3.61 \(\int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx\)

Optimal. Leaf size=46 \[ -\frac{256 \cos ^{13}(a+b x)}{13 b}+\frac{512 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b} \]

[Out]

(-256*Cos[a + b*x]^9)/(9*b) + (512*Cos[a + b*x]^11)/(11*b) - (256*Cos[a + b*x]^13)/(13*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0672123, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2565, 270} \[ -\frac{256 \cos ^{13}(a+b x)}{13 b}+\frac{512 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^8,x]

[Out]

(-256*Cos[a + b*x]^9)/(9*b) + (512*Cos[a + b*x]^11)/(11*b) - (256*Cos[a + b*x]^13)/(13*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 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx &=256 \int \cos ^8(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac{256 \operatorname{Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{256 \operatorname{Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{256 \cos ^9(a+b x)}{9 b}+\frac{512 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^{13}(a+b x)}{13 b}\\ \end{align*}

Mathematica [B]  time = 0.0982023, size = 104, normalized size = 2.26 \[ -\frac{5 \cos (a+b x)}{4 b}-\frac{25 \cos (3 (a+b x))}{48 b}+\frac{\cos (5 (a+b x))}{16 b}+\frac{\cos (7 (a+b x))}{8 b}+\frac{\cos (9 (a+b x))}{72 b}-\frac{3 \cos (11 (a+b x))}{176 b}-\frac{\cos (13 (a+b x))}{208 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^8,x]

[Out]

(-5*Cos[a + b*x])/(4*b) - (25*Cos[3*(a + b*x)])/(48*b) + Cos[5*(a + b*x)]/(16*b) + Cos[7*(a + b*x)]/(8*b) + Co
s[9*(a + b*x)]/(72*b) - (3*Cos[11*(a + b*x)])/(176*b) - Cos[13*(a + b*x)]/(208*b)

________________________________________________________________________________________

Maple [A]  time = 0.032, size = 53, normalized size = 1.2 \begin{align*} 256\,{\frac{1}{b} \left ( -1/13\, \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{9}-{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{143}}-{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{1287}} \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)^8,x)

[Out]

256/b*(-1/13*sin(b*x+a)^4*cos(b*x+a)^9-4/143*sin(b*x+a)^2*cos(b*x+a)^9-8/1287*cos(b*x+a)^9)

________________________________________________________________________________________

Maxima [A]  time = 1.05838, size = 108, normalized size = 2.35 \begin{align*} -\frac{99 \, \cos \left (13 \, b x + 13 \, a\right ) + 351 \, \cos \left (11 \, b x + 11 \, a\right ) - 286 \, \cos \left (9 \, b x + 9 \, a\right ) - 2574 \, \cos \left (7 \, b x + 7 \, a\right ) - 1287 \, \cos \left (5 \, b x + 5 \, a\right ) + 10725 \, \cos \left (3 \, b x + 3 \, a\right ) + 25740 \, \cos \left (b x + a\right )}{20592 \, 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)^8,x, algorithm="maxima")

[Out]

-1/20592*(99*cos(13*b*x + 13*a) + 351*cos(11*b*x + 11*a) - 286*cos(9*b*x + 9*a) - 2574*cos(7*b*x + 7*a) - 1287
*cos(5*b*x + 5*a) + 10725*cos(3*b*x + 3*a) + 25740*cos(b*x + a))/b

________________________________________________________________________________________

Fricas [A]  time = 0.523297, size = 104, normalized size = 2.26 \begin{align*} -\frac{256 \,{\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, 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)^8,x, algorithm="fricas")

[Out]

-256/1287*(99*cos(b*x + a)^13 - 234*cos(b*x + a)^11 + 143*cos(b*x + a)^9)/b

________________________________________________________________________________________

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)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.8694, size = 335, normalized size = 7.28 \begin{align*} -\frac{4096 \,{\left (\frac{13 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{78 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac{572 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac{3718 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac{7722 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac{13728 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac{12012 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac{9009 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - \frac{3003 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} - \frac{858 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} - 1\right )}}{1287 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{13}} \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)^8,x, algorithm="giac")

[Out]

-4096/1287*(13*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 78*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 - 572*(cos
(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 3718*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 - 7722*(cos(b*x + a) -
1)^5/(cos(b*x + a) + 1)^5 - 13728*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 - 12012*(cos(b*x + a) - 1)^7/(cos(
b*x + a) + 1)^7 - 9009*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 - 3003*(cos(b*x + a) - 1)^9/(cos(b*x + a) + 1
)^9 - 858*(cos(b*x + a) - 1)^10/(cos(b*x + a) + 1)^10 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^13)