3.68 \(\int \csc ^3(a+b x) \sin (2 a+2 b x) \, dx\)

Optimal. Leaf size=11 \[ -\frac{2 \csc (a+b x)}{b} \]

[Out]

(-2*Csc[a + b*x])/b

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Rubi [A]  time = 0.0255341, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4288, 2606, 8} \[ -\frac{2 \csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x],x]

[Out]

(-2*Csc[a + b*x])/b

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 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

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

Rubi steps

\begin{align*} \int \csc ^3(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac{2 \operatorname{Subst}(\int 1 \, dx,x,\csc (a+b x))}{b}\\ &=-\frac{2 \csc (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0103081, size = 11, normalized size = 1. \[ -\frac{2 \csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x],x]

[Out]

(-2*Csc[a + b*x])/b

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Maple [A]  time = 0.019, size = 14, normalized size = 1.3 \begin{align*} -2\,{\frac{1}{b\sin \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a),x)

[Out]

-2/b/sin(b*x+a)

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Maxima [B]  time = 1.03779, size = 113, normalized size = 10.27 \begin{align*} -\frac{4 \,{\left (\cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + \sin \left (b x + a\right )\right )}}{b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a),x, algorithm="maxima")

[Out]

-4*(cos(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))/(b*cos(2*b*x + 2*a)^2 + b*si
n(2*b*x + 2*a)^2 - 2*b*cos(2*b*x + 2*a) + b)

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Fricas [A]  time = 0.453849, size = 28, normalized size = 2.55 \begin{align*} -\frac{2}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a),x, algorithm="fricas")

[Out]

-2/(b*sin(b*x + a))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**3*sin(2*b*x+2*a),x)

[Out]

Timed out

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Giac [A]  time = 1.2038, size = 18, normalized size = 1.64 \begin{align*} -\frac{2}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a),x, algorithm="giac")

[Out]

-2/(b*sin(b*x + a))