Integrand size = 41, antiderivative size = 314 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {b^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d} \]
1/8*(3*B*a^4+24*B*a^2*b^2+8*B*b^4+16*a*b^3*(A+2*C)+4*a^3*b*(3*A+4*C))*x+b^ 4*C*arctanh(sin(d*x+c))/d+1/30*(12*A*b^4+80*B*a^3*b+95*B*a*b^3+4*a^4*(4*A+ 5*C)+2*a^2*b^2*(56*A+85*C))*sin(d*x+c)/d+1/120*a*(24*A*b^3+45*B*a^3+130*B* a*b^2+4*a^2*b*(29*A+40*C))*cos(d*x+c)*sin(d*x+c)/d+1/60*(12*A*b^2+35*B*a*b +4*a^2*(4*A+5*C))*cos(d*x+c)^2*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/20*(4*A*b +5*B*a)*cos(d*x+c)^3*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/5*A*cos(d*x+c)^4*(a +b*sec(d*x+c))^4*sin(d*x+c)/d
Time = 3.00 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.22 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {720 a^3 A b c+960 a A b^3 c+180 a^4 B c+1440 a^2 b^2 B c+480 b^4 B c+960 a^3 b c C+1920 a b^3 c C+720 a^3 A b d x+960 a A b^3 d x+180 a^4 B d x+1440 a^2 b^2 B d x+480 b^4 B d x+960 a^3 b C d x+1920 a b^3 C d x-480 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \left (8 A b^4+24 a^3 b B+32 a b^3 B+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \sin (c+d x)+120 a \left (4 A b^3+a^3 B+6 a b^2 B+4 a^2 b (A+C)\right ) \sin (2 (c+d x))+50 a^4 A \sin (3 (c+d x))+240 a^2 A b^2 \sin (3 (c+d x))+160 a^3 b B \sin (3 (c+d x))+40 a^4 C \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+15 a^4 B \sin (4 (c+d x))+6 a^4 A \sin (5 (c+d x))}{480 d} \]
(720*a^3*A*b*c + 960*a*A*b^3*c + 180*a^4*B*c + 1440*a^2*b^2*B*c + 480*b^4* B*c + 960*a^3*b*c*C + 1920*a*b^3*c*C + 720*a^3*A*b*d*x + 960*a*A*b^3*d*x + 180*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 480*b^4*B*d*x + 960*a^3*b*C*d*x + 19 20*a*b^3*C*d*x - 480*b^4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480* b^4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sin[c + d*x] + 12 0*a*(4*A*b^3 + a^3*B + 6*a*b^2*B + 4*a^2*b*(A + C))*Sin[2*(c + d*x)] + 50* a^4*A*Sin[3*(c + d*x)] + 240*a^2*A*b^2*Sin[3*(c + d*x)] + 160*a^3*b*B*Sin[ 3*(c + d*x)] + 40*a^4*C*Sin[3*(c + d*x)] + 60*a^3*A*b*Sin[4*(c + d*x)] + 1 5*a^4*B*Sin[4*(c + d*x)] + 6*a^4*A*Sin[5*(c + d*x)])/(480*d)
Time = 2.23 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 4582, 3042, 4582, 3042, 4582, 3042, 4562, 25, 3042, 4535, 24, 3042, 4533, 3042, 4257}
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.
| \(\displaystyle \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (5 b C \sec ^2(c+d x)+(4 a A+5 b B+5 a C) \sec (c+d x)+4 A b+5 a B\right )dx+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (5 b C \csc \left (c+d x+\frac {\pi }{2}\right )^2+(4 a A+5 b B+5 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 A b+5 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (4 (4 A+5 C) a^2+35 b B a+12 A b^2+20 b^2 C \sec ^2(c+d x)+\left (15 B a^2+28 A b a+40 b C a+20 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (4 (4 A+5 C) a^2+35 b B a+12 A b^2+20 b^2 C \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (15 B a^2+28 A b a+40 b C a+20 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (45 B a^3+4 b (29 A+40 C) a^2+130 b^2 B a+24 A b^3+60 b^3 C \sec ^2(c+d x)+\left (8 (4 A+5 C) a^3+115 b B a^2+36 b^2 (3 A+5 C) a+60 b^3 B\right ) \sec (c+d x)\right )dx+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (45 B a^3+4 b (29 A+40 C) a^2+130 b^2 B a+24 A b^3+60 b^3 C \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (8 (4 A+5 C) a^3+115 b B a^2+36 b^2 (3 A+5 C) a+60 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}-\frac {1}{2} \int -\cos (c+d x) \left (120 C \sec ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \sec (c+d x)\right )dx\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \cos (c+d x) \left (120 C \sec ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \sec (c+d x)\right )dx+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {120 C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (120 C \sec ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )\right )dx+15 \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \int 1dx\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \cos (c+d x) \left (120 C \sec ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )\right )dx+15 x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {120 C \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+15 x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (120 b^4 C \int \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}+15 x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (120 b^4 C \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 \sin (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}+15 x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )\right )+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}+\frac {1}{2} \left (\frac {4 \sin (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}+15 x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac {120 b^4 C \text {arctanh}(\sin (c+d x))}{d}\right )\right )\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}\) |
(A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + (((4*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + (((12*A *b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2 *Sin[c + d*x])/(3*d) + ((a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(2 9*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (15*(3*a^4*B + 24*a^2*b^2* B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x + (120*b^4*C*Arc Tanh[Sin[c + d*x]])/d + (4*(12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4* A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/d)/2)/3)/4)/5
3.9.94.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Time = 1.00 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(\frac {-480 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{4}+480 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{4}+480 a \left (\frac {B \,a^{3}}{4}+b \left (A +C \right ) a^{2}+\frac {3 B a \,b^{2}}{2}+A \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (\left (50 A +40 C \right ) a^{4}+160 B \,a^{3} b +240 A \,a^{2} b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (60 A \,a^{3} b +15 B \,a^{4}\right ) \sin \left (4 d x +4 c \right )+6 a^{4} A \sin \left (5 d x +5 c \right )+\left (\left (300 A +360 C \right ) a^{4}+1440 B \,a^{3} b +2160 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}+1920 B a \,b^{3}+480 A \,b^{4}\right ) \sin \left (d x +c \right )+720 d \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {4 C}{3}\right ) a^{3}+2 B \,a^{2} b^{2}+\frac {4 a \,b^{3} \left (A +2 C \right )}{3}+\frac {2 B \,b^{4}}{3}\right ) x}{480 d}\) | \(258\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(355\) |
| default | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(355\) |
| risch | \(\frac {5 a^{4} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3} b}{2 d}+\frac {3 B \,a^{4} x}{8}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{8 d}+\frac {3 a^{3} A b x}{2}-\frac {5 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {5 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{3} b}{8 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3} b}{3 d}+x B \,b^{4}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{80 d}+4 x C a \,b^{3}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3} b}{2 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}+2 A a \,b^{3} x +3 B \,a^{2} b^{2} x +2 C \,a^{3} b x +\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{4}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{4}}{d}\) | \(606\) |
1/480*(-480*C*ln(tan(1/2*d*x+1/2*c)-1)*b^4+480*C*ln(tan(1/2*d*x+1/2*c)+1)* b^4+480*a*(1/4*B*a^3+b*(A+C)*a^2+3/2*B*a*b^2+A*b^3)*sin(2*d*x+2*c)+((50*A+ 40*C)*a^4+160*B*a^3*b+240*A*a^2*b^2)*sin(3*d*x+3*c)+(60*A*a^3*b+15*B*a^4)* sin(4*d*x+4*c)+6*a^4*A*sin(5*d*x+5*c)+((300*A+360*C)*a^4+1440*B*a^3*b+2160 *b^2*(A+4/3*C)*a^2+1920*B*a*b^3+480*A*b^4)*sin(d*x+c)+720*d*(1/4*B*a^4+b*( A+4/3*C)*a^3+2*B*a^2*b^2+4/3*a*b^3*(A+2*C)+2/3*B*b^4)*x)/d
Time = 0.33 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.83 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {60 \, C b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, {\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 16 \, {\left (4 \, A + 5 \, C\right )} a^{4} + 320 \, B a^{3} b + 240 \, {\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 480 \, B a b^{3} + 120 \, A b^{4} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/120*(60*C*b^4*log(sin(d*x + c) + 1) - 60*C*b^4*log(-sin(d*x + c) + 1) + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*(A + 2*C)*a*b^3 + 8* B*b^4)*d*x + (24*A*a^4*cos(d*x + c)^4 + 16*(4*A + 5*C)*a^4 + 320*B*a^3*b + 240*(2*A + 3*C)*a^2*b^2 + 480*B*a*b^3 + 120*A*b^4 + 30*(B*a^4 + 4*A*a^3*b )*cos(d*x + c)^3 + 8*((4*A + 5*C)*a^4 + 20*B*a^3*b + 30*A*a^2*b^2)*cos(d*x + c)^2 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3)*c os(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.11 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 1920 \, {\left (d x + c\right )} C a b^{3} + 480 \, {\left (d x + c\right )} B b^{4} + 240 \, C b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2880 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 1920 \, B a b^{3} \sin \left (d x + c\right ) + 480 \, A b^{4} \sin \left (d x + c\right )}{480 \, d} \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 160*(s in(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 60*(12*d*x + 12*c + sin(4*d*x + 4* c) + 8*sin(2*d*x + 2*c))*A*a^3*b - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B *a^3*b + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3*b - 960*(sin(d*x + c)^ 3 - 3*sin(d*x + c))*A*a^2*b^2 + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2 *b^2 + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^3 + 1920*(d*x + c)*C*a*b ^3 + 480*(d*x + c)*B*b^4 + 240*C*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 2880*C*a^2*b^2*sin(d*x + c) + 1920*B*a*b^3*sin(d*x + c) + 480 *A*b^4*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1094 vs. \(2 (302) = 604\).
Time = 0.39 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.48 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/120*(120*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*C*b^4*log(abs(ta n(1/2*d*x + 1/2*c) - 1)) + 15*(3*B*a^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^ 2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*(d*x + c) + 2*(120*A*a^4*tan(1/ 2*d*x + 1/2*c)^9 - 75*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 300*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1/2*d *x + 1/2*c)^9 - 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1 /2*d*x + 1/2*c)^9 - 240*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3*tan(1 /2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 160*A*a^4*tan(1/2*d *x + 1/2*c)^7 - 30*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 320*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 120*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 1280*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 1920*A*a^2*b^2*tan(1/2*d* x + 1/2*c)^7 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 2880*C*a^2*b^2*tan(1 /2*d*x + 1/2*c)^7 - 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 1920*B*a*b^3*tan( 1/2*d*x + 1/2*c)^7 + 480*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^4*tan(1/2* d*x + 1/2*c)^5 + 400*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a^3*b*tan(1/2*d *x + 1/2*c)^5 + 2400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 4320*C*a^2*b^2*tan (1/2*d*x + 1/2*c)^5 + 2880*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 720*A*b^4*tan( 1/2*d*x + 1/2*c)^5 + 160*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^4*tan(1/2*d *x + 1/2*c)^3 + 320*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^3*b*tan(1/2*...
Time = 20.42 (sec) , antiderivative size = 4118, normalized size of antiderivative = 13.11 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
(tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + (5*B*a^4)/4 + 2*C*a^4 + 12*A*a^2* b^2 + 6*B*a^2*b^2 + 12*C*a^2*b^2 + 4*A*a*b^3 + 5*A*a^3*b + 8*B*a*b^3 + 8*B *a^3*b + 4*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(2*A*a^4 + 2*A*b^4 - (5*B*a^4)/ 4 + 2*C*a^4 + 12*A*a^2*b^2 - 6*B*a^2*b^2 + 12*C*a^2*b^2 - 4*A*a*b^3 - 5*A* a^3*b + 8*B*a*b^3 + 8*B*a^3*b - 4*C*a^3*b) + tan(c/2 + (d*x)/2)^3*((8*A*a^ 4)/3 + 8*A*b^4 + (B*a^4)/2 + (16*C*a^4)/3 + 32*A*a^2*b^2 + 12*B*a^2*b^2 + 48*C*a^2*b^2 + 8*A*a*b^3 + 2*A*a^3*b + 32*B*a*b^3 + (64*B*a^3*b)/3 + 8*C*a ^3*b) + tan(c/2 + (d*x)/2)^7*((8*A*a^4)/3 + 8*A*b^4 - (B*a^4)/2 + (16*C*a^ 4)/3 + 32*A*a^2*b^2 - 12*B*a^2*b^2 + 48*C*a^2*b^2 - 8*A*a*b^3 - 2*A*a^3*b + 32*B*a*b^3 + (64*B*a^3*b)/3 - 8*C*a^3*b) + tan(c/2 + (d*x)/2)^5*((116*A* a^4)/15 + 12*A*b^4 + (20*C*a^4)/3 + 40*A*a^2*b^2 + 72*C*a^2*b^2 + 48*B*a*b ^3 + (80*B*a^3*b)/3))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^1 0 + 1)) + (atan(((tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2* b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^ 7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^ 4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + ((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C...