Integrand size = 21, antiderivative size = 127 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {10 \cot ^7(c+d x)}{7 a^3 d}+\frac {11 \cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}-\frac {3 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d} \]
3/5*cot(d*x+c)^5/a^3/d+10/7*cot(d*x+c)^7/a^3/d+11/9*cot(d*x+c)^9/a^3/d+4/1 1*cot(d*x+c)^11/a^3/d-3/7*csc(d*x+c)^7/a^3/d+7/9*csc(d*x+c)^9/a^3/d-4/11*c sc(d*x+c)^11/a^3/d
Time = 1.41 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.76 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\csc (c) \csc ^5(c+d x) \sec ^3(c+d x) (-3886080 \sin (c)+563200 \sin (d x)+524150 \sin (c+d x)+314490 \sin (2 (c+d x))-162010 \sin (3 (c+d x))-238250 \sin (4 (c+d x))-47650 \sin (5 (c+d x))+47650 \sin (6 (c+d x))+28590 \sin (7 (c+d x))+4765 \sin (8 (c+d x))-2027520 \sin (2 c+d x)+1486848 \sin (c+2 d x)-2365440 \sin (3 c+2 d x)+452608 \sin (2 c+3 d x)+665600 \sin (3 c+4 d x)+133120 \sin (4 c+5 d x)-133120 \sin (5 c+6 d x)-79872 \sin (6 c+7 d x)-13312 \sin (7 c+8 d x))}{56770560 a^3 d (1+\sec (c+d x))^3} \]
(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^3*(-3886080*Sin[c] + 563200*Sin[d*x] + 524150*Sin[c + d*x] + 314490*Sin[2*(c + d*x)] - 162010*Sin[3*(c + d*x)] - 238250*Sin[4*(c + d*x)] - 47650*Sin[5*(c + d*x)] + 47650*Sin[6*(c + d*x)] + 28590*Sin[7*(c + d*x)] + 4765*Sin[8*(c + d*x)] - 2027520*Sin[2*c + d*x] + 1486848*Sin[c + 2*d*x] - 2365440*Sin[3*c + 2*d*x] + 452608*Sin[2*c + 3* d*x] + 665600*Sin[3*c + 4*d*x] + 133120*Sin[4*c + 5*d*x] - 133120*Sin[5*c + 6*d*x] - 79872*Sin[6*c + 7*d*x] - 13312*Sin[7*c + 8*d*x]))/(56770560*a^3 *d*(1 + Sec[c + d*x])^3)
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.
| \(\displaystyle \int \frac {\csc ^6(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cot ^3(c+d x) \csc ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^6 \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle -\frac {\int -(a-a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^9(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (a-a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^9(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^{12}}dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \left (\sin \left (\frac {1}{2} (2 c-\pi )+d x\right ) a+a\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^{12}}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle -\frac {\int \left (a^3 \cos ^6(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a^3 \cos ^5(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )+3 a^3 \cos ^4(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )-a^3 \cos ^3(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \int \cos ^6(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )dx-3 a^3 \int \cos ^5(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )dx+3 a^3 \int \cos ^4(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )dx+a^3 \left (-\int \cos ^3(c+d x) \sec ^{12}\left (\frac {1}{2} (2 c-\pi )+d x\right )dx\right )}{a^6}\) |
3.2.6.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.90 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87
| method | result | size |
| parallelrisch | \(\frac {-315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-770 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+990 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+4158 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-693 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-2310 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-20790 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6930 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{887040 a^{3} d}\) | \(110\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\) | \(112\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d \,a^{3}}\) | \(112\) |
| risch | \(-\frac {16 i \left (2310 \,{\mathrm e}^{10 i \left (d x +c \right )}+1980 \,{\mathrm e}^{9 i \left (d x +c \right )}+3795 \,{\mathrm e}^{8 i \left (d x +c \right )}+550 \,{\mathrm e}^{7 i \left (d x +c \right )}+1452 \,{\mathrm e}^{6 i \left (d x +c \right )}+442 \,{\mathrm e}^{5 i \left (d x +c \right )}+650 \,{\mathrm e}^{4 i \left (d x +c \right )}+130 \,{\mathrm e}^{3 i \left (d x +c \right )}-130 \,{\mathrm e}^{2 i \left (d x +c \right )}-78 \,{\mathrm e}^{i \left (d x +c \right )}-13\right )}{3465 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(148\) |
| norman | \(\frac {-\frac {1}{1280 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{896 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{1152 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{2816 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{128 d a}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 d a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{640 d a}}{a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\) | \(158\) |
1/887040*(-315*tan(1/2*d*x+1/2*c)^11-770*tan(1/2*d*x+1/2*c)^9+990*tan(1/2* d*x+1/2*c)^7+4158*tan(1/2*d*x+1/2*c)^5-693*cot(1/2*d*x+1/2*c)^5-2310*cot(1 /2*d*x+1/2*c)^3-20790*tan(1/2*d*x+1/2*c)+6930*cot(1/2*d*x+1/2*c))/a^3/d
Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.50 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {104 \, \cos \left (d x + c\right )^{8} + 312 \, \cos \left (d x + c\right )^{7} + 52 \, \cos \left (d x + c\right )^{6} - 676 \, \cos \left (d x + c\right )^{5} - 585 \, \cos \left (d x + c\right )^{4} + 325 \, \cos \left (d x + c\right )^{3} - 25 \, \cos \left (d x + c\right )^{2} - 150 \, \cos \left (d x + c\right ) - 50}{3465 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]
1/3465*(104*cos(d*x + c)^8 + 312*cos(d*x + c)^7 + 52*cos(d*x + c)^6 - 676* cos(d*x + c)^5 - 585*cos(d*x + c)^4 + 325*cos(d*x + c)^3 - 25*cos(d*x + c) ^2 - 150*cos(d*x + c) - 50)/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^ 6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))
\[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.37 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {20790 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4158 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {990 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{3}} + \frac {231 \, {\left (\frac {10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {30 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \]
-1/887040*((20790*sin(d*x + c)/(cos(d*x + c) + 1) - 4158*sin(d*x + c)^5/(c os(d*x + c) + 1)^5 - 990*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/ a^3 + 231*(10*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 30*sin(d*x + c)^4/(cos (d*x + c) + 1)^4 + 3)*(cos(d*x + c) + 1)^5/(a^3*sin(d*x + c)^5))/d
Time = 0.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {231 \, {\left (30 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {315 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 770 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 990 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4158 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 20790 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{33}}}{887040 \, d} \]
1/887040*(231*(30*tan(1/2*d*x + 1/2*c)^4 - 10*tan(1/2*d*x + 1/2*c)^2 - 3)/ (a^3*tan(1/2*d*x + 1/2*c)^5) - (315*a^30*tan(1/2*d*x + 1/2*c)^11 + 770*a^3 0*tan(1/2*d*x + 1/2*c)^9 - 990*a^30*tan(1/2*d*x + 1/2*c)^7 - 4158*a^30*tan (1/2*d*x + 1/2*c)^5 + 20790*a^30*tan(1/2*d*x + 1/2*c))/a^33)/d
Time = 13.93 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.58 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+2310\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-6930\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20790\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4158\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-990\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{887040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]
-(693*cos(c/2 + (d*x)/2)^16 + 315*sin(c/2 + (d*x)/2)^16 + 770*cos(c/2 + (d *x)/2)^2*sin(c/2 + (d*x)/2)^14 - 990*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/ 2)^12 - 4158*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 20790*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 - 6930*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d *x)/2)^4 + 2310*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2)/(887040*a^3*d* cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^5)