\(\int \csc (3 x) \sin (6 x) \, dx\) [96]

Optimal result

Integrand size = 9, antiderivative size = 8 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \sin (3 x) \]

[Out]

2/3*sin(3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4373, 2717} \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \sin (3 x) \]

[In]

Int[Csc[3*x]*Sin[6*x],x]

[Out]

(2*Sin[3*x])/3

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4373

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = 2 \int \cos (3 x) \, dx \\ & = \frac {2}{3} \sin (3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \sin (3 x) \]

[In]

Integrate[Csc[3*x]*Sin[6*x],x]

[Out]

(2*Sin[3*x])/3

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {2 \sin \left (3 x \right )}{3}\)	\(7\)
derivativedivides	\(\frac {2}{3 \csc \left (3 x \right )}\)	\(9\)
default	\(\frac {2}{3 \csc \left (3 x \right )}\)	\(9\)

[In]

int(csc(3*x)*sin(6*x),x,method=_RETURNVERBOSE)

[Out]

2/3*sin(3*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \, \sin \left (3 \, x\right ) \]

[In]

integrate(csc(3*x)*sin(6*x),x, algorithm="fricas")

[Out]

2/3*sin(3*x)

Sympy [A] (verification not implemented)

Time = 1.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2 \sin {\left (3 x \right )}}{3} \]

[In]

integrate(csc(3*x)*sin(6*x),x)

[Out]

2*sin(3*x)/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \, \sin \left (3 \, x\right ) \]

[In]

integrate(csc(3*x)*sin(6*x),x, algorithm="maxima")

[Out]

2/3*sin(3*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2}{3} \, \sin \left (3 \, x\right ) \]

[In]

integrate(csc(3*x)*sin(6*x),x, algorithm="giac")

[Out]

2/3*sin(3*x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \csc (3 x) \sin (6 x) \, dx=\frac {2\,\sin \left (3\,x\right )}{3} \]

[In]

int(sin(6*x)/sin(3*x),x)

[Out]

(2*sin(3*x))/3