Integrand size = 18, antiderivative size = 61 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {256 \cos ^9(a+b x)}{9 b}+\frac {768 \cos ^{11}(a+b x)}{11 b}-\frac {768 \cos ^{13}(a+b x)}{13 b}+\frac {256 \cos ^{15}(a+b x)}{15 b} \] Output:
-256/9*cos(b*x+a)^9/b+768/11*cos(b*x+a)^11/b-768/13*cos(b*x+a)^13/b+256/15 *cos(b*x+a)^15/b
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {35 \cos (a+b x)}{64 b}-\frac {35 \cos (3 (a+b x))}{192 b}+\frac {21 \cos (5 (a+b x))}{320 b}+\frac {3 \cos (7 (a+b x))}{64 b}-\frac {7 \cos (9 (a+b x))}{576 b}-\frac {7 \cos (11 (a+b x))}{704 b}+\frac {\cos (13 (a+b x))}{832 b}+\frac {\cos (15 (a+b x))}{960 b} \] Input:
Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^8,x]
Output:
(-35*Cos[a + b*x])/(64*b) - (35*Cos[3*(a + b*x)])/(192*b) + (21*Cos[5*(a + b*x)])/(320*b) + (3*Cos[7*(a + b*x)])/(64*b) - (7*Cos[9*(a + b*x)])/(576* b) - (7*Cos[11*(a + b*x)])/(704*b) + Cos[13*(a + b*x)]/(832*b) + Cos[15*(a + b*x)]/(960*b)
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4776, 3042, 3045, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^8(2 a+2 b x) \csc (a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (2 a+2 b x)^8}{\sin (a+b x)}dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle 256 \int \cos ^8(a+b x) \sin ^7(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 256 \int \cos (a+b x)^8 \sin (a+b x)^7dx\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle -\frac {256 \int \cos ^8(a+b x) \left (1-\cos ^2(a+b x)\right )^3d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {256 \int \left (-\cos ^{14}(a+b x)+3 \cos ^{12}(a+b x)-3 \cos ^{10}(a+b x)+\cos ^8(a+b x)\right )d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {256 \left (-\frac {1}{15} \cos ^{15}(a+b x)+\frac {3}{13} \cos ^{13}(a+b x)-\frac {3}{11} \cos ^{11}(a+b x)+\frac {1}{9} \cos ^9(a+b x)\right )}{b}\) |
Input:
Int[Csc[a + b*x]*Sin[2*a + 2*b*x]^8,x]
Output:
(-256*(Cos[a + b*x]^9/9 - (3*Cos[a + b*x]^11)/11 + (3*Cos[a + b*x]^13)/13 - Cos[a + b*x]^15/15))/b
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(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])
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_ Symbol] :> Simp[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] && I ntegerQ[p]
Time = 13.58 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\frac {256 \cos \left (b x +a \right )^{15}}{15}-\frac {768 \cos \left (b x +a \right )^{13}}{13}+\frac {768 \cos \left (b x +a \right )^{11}}{11}-\frac {256 \cos \left (b x +a \right )^{9}}{9}}{b}\) | \(47\) |
risch | \(-\frac {35 \cos \left (b x +a \right )}{64 b}+\frac {\cos \left (15 b x +15 a \right )}{960 b}+\frac {\cos \left (13 b x +13 a \right )}{832 b}-\frac {7 \cos \left (11 b x +11 a \right )}{704 b}-\frac {7 \cos \left (9 b x +9 a \right )}{576 b}+\frac {3 \cos \left (7 b x +7 a \right )}{64 b}+\frac {21 \cos \left (5 b x +5 a \right )}{320 b}-\frac {35 \cos \left (3 b x +3 a \right )}{192 b}\) | \(111\) |
Input:
int(csc(b*x+a)*sin(2*b*x+2*a)^8,x,method=_RETURNVERBOSE)
Output:
256/b*(1/15*cos(b*x+a)^15-3/13*cos(b*x+a)^13+3/11*cos(b*x+a)^11-1/9*cos(b* x+a)^9)
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {256 \, {\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \] Input:
integrate(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="fricas")
Output:
256/6435*(429*cos(b*x + a)^15 - 1485*cos(b*x + a)^13 + 1755*cos(b*x + a)^1 1 - 715*cos(b*x + a)^9)/b
Timed out. \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)*sin(2*b*x+2*a)**8,x)
Output:
Timed out
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {429 \, \cos \left (15 \, b x + 15 \, a\right ) + 495 \, \cos \left (13 \, b x + 13 \, a\right ) - 4095 \, \cos \left (11 \, b x + 11 \, a\right ) - 5005 \, \cos \left (9 \, b x + 9 \, a\right ) + 19305 \, \cos \left (7 \, b x + 7 \, a\right ) + 27027 \, \cos \left (5 \, b x + 5 \, a\right ) - 75075 \, \cos \left (3 \, b x + 3 \, a\right ) - 225225 \, \cos \left (b x + a\right )}{411840 \, b} \] Input:
integrate(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="maxima")
Output:
1/411840*(429*cos(15*b*x + 15*a) + 495*cos(13*b*x + 13*a) - 4095*cos(11*b* x + 11*a) - 5005*cos(9*b*x + 9*a) + 19305*cos(7*b*x + 7*a) + 27027*cos(5*b *x + 5*a) - 75075*cos(3*b*x + 3*a) - 225225*cos(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (53) = 106\).
Time = 0.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.43 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {8192 \, {\left (\frac {15 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {105 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {455 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {5070 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {30030 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {70070 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {115830 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {109395 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac {75075 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} + \frac {27027 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} + \frac {6435 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{11}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{11}} - 1\right )}}{6435 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{15}} \] Input:
integrate(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="giac")
Output:
-8192/6435*(15*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 105*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 455*(cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 5070*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 30030*(cos(b*x + a) - 1)^5/(cos(b*x + a) + 1)^5 + 70070*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^ 6 + 115830*(cos(b*x + a) - 1)^7/(cos(b*x + a) + 1)^7 + 109395*(cos(b*x + a ) - 1)^8/(cos(b*x + a) + 1)^8 + 75075*(cos(b*x + a) - 1)^9/(cos(b*x + a) + 1)^9 + 27027*(cos(b*x + a) - 1)^10/(cos(b*x + a) + 1)^10 + 6435*(cos(b*x + a) - 1)^11/(cos(b*x + a) + 1)^11 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^15)
Time = 18.72 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {-\frac {256\,{\cos \left (a+b\,x\right )}^{15}}{15}+\frac {768\,{\cos \left (a+b\,x\right )}^{13}}{13}-\frac {768\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {256\,{\cos \left (a+b\,x\right )}^9}{9}}{b} \] Input:
int(sin(2*a + 2*b*x)^8/sin(a + b*x),x)
Output:
-((256*cos(a + b*x)^9)/9 - (768*cos(a + b*x)^11)/11 + (768*cos(a + b*x)^13 )/13 - (256*cos(a + b*x)^15)/15)/b
\[ \int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx=\int \csc \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{8}d x \] Input:
int(csc(b*x+a)*sin(2*b*x+2*a)^8,x)
Output:
int(csc(a + b*x)*sin(2*a + 2*b*x)**8,x)