\(\int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx\) [61]

Optimal result

Integrand size = 20, antiderivative size = 46 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256 \cos ^9(a+b x)}{9 b}+\frac {512 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^{13}(a+b x)}{13 b} \]

[Out]

-256/9*cos(b*x+a)^9/b+512/11*cos(b*x+a)^11/b-256/13*cos(b*x+a)^13/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2645, 276} \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256 \cos ^{13}(a+b x)}{13 b}+\frac {512 \cos ^{11}(a+b x)}{11 b}-\frac {256 \cos ^9(a+b x)}{9 b} \]

[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 276

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]

Rule 2645

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 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}& = 256 \int \cos ^8(a+b x) \sin ^5(a+b x) \, dx \\ & = -\frac {256 \text {Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {256 \text {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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(46)=92\).

Time = 0.79 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.26 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {5 \cos (a+b x)}{4 b}-\frac {25 \cos (3 (a+b x))}{48 b}+\frac {\cos (5 (a+b x))}{16 b}+\frac {\cos (7 (a+b x))}{8 b}+\frac {\cos (9 (a+b x))}{72 b}-\frac {3 \cos (11 (a+b x))}{176 b}-\frac {\cos (13 (a+b x))}{208 b} \]

[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] (verified)

Time = 68.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {256 \left (\frac {\cos \left (x b +a \right )^{13}}{13}-\frac {2 \cos \left (x b +a \right )^{11}}{11}+\frac {\cos \left (x b +a \right )^{9}}{9}\right )}{b}\)	\(37\)
risch	\(-\frac {5 \cos \left (x b +a \right )}{4 b}-\frac {\cos \left (13 x b +13 a \right )}{208 b}-\frac {3 \cos \left (11 x b +11 a \right )}{176 b}+\frac {\cos \left (9 x b +9 a \right )}{72 b}+\frac {\cos \left (7 x b +7 a \right )}{8 b}+\frac {\cos \left (5 x b +5 a \right )}{16 b}-\frac {25 \cos \left (3 x b +3 a \right )}{48 b}\)	\(97\)

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a)^8,x,method=_RETURNVERBOSE)

[Out]

-256/b*(1/13*cos(b*x+a)^13-2/11*cos(b*x+a)^11+1/9*cos(b*x+a)^9)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \]

[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)]

Timed out. \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+a)**3*sin(2*b*x+2*a)**8,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {99 \, \cos \left (13 \, b x + 13 \, a\right ) + 351 \, \cos \left (11 \, b x + 11 \, a\right ) - 286 \, \cos \left (9 \, b x + 9 \, a\right ) - 2574 \, \cos \left (7 \, b x + 7 \, a\right ) - 1287 \, \cos \left (5 \, b x + 5 \, a\right ) + 10725 \, \cos \left (3 \, b x + 3 \, a\right ) + 25740 \, \cos \left (b x + a\right )}{20592 \, b} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (40) = 80\).

Time = 0.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 5.39 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {4096 \, {\left (\frac {13 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {78 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {572 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {3718 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {7722 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac {13728 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac {12012 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac {9009 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - \frac {3003 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} - \frac {858 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} - 1\right )}}{1287 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{13}} \]

[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)

Mupad [B] (verification not implemented)

Time = 19.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256\,\left (99\,{\cos \left (a+b\,x\right )}^{13}-234\,{\cos \left (a+b\,x\right )}^{11}+143\,{\cos \left (a+b\,x\right )}^9\right )}{1287\,b} \]

[In]

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

[Out]

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