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

Optimal. Leaf size=27 \[ \frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b} \]

[Out]

(8*Sin[a + b*x])/b - (8*Sin[a + b*x]^3)/(3*b)

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Rubi [A]  time = 0.0389608, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4288, 2633} \[ \frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Sin[a + b*x])/b - (8*Sin[a + b*x]^3)/(3*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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \csc ^3(a+b x) \sin ^3(2 a+2 b x) \, dx &=8 \int \cos ^3(a+b x) \, dx\\ &=-\frac{8 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0114353, size = 28, normalized size = 1.04 \[ 8 \left (\frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*(Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b))

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Maple [A]  time = 0.025, size = 22, normalized size = 0.8 \begin{align*}{\frac{ \left ( 16+8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,b}} \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)^3,x)

[Out]

8/3/b*(2+cos(b*x+a)^2)*sin(b*x+a)

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Maxima [A]  time = 1.06617, size = 31, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (\sin \left (3 \, b x + 3 \, a\right ) + 9 \, \sin \left (b x + a\right )\right )}}{3 \, 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)^3,x, algorithm="maxima")

[Out]

2/3*(sin(3*b*x + 3*a) + 9*sin(b*x + a))/b

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Fricas [A]  time = 0.473609, size = 55, normalized size = 2.04 \begin{align*} \frac{8 \,{\left (\cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{3 \, 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)^3,x, algorithm="fricas")

[Out]

8/3*(cos(b*x + a)^2 + 2)*sin(b*x + a)/b

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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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.35142, size = 30, normalized size = 1.11 \begin{align*} -\frac{8 \,{\left (\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )\right )}}{3 \, 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)^3,x, algorithm="giac")

[Out]

-8/3*(sin(b*x + a)^3 - 3*sin(b*x + a))/b