3 x) \sin (3 x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}-\frac {1}{6} \cos (3 x) \sin (3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.78 \begin {gather*} \frac {x}{2}-\frac {1}{12} \sin (6 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[3*x]^2,x]

[Out]

x/2 - Sin[6*x]/12

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Mathics [A]
time = 1.77, size = 10, normalized size = 0.56 \begin {gather*} \frac {x}{2}-\frac {\text {Sin}\left [6 x\right ]}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[3*x]^2,x]')

[Out]

x / 2 - Sin[6 x] / 12

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Maple [A]
time = 0.02, size = 15, normalized size = 0.83

method	result	size
risch	\(\frac {x}{2}-\frac {\sin \left (6 x \right )}{12}\)	\(11\)
derivativedivides	\(\frac {x}{2}-\frac {\cos \left (3 x \right ) \sin \left (3 x \right )}{6}\)	\(15\)
default	\(\frac {x}{2}-\frac {\cos \left (3 x \right ) \sin \left (3 x \right )}{6}\)	\(15\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {6 x}{\sqrt {\pi }}-\frac {\sin \left (6 x \right )}{\sqrt {\pi }}\right )}{12}\)	\(22\)
norman	\(\frac {x \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )+\frac {x}{2}+\frac {\left (\tan ^{3}\left (\frac {3 x}{2}\right )\right )}{3}+\frac {x \left (\tan ^{4}\left (\frac {3 x}{2}\right )\right )}{2}-\frac {\tan \left (\frac {3 x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right )^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/2*x-1/6*cos(3*x)*sin(3*x)

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Maxima [A]
time = 0.26, size = 10, normalized size = 0.56 \begin {gather*} \frac {1}{2} \, x - \frac {1}{12} \, \sin \left (6 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)^2,x, algorithm="maxima")

[Out]

1/2*x - 1/12*sin(6*x)

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Fricas [A]
time = 0.34, size = 14, normalized size = 0.78 \begin {gather*} -\frac {1}{6} \, \cos \left (3 \, x\right ) \sin \left (3 \, x\right ) + \frac {1}{2} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)^2,x, algorithm="fricas")

[Out]

-1/6*cos(3*x)*sin(3*x) + 1/2*x

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.78 \begin {gather*} \frac {x}{2} - \frac {\sin {\left (3 x \right )} \cos {\left (3 x \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)**2,x)

[Out]

x/2 - sin(3*x)*cos(3*x)/6

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Giac [A]
time = 0.00, size = 13, normalized size = 0.72 \begin {gather*} \frac {x}{2}-\frac {\sin \left (6 x\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)^2,x)

[Out]

1/2*x - 1/12*sin(6*x)

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Mupad [B]
time = 0.05, size = 10, normalized size = 0.56 \begin {gather*} \frac {x}{2}-\frac {\sin \left (6\,x\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*x)^2,x)

[Out]

x/2 - sin(6*x)/12

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