3.96 \(\int \csc (3 x) \sin (6 x) \, dx\)

Optimal. Leaf size=8 \[ \frac{2}{3} \sin (3 x) \]

[Out]

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

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Rubi [A]  time = 0.0135801, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4288, 2637} \[ \frac{2}{3} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4288

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]

Rule 2637

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

Rubi steps

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

Mathematica [A]  time = 0.0030499, size = 8, normalized size = 1. \[ \frac{2}{3} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 9, normalized size = 1.1 \begin{align*}{\frac{2}{3\,\csc \left ( 3\,x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/csc(3*x)

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Maxima [A]  time = 0.992151, size = 8, normalized size = 1. \begin{align*} \frac{2}{3} \, \sin \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(3*x)

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Fricas [A]  time = 2.47485, size = 19, normalized size = 2.38 \begin{align*} \frac{2}{3} \, \sin \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(3*x)

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Sympy [A]  time = 4.92115, size = 7, normalized size = 0.88 \begin{align*} \frac{2 \sin{\left (3 x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(3*x)/3

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Giac [A]  time = 1.15465, size = 8, normalized size = 1. \begin{align*} \frac{2}{3} \, \sin \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(3*x)