3.58 \(\int \sin ^2(3 x) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, 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 = {2635, 8} \[ \frac {x}{2}-\frac {1}{6} \sin (3 x) \cos (3 x) \]

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 2635

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] && Integer
Q[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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

x/2 - Sin[6*x]/12

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 14, normalized size = 0.78 \[ -\frac {1}{6} \, \cos \left (3 \, x\right ) \sin \left (3 \, x\right ) + \frac {1}{2} \, x \]

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

________________________________________________________________________________________

giac [A]  time = 0.84, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x - \frac {1}{12} \, \sin \left (6 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x - 1/12*sin(6*x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 15, normalized size = 0.83 \[ -\frac {\cos \left (3 x \right ) \sin \left (3 x \right )}{6}+\frac {x}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x - \frac {1}{12} \, \sin \left (6 \, x\right ) \]

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/2 - sin(6*x)/12

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 14, normalized size = 0.78 \[ \frac {x}{2} - \frac {\sin {\left (3 x \right )} \cos {\left (3 x \right )}}{6} \]

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

________________________________________________________________________________________