Integrand size = 21, antiderivative size = 109 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {4 \cot ^5(c+d x)}{5 a^2 d}-\frac {5 \cot ^7(c+d x)}{7 a^2 d}-\frac {2 \cot ^9(c+d x)}{9 a^2 d}-\frac {2 \csc ^7(c+d x)}{7 a^2 d}+\frac {2 \csc ^9(c+d x)}{9 a^2 d} \]
-1/3*cot(d*x+c)^3/a^2/d-4/5*cot(d*x+c)^5/a^2/d-5/7*cot(d*x+c)^7/a^2/d-2/9* cot(d*x+c)^9/a^2/d-2/7*csc(d*x+c)^7/a^2/d+2/9*csc(d*x+c)^9/a^2/d
Time = 1.32 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.75 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\csc (c) \csc ^5(c+d x) \sec ^2(c+d x) (-61440 \sin (c)+84480 \sin (d x)+25875 \sin (c+d x)+11500 \sin (2 (c+d x))-10925 \sin (3 (c+d x))-9200 \sin (4 (c+d x))+575 \sin (5 (c+d x))+2300 \sin (6 (c+d x))+575 \sin (7 (c+d x))-107520 \sin (2 c+d x)-10240 \sin (c+2 d x)+9728 \sin (2 c+3 d x)+8192 \sin (3 c+4 d x)-512 \sin (4 c+5 d x)-2048 \sin (5 c+6 d x)-512 \sin (6 c+7 d x))}{1290240 a^2 d (1+\sec (c+d x))^2} \]
(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^2*(-61440*Sin[c] + 84480*Sin[d*x] + 25 875*Sin[c + d*x] + 11500*Sin[2*(c + d*x)] - 10925*Sin[3*(c + d*x)] - 9200* Sin[4*(c + d*x)] + 575*Sin[5*(c + d*x)] + 2300*Sin[6*(c + d*x)] + 575*Sin[ 7*(c + d*x)] - 107520*Sin[2*c + d*x] - 10240*Sin[c + 2*d*x] + 9728*Sin[2*c + 3*d*x] + 8192*Sin[3*c + 4*d*x] - 512*Sin[4*c + 5*d*x] - 2048*Sin[5*c + 6*d*x] - 512*Sin[6*c + 7*d*x]))/(1290240*a^2*d*(1 + Sec[c + d*x])^2)
Time = 0.66 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 3042, 3354, 3042, 3352, 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.
| \(\displaystyle \int \frac {\csc ^6(c+d x)}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^4(c+d x)}{(a (-\cos (c+d x))-a)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int (a-a \cos (c+d x))^2 \cot ^2(c+d x) \csc ^8(c+d x)dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{10}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^2(c+d x) \csc ^8(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^7(c+d x)+a^2 \cot ^4(c+d x) \csc ^6(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {5 a^2 \cot ^7(c+d x)}{7 d}-\frac {4 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {2 a^2 \csc ^9(c+d x)}{9 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}}{a^4}\) |
(-1/3*(a^2*Cot[c + d*x]^3)/d - (4*a^2*Cot[c + d*x]^5)/(5*d) - (5*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (2*a^2*Csc[c + d*x]^7)/( 7*d) + (2*a^2*Csc[c + d*x]^9)/(9*d))/a^4
3.1.89.3.1 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 = 0.72 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01
| method | result | size |
| parallelrisch | \(\frac {35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+135 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+63 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-63 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-525 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-315 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-1575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-315 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{40320 a^{2} d}\) | \(110\) |
| derivativedivides | \(\frac {-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{2}}\) | \(112\) |
| default | \(\frac {-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 d \,a^{2}}\) | \(112\) |
| risch | \(-\frac {16 i \left (210 \,{\mathrm e}^{8 i \left (d x +c \right )}+120 \,{\mathrm e}^{7 i \left (d x +c \right )}+165 \,{\mathrm e}^{6 i \left (d x +c \right )}-20 \,{\mathrm e}^{5 i \left (d x +c \right )}+19 \,{\mathrm e}^{4 i \left (d x +c \right )}+16 \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{315 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(126\) |
| norman | \(\frac {-\frac {1}{640 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{896 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{1152 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{128 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{384 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{640 d a}}{a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\) | \(158\) |
1/40320*(35*tan(1/2*d*x+1/2*c)^9+135*tan(1/2*d*x+1/2*c)^7+63*tan(1/2*d*x+1 /2*c)^5-63*cot(1/2*d*x+1/2*c)^5-525*tan(1/2*d*x+1/2*c)^3-315*cot(1/2*d*x+1 /2*c)^3-1575*tan(1/2*d*x+1/2*c)-315*cot(1/2*d*x+1/2*c))/a^2/d
Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.55 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {8 \, \cos \left (d x + c\right )^{7} + 16 \, \cos \left (d x + c\right )^{6} - 12 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2} - 40 \, \cos \left (d x + c\right ) - 20}{315 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )} \]
1/315*(8*cos(d*x + c)^7 + 16*cos(d*x + c)^6 - 12*cos(d*x + c)^5 - 40*cos(d *x + c)^4 - 5*cos(d*x + c)^3 + 30*cos(d*x + c)^2 - 40*cos(d*x + c) - 20)/( (a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4* a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d )*sin(d*x + c))
\[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.60 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {\frac {1575 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {525 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {135 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{2}} + \frac {63 \, {\left (\frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{2} \sin \left (d x + c\right )^{5}}}{40320 \, d} \]
-1/40320*((1575*sin(d*x + c)/(cos(d*x + c) + 1) + 525*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 - 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 135*sin(d*x + c )^7/(cos(d*x + c) + 1)^7 - 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a^2 + 6 3*(5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1)*(cos(d*x + c) + 1)^5/(a^2*sin(d*x + c)^5))/d
Time = 0.41 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {63 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {35 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 135 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 525 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, a^{16} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{40320 \, d} \]
-1/40320*(63*(5*tan(1/2*d*x + 1/2*c)^4 + 5*tan(1/2*d*x + 1/2*c)^2 + 1)/(a^ 2*tan(1/2*d*x + 1/2*c)^5) - (35*a^16*tan(1/2*d*x + 1/2*c)^9 + 135*a^16*tan (1/2*d*x + 1/2*c)^7 + 63*a^16*tan(1/2*d*x + 1/2*c)^5 - 525*a^16*tan(1/2*d* x + 1/2*c)^3 - 1575*a^16*tan(1/2*d*x + 1/2*c))/a^18)/d
Time = 14.39 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {375\,\cos \left (c+d\,x\right )}{8}-\frac {5\,\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {19\,\cos \left (3\,c+3\,d\,x\right )}{8}+2\,\cos \left (4\,c+4\,d\,x\right )-\frac {\cos \left (5\,c+5\,d\,x\right )}{8}-\frac {\cos \left (6\,c+6\,d\,x\right )}{2}-\frac {\cos \left (7\,c+7\,d\,x\right )}{8}+15}{40320\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \]