Integrand size = 41, antiderivative size = 320 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac {\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {(A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \]
1/16*(18*B*a^2*b+8*B*b^3+6*a*b^2*(3*A+4*C)+a^3*(5*A+6*C))*x+1/15*(12*B*a^3 +42*B*a*b^2+9*a^2*b*(4*A+5*C)+b^3*(11*A+15*C))*sin(d*x+c)/d+1/16*(18*B*a^2 *b+8*B*b^3+6*a*b^2*(3*A+4*C)+a^3*(5*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/120* a*(6*A*b^2+42*B*a*b+5*a^2*(5*A+6*C))*cos(d*x+c)^3*sin(d*x+c)/d+1/10*(A*b+2 *B*a)*cos(d*x+c)^4*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+b *sec(d*x+c))^3*sin(d*x+c)/d-1/15*(A*b^3+4*B*a^3+12*B*a*b^2+3*a^2*b*(4*A+5* C))*sin(d*x+c)^3/d
Time = 1.99 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.15 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^3 A c+1080 a A b^2 c+1080 a^2 b B c+480 b^3 B c+360 a^3 c C+1440 a b^2 c C+300 a^3 A d x+1080 a A b^2 d x+1080 a^2 b B d x+480 b^3 B d x+360 a^3 C d x+1440 a b^2 C d x+120 \left (5 a^3 B+18 a b^2 B+2 b^3 (3 A+4 C)+3 a^2 b (5 A+6 C)\right ) \sin (c+d x)+15 \left (48 a^2 b B+16 b^3 B+48 a b^2 (A+C)+a^3 (15 A+16 C)\right ) \sin (2 (c+d x))+300 a^2 A b \sin (3 (c+d x))+80 A b^3 \sin (3 (c+d x))+100 a^3 B \sin (3 (c+d x))+240 a b^2 B \sin (3 (c+d x))+240 a^2 b C \sin (3 (c+d x))+45 a^3 A \sin (4 (c+d x))+90 a A b^2 \sin (4 (c+d x))+90 a^2 b B \sin (4 (c+d x))+30 a^3 C \sin (4 (c+d x))+36 a^2 A b \sin (5 (c+d x))+12 a^3 B \sin (5 (c+d x))+5 a^3 A \sin (6 (c+d x))}{960 d} \]
(300*a^3*A*c + 1080*a*A*b^2*c + 1080*a^2*b*B*c + 480*b^3*B*c + 360*a^3*c*C + 1440*a*b^2*c*C + 300*a^3*A*d*x + 1080*a*A*b^2*d*x + 1080*a^2*b*B*d*x + 480*b^3*B*d*x + 360*a^3*C*d*x + 1440*a*b^2*C*d*x + 120*(5*a^3*B + 18*a*b^2 *B + 2*b^3*(3*A + 4*C) + 3*a^2*b*(5*A + 6*C))*Sin[c + d*x] + 15*(48*a^2*b* B + 16*b^3*B + 48*a*b^2*(A + C) + a^3*(15*A + 16*C))*Sin[2*(c + d*x)] + 30 0*a^2*A*b*Sin[3*(c + d*x)] + 80*A*b^3*Sin[3*(c + d*x)] + 100*a^3*B*Sin[3*( c + d*x)] + 240*a*b^2*B*Sin[3*(c + d*x)] + 240*a^2*b*C*Sin[3*(c + d*x)] + 45*a^3*A*Sin[4*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 90*a^2*b*B*Sin[4 *(c + d*x)] + 30*a^3*C*Sin[4*(c + d*x)] + 36*a^2*A*b*Sin[5*(c + d*x)] + 12 *a^3*B*Sin[5*(c + d*x)] + 5*a^3*A*Sin[6*(c + d*x)])/(960*d)
Time = 1.89 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.92, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 4582, 3042, 4582, 3042, 4562, 25, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 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.
| \(\displaystyle \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (2 b (A+3 C) \sec ^2(c+d x)+(5 a A+6 b B+6 a C) \sec (c+d x)+3 (A b+2 a B)\right )dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+6 b B+6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (A b+2 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (5 (5 A+6 C) a^2+42 b B a+6 A b^2+2 b (8 A b+15 C b+6 a B) \sec ^2(c+d x)+\left (24 B a^2+b (47 A+60 C) a+30 b^2 B\right ) \sec (c+d x)\right )dx+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+6 C) a^2+42 b B a+6 A b^2+2 b (8 A b+15 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (24 B a^2+b (47 A+60 C) a+30 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 b^2 (8 A b+15 C b+6 a B) \sec ^2(c+d x)+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \sec (c+d x)+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )\right )dx\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 b^2 (8 A b+15 C b+6 a B) \sec ^2(c+d x)+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \sec (c+d x)+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )\right )dx+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 b^2 (8 A b+15 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \cos ^2(c+d x)dx+\int \cos ^3(c+d x) \left (8 b^2 (8 A b+15 C b+6 a B) \sec ^2(c+d x)+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )\right )dx\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\int \frac {8 b^2 (8 A b+15 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 (8 A b+15 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 (8 A b+15 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4532 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 (8 A b+15 C b+6 a B) b^2+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right ) \cos ^2(c+d x)\right )dx+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 (8 A b+15 C b+6 a B) b^2+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3492 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (8 \left (12 B a^3+9 b (4 A+5 C) a^2+42 b^2 B a+b^3 (11 A+15 C)\right )-24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right ) \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}+\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 \sin ^3(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )-8 \sin (c+d x) \left (12 a^3 B+9 a^2 b (4 A+5 C)+42 a b^2 B+b^3 (11 A+15 C)\right )}{d}\right )\right )+\frac {3 (2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
(A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*(A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((a*(6* A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (15*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (-8*(12*a^3*B + 42*a*b^2*B + 9*a^2*b *(4*A + 5*C) + b^3*(11*A + 15*C))*Sin[c + d*x] + 8*(A*b^3 + 4*a^3*B + 12*a *b^2*B + 3*a^2*b*(4*A + 5*C))*Sin[c + d*x]^3)/d)/4)/5)/6
3.9.86.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
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 = 0.99 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.76
| method | result | size |
| parallelrisch | \(\frac {\left (\left (225 A +240 C \right ) a^{3}+720 B \,a^{2} b +720 b^{2} \left (A +C \right ) a +240 B \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (100 B \,a^{3}+300 b \left (A +\frac {4 C}{5}\right ) a^{2}+240 B a \,b^{2}+80 A \,b^{3}\right ) \sin \left (3 d x +3 c \right )+45 \left (a^{2} \left (A +\frac {2 C}{3}\right )+2 B a b +2 A \,b^{2}\right ) a \sin \left (4 d x +4 c \right )+\left (36 A \,a^{2} b +12 B \,a^{3}\right ) \sin \left (5 d x +5 c \right )+5 a^{3} A \sin \left (6 d x +6 c \right )+\left (600 B \,a^{3}+1800 b \left (A +\frac {6 C}{5}\right ) a^{2}+2160 B a \,b^{2}+720 b^{3} \left (A +\frac {4 C}{3}\right )\right ) \sin \left (d x +c \right )+300 d x \left (a^{3} \left (A +\frac {6 C}{5}\right )+\frac {18 B \,a^{2} b}{5}+\frac {18 \left (A +\frac {4 C}{3}\right ) a \,b^{2}}{5}+\frac {8 B \,b^{3}}{5}\right )}{960 d}\) | \(244\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(370\) |
| default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 B \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(370\) |
| risch | \(\frac {9 A a \,b^{2} x}{8}+\frac {15 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{3}}{4 d}+\frac {5 a^{3} A x}{16}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{2} b}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {3 a^{3} x C}{8}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b}{4 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {9 B \,a^{2} b x}{8}+\frac {5 B \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {3 x C a \,b^{2}}{2}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b}{8 d}+\frac {9 \sin \left (d x +c \right ) B a \,b^{2}}{4 d}+\frac {9 \sin \left (d x +c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (5 d x +5 c \right ) A \,a^{2} b}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A \,b^{2}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{2} b}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B a \,b^{2}}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C a \,b^{2}}{4 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {x B \,b^{3}}{2}+\frac {3 \sin \left (d x +c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {5 a^{3} B \sin \left (d x +c \right )}{8 d}\) | \(472\) |
1/960*(((225*A+240*C)*a^3+720*B*a^2*b+720*b^2*(A+C)*a+240*B*b^3)*sin(2*d*x +2*c)+(100*B*a^3+300*b*(A+4/5*C)*a^2+240*B*a*b^2+80*A*b^3)*sin(3*d*x+3*c)+ 45*(a^2*(A+2/3*C)+2*B*a*b+2*A*b^2)*a*sin(4*d*x+4*c)+(36*A*a^2*b+12*B*a^3)* sin(5*d*x+5*c)+5*a^3*A*sin(6*d*x+6*c)+(600*B*a^3+1800*b*(A+6/5*C)*a^2+2160 *B*a*b^2+720*b^3*(A+4/3*C))*sin(d*x+c)+300*d*x*(a^3*(A+6/5*C)+18/5*B*a^2*b +18/5*(A+4/3*C)*a*b^2+8/5*B*b^3))/d
Time = 0.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.80 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 128 \, B a^{3} + 96 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 480 \, B a b^{2} + 80 \, {\left (2 \, A + 3 \, C\right )} b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (4 \, B a^{3} + 3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/240*(15*((5*A + 6*C)*a^3 + 18*B*a^2*b + 6*(3*A + 4*C)*a*b^2 + 8*B*b^3)*d *x + (40*A*a^3*cos(d*x + c)^5 + 48*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^4 + 12 8*B*a^3 + 96*(4*A + 5*C)*a^2*b + 480*B*a*b^2 + 80*(2*A + 3*C)*b^3 + 10*((5 *A + 6*C)*a^3 + 18*B*a^2*b + 18*A*a*b^2)*cos(d*x + c)^3 + 16*(4*B*a^3 + 3* (4*A + 5*C)*a^2*b + 15*B*a*b^2 + 5*A*b^3)*cos(d*x + c)^2 + 15*((5*A + 6*C) *a^3 + 18*B*a^2*b + 6*(3*A + 4*C)*a*b^2 + 8*B*b^3)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.12 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
-1/960*(5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48* sin(2*d*x + 2*c))*A*a^3 - 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*si n(d*x + c))*B*a^3 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2 *c))*C*a^3 - 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))* A*a^2*b - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^2 *b + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b - 90*(12*d*x + 12*c + s in(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^2 + 960*(sin(d*x + c)^3 - 3*si n(d*x + c))*B*a*b^2 - 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b^2 + 320*( sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^3 - 240*(2*d*x + 2*c + sin(2*d*x + 2* c))*B*b^3 - 960*C*b^3*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1307 vs. \(2 (306) = 612\).
Time = 0.37 (sec) , antiderivative size = 1307, normalized size of antiderivative = 4.08 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/240*(15*(5*A*a^3 + 6*C*a^3 + 18*B*a^2*b + 18*A*a*b^2 + 24*C*a*b^2 + 8*B* b^3)*(d*x + c) - 2*(165*A*a^3*tan(1/2*d*x + 1/2*c)^11 - 240*B*a^3*tan(1/2* d*x + 1/2*c)^11 + 150*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 720*A*a^2*b*tan(1/2* d*x + 1/2*c)^11 + 450*B*a^2*b*tan(1/2*d*x + 1/2*c)^11 - 720*C*a^2*b*tan(1/ 2*d*x + 1/2*c)^11 + 450*A*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 720*B*a*b^2*tan( 1/2*d*x + 1/2*c)^11 + 360*C*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 240*A*b^3*tan( 1/2*d*x + 1/2*c)^11 + 120*B*b^3*tan(1/2*d*x + 1/2*c)^11 - 240*C*b^3*tan(1/ 2*d*x + 1/2*c)^11 - 25*A*a^3*tan(1/2*d*x + 1/2*c)^9 - 560*B*a^3*tan(1/2*d* x + 1/2*c)^9 + 210*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 1680*A*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*B*a^2*b*tan(1/2*d*x + 1/2*c)^9 - 2640*C*a^2*b*tan(1/2*d* x + 1/2*c)^9 + 630*A*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 2640*B*a*b^2*tan(1/2*d *x + 1/2*c)^9 + 1080*C*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 880*A*b^3*tan(1/2*d* x + 1/2*c)^9 + 360*B*b^3*tan(1/2*d*x + 1/2*c)^9 - 1200*C*b^3*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 1248*B*a^3*tan(1/2*d*x + 1/ 2*c)^7 + 60*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 3744*A*a^2*b*tan(1/2*d*x + 1/2* c)^7 + 180*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 4320*C*a^2*b*tan(1/2*d*x + 1/2 *c)^7 + 180*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 4320*B*a*b^2*tan(1/2*d*x + 1/ 2*c)^7 + 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 1440*A*b^3*tan(1/2*d*x + 1/2 *c)^7 + 240*B*b^3*tan(1/2*d*x + 1/2*c)^7 - 2400*C*b^3*tan(1/2*d*x + 1/2*c) ^7 - 450*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 1248*B*a^3*tan(1/2*d*x + 1/2*c)...
Time = 18.72 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.47 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,a^3\,x}{16}+\frac {B\,b^3\,x}{2}+\frac {3\,C\,a^3\,x}{8}+\frac {9\,A\,a\,b^2\,x}{8}+\frac {9\,B\,a^2\,b\,x}{8}+\frac {3\,C\,a\,b^2\,x}{2}+\frac {3\,A\,b^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,B\,a^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {15\,A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^3\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {5\,B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{16\,d}+\frac {3\,A\,a\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a^2\,b\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {15\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{8\,d}+\frac {9\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {9\,C\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d} \]
(5*A*a^3*x)/16 + (B*b^3*x)/2 + (3*C*a^3*x)/8 + (9*A*a*b^2*x)/8 + (9*B*a^2* b*x)/8 + (3*C*a*b^2*x)/2 + (3*A*b^3*sin(c + d*x))/(4*d) + (5*B*a^3*sin(c + d*x))/(8*d) + (C*b^3*sin(c + d*x))/d + (15*A*a^3*sin(2*c + 2*d*x))/(64*d) + (3*A*a^3*sin(4*c + 4*d*x))/(64*d) + (A*a^3*sin(6*c + 6*d*x))/(192*d) + (A*b^3*sin(3*c + 3*d*x))/(12*d) + (5*B*a^3*sin(3*c + 3*d*x))/(48*d) + (B*a ^3*sin(5*c + 5*d*x))/(80*d) + (B*b^3*sin(2*c + 2*d*x))/(4*d) + (C*a^3*sin( 2*c + 2*d*x))/(4*d) + (C*a^3*sin(4*c + 4*d*x))/(32*d) + (3*A*a*b^2*sin(2*c + 2*d*x))/(4*d) + (5*A*a^2*b*sin(3*c + 3*d*x))/(16*d) + (3*A*a*b^2*sin(4* c + 4*d*x))/(32*d) + (3*A*a^2*b*sin(5*c + 5*d*x))/(80*d) + (3*B*a^2*b*sin( 2*c + 2*d*x))/(4*d) + (B*a*b^2*sin(3*c + 3*d*x))/(4*d) + (3*B*a^2*b*sin(4* c + 4*d*x))/(32*d) + (3*C*a*b^2*sin(2*c + 2*d*x))/(4*d) + (C*a^2*b*sin(3*c + 3*d*x))/(4*d) + (15*A*a^2*b*sin(c + d*x))/(8*d) + (9*B*a*b^2*sin(c + d* x))/(4*d) + (9*C*a^2*b*sin(c + d*x))/(4*d)