Integrand size = 41, antiderivative size = 336 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (8 a^3 B+30 a b^2 B+5 b^3 (2 A+3 C)+6 a^2 b (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(A b+2 a B) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \] Output:
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))*arctanh(sin(d*x+ c))/d+1/15*(8*B*a^3+30*B*a*b^2+5*b^3*(2*A+3*C)+6*a^2*b*(4*A+5*C))*tan(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))*sec(d*x+c)* tan(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))*sec(d*x+c)^ 2*tan(d*x+c)/d+1/120*a*(6*A*b^2+42*B*a*b+5*a^2*(5*A+6*C))*sec(d*x+c)^3*tan (d*x+c)/d+1/10*(A*b+2*B*a)*(a+b*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d+1/ 6*A*(a+b*cos(d*x+c))^3*sec(d*x+c)^5*tan(d*x+c)/d
Time = 3.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.75 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \sec (c+d x)+10 a \left (18 A b^2+18 a b B+a^2 (5 A+6 C)\right ) \sec ^3(c+d x)+40 a^3 A \sec ^5(c+d x)+16 \left (15 \left (a^3 B+3 a b^2 B+3 a^2 b (A+C)+b^3 (A+C)\right )+5 \left (A b^3+2 a^3 B+3 a b^2 B+3 a^2 b (2 A+C)\right ) \tan ^2(c+d x)+3 a^2 (3 A b+a B) \tan ^4(c+d x)\right )\right )}{240 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^7,x]
Output:
(15*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*ArcTanh [Sin[c + d*x]] + Tan[c + d*x]*(15*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4 *C) + a^3*(5*A + 6*C))*Sec[c + d*x] + 10*a*(18*A*b^2 + 18*a*b*B + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 40*a^3*A*Sec[c + d*x]^5 + 16*(15*(a^3*B + 3*a*b^ 2*B + 3*a^2*b*(A + C) + b^3*(A + C)) + 5*(A*b^3 + 2*a^3*B + 3*a*b^2*B + 3* a^2*b*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*(3*A*b + a*B)*Tan[c + d*x]^4)))/(2 40*d)
Time = 2.10 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.93, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3500, 27, 3042, 3227, 3042, 4254, 24, 4255, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \int (a+b \cos (c+d x))^2 \left (2 b (A+3 C) \cos ^2(c+d x)+(5 a A+6 b B+6 a C) \cos (c+d x)+3 (A b+2 a B)\right ) \sec ^6(c+d x)dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+6 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 (A b+2 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int (a+b \cos (c+d x)) \left (5 (5 A+6 C) a^2+42 b B a+6 A b^2+2 b (8 A b+15 C b+6 a B) \cos ^2(c+d x)+\left (24 B a^2+b (47 A+60 C) a+30 b^2 B\right ) \cos (c+d x)\right ) \sec ^5(c+d x)dx+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+6 C) a^2+42 b B a+6 A b^2+2 b (8 A b+15 C b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (24 B a^2+b (47 A+60 C) a+30 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}-\frac {1}{4} \int -\left (\left (8 b^2 (8 A b+15 C b+6 a B) \cos ^2(c+d x)+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \cos (c+d x)+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )\right ) \sec ^4(c+d x)\right )dx\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \left (8 b^2 (8 A b+15 C b+6 a B) \cos ^2(c+d x)+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \cos (c+d x)+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )\right ) \sec ^4(c+d x)dx+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 b^2 (8 A b+15 C b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+24 \left (4 B a^3+3 b (4 A+5 C) a^2+12 b^2 B a+A b^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 \left (15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right )+8 \left (8 B a^3+6 b (4 A+5 C) a^2+30 b^2 B a+5 b^3 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \left (15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right )+8 \left (8 B a^3+6 b (4 A+5 C) a^2+30 b^2 B a+5 b^3 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {15 \left ((5 A+6 C) a^3+18 b B a^2+6 b^2 (3 A+4 C) a+8 b^3 B\right )+8 \left (8 B a^3+6 b (4 A+5 C) a^2+30 b^2 B a+5 b^3 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \sec ^3(c+d x)dx+8 \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right ) \int \sec ^2(c+d x)dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (8 \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {8 \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right ) \int 1d(-\tan (c+d x))}{d}+15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 \tan (c+d x) \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right )}{d}+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 \tan (c+d x) \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right )}{d}+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 \tan (c+d x) \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right )}{d}+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )+\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{4 d}+\frac {1}{4} \left (15 \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 \tan (c+d x) \left (8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right )}{d}+\frac {8 \tan (c+d x) \sec ^2(c+d x) \left (4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right )}{d}\right )\right )+\frac {3 (2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]
Output:
(A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((3*(A*b + 2*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((a*(6* A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((8*(8*a^3*B + 30*a*b^2*B + 5*b^3*(2*A + 3*C) + 6*a^2*b*(4*A + 5*C))*Tan[ c + d*x])/d + (8*(A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sec[ c + d*x]^2*Tan[c + d*x])/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))*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4)/5)/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.77 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.84
| method | result | size |
| parts | \(\frac {A \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,b^{3} \tan \left (d x +c \right )}{d}\) | \(283\) |
| derivativedivides | \(\frac {A \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 A \,a^{2} b \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 B \,a^{2} b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b^{3}}{d}\) | \(444\) |
| default | \(\frac {A \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+a^{3} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 A \,a^{2} b \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+3 B \,a^{2} b \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a^{2} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a A \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B a \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \tan \left (d x +c \right ) b^{3}}{d}\) | \(444\) |
| parallelrisch | \(\frac {-450 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{3}+\frac {18 B \,a^{2} b}{5}+\frac {18 b^{2} \left (A +\frac {4 C}{3}\right ) a}{5}+\frac {8 B \,b^{3}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+450 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{3}+\frac {18 B \,a^{2} b}{5}+\frac {18 b^{2} \left (A +\frac {4 C}{3}\right ) a}{5}+\frac {8 B \,b^{3}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1920 B \,a^{3}+5760 a^{2} \left (A +\frac {3 C}{4}\right ) b +4320 B a \,b^{2}+1440 \left (A +\frac {5 C}{6}\right ) b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (\left (850 A +1020 C \right ) a^{3}+3060 B \,a^{2} b +3060 b^{2} \left (A +\frac {12 C}{17}\right ) a +720 B \,b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (768 B \,a^{3}+2304 b \left (A +\frac {5 C}{4}\right ) a^{2}+2880 B a \,b^{2}+960 b^{3} \left (A +C \right )\right ) \sin \left (4 d x +4 c \right )+\left (\left (150 A +180 C \right ) a^{3}+540 B \,a^{2} b +540 b^{2} \left (A +\frac {4 C}{3}\right ) a +240 B \,b^{3}\right ) \sin \left (5 d x +5 c \right )+\left (128 B \,a^{3}+384 b \left (A +\frac {5 C}{4}\right ) a^{2}+480 B a \,b^{2}+160 b^{3} \left (A +\frac {3 C}{2}\right )\right ) \sin \left (6 d x +6 c \right )+1980 \sin \left (d x +c \right ) \left (\left (A +\frac {14 C}{33}\right ) a^{3}+\frac {14 B \,a^{2} b}{11}+\frac {14 b^{2} \left (A +\frac {4 C}{7}\right ) a}{11}+\frac {8 B \,b^{3}}{33}\right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(471\) |
| risch | \(\text {Expression too large to display}\) | \(1245\) |
Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x,meth od=_RETURNVERBOSE)
Output:
A*a^3/d*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c) +5/16*ln(sec(d*x+c)+tan(d*x+c)))-(3*A*a^2*b+B*a^3)/d*(-8/15-1/5*sec(d*x+c) ^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+(B*b^3+3*C*a*b^2)/d*(1/2*sec(d*x+c)*tan(d *x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-(A*b^3+3*B*a*b^2+3*C*a^2*b)/d*(-2/3-1 /3*sec(d*x+c)^2)*tan(d*x+c)+(3*A*a*b^2+3*B*a^2*b+C*a^3)/d*(-(-1/4*sec(d*x+ c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))+C*b^3/d*tan (d*x+c)
Time = 0.11 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.02 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 30 \, B a b^{2} + 5 \, {\left (2 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, A a^{3} + 16 \, {\left (4 \, B a^{3} + 3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="fricas")
Output:
1/480*(15*((5*A + 6*C)*a^3 + 18*B*a^2*b + 6*(3*A + 4*C)*a*b^2 + 8*B*b^3)*c os(d*x + c)^6*log(sin(d*x + c) + 1) - 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)^6*log(-sin(d*x + c) + 1) + 2*(1 6*(8*B*a^3 + 6*(4*A + 5*C)*a^2*b + 30*B*a*b^2 + 5*(2*A + 3*C)*b^3)*cos(d*x + c)^5 + 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)^4 + 40*A*a^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)^3 + 10*((5*A + 6*C)*a^3 + 18*B*a^2*b + 18*A*a*b ^2)*cos(d*x + c)^2 + 48*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*sin(d*x + c))/(d *cos(d*x + c)^6)
Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)* *7,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.68 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="maxima")
Output:
1/480*(32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^3 + 96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^2*b + 480 *(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b + 480*(tan(d*x + c)^3 + 3*tan(d *x + c))*B*a*b^2 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^3 - 5*A*a^3*( 2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^ 6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 30*C*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c) )/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*lo g(sin(d*x + c) - 1)) - 90*B*a^2*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/( sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(s in(d*x + c) - 1)) - 90*A*a*b^2*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin (d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin( d*x + c) - 1)) - 360*C*a*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(si n(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 120*B*b^3*(2*sin(d*x + c)/(sin( d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*C*b ^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (322) = 644\).
Time = 0.23 (sec) , antiderivative size = 1370, normalized size of antiderivative = 4.08 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="giac")
Output:
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)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 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)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 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)^1 1 + 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)^1 1 + 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 + 45 0*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...
Time = 1.95 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.28 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^7,x)
Output:
(atanh((4*tan(c/2 + (d*x)/2)*((5*A*a^3)/16 + (B*b^3)/2 + (3*C*a^3)/8 + (9* A*a*b^2)/8 + (9*B*a^2*b)/8 + (3*C*a*b^2)/2))/((5*A*a^3)/4 + 2*B*b^3 + (3*C *a^3)/2 + (9*A*a*b^2)/2 + (9*B*a^2*b)/2 + 6*C*a*b^2))*((5*A*a^3)/8 + B*b^3 + (3*C*a^3)/4 + (9*A*a*b^2)/4 + (9*B*a^2*b)/4 + 3*C*a*b^2))/d + (tan(c/2 + (d*x)/2)*((11*A*a^3)/8 + 2*A*b^3 + 2*B*a^3 + B*b^3 + (5*C*a^3)/4 + 2*C*b ^3 + (15*A*a*b^2)/4 + 6*A*a^2*b + 6*B*a*b^2 + (15*B*a^2*b)/4 + 3*C*a*b^2 + 6*C*a^2*b) + tan(c/2 + (d*x)/2)^11*((11*A*a^3)/8 - 2*A*b^3 - 2*B*a^3 + B* b^3 + (5*C*a^3)/4 - 2*C*b^3 + (15*A*a*b^2)/4 - 6*A*a^2*b - 6*B*a*b^2 + (15 *B*a^2*b)/4 + 3*C*a*b^2 - 6*C*a^2*b) - tan(c/2 + (d*x)/2)^3*((22*A*b^3)/3 - (5*A*a^3)/24 + (14*B*a^3)/3 + 3*B*b^3 + (7*C*a^3)/4 + 10*C*b^3 + (21*A*a *b^2)/4 + 14*A*a^2*b + 22*B*a*b^2 + (21*B*a^2*b)/4 + 9*C*a*b^2 + 22*C*a^2* b) + tan(c/2 + (d*x)/2)^9*((5*A*a^3)/24 + (22*A*b^3)/3 + (14*B*a^3)/3 - 3* B*b^3 - (7*C*a^3)/4 + 10*C*b^3 - (21*A*a*b^2)/4 + 14*A*a^2*b + 22*B*a*b^2 - (21*B*a^2*b)/4 - 9*C*a*b^2 + 22*C*a^2*b) + tan(c/2 + (d*x)/2)^5*((15*A*a ^3)/4 + 12*A*b^3 + (52*B*a^3)/5 + 2*B*b^3 + (C*a^3)/2 + 20*C*b^3 + (3*A*a* b^2)/2 + (156*A*a^2*b)/5 + 36*B*a*b^2 + (3*B*a^2*b)/2 + 6*C*a*b^2 + 36*C*a ^2*b) + tan(c/2 + (d*x)/2)^7*((15*A*a^3)/4 - 12*A*b^3 - (52*B*a^3)/5 + 2*B *b^3 + (C*a^3)/2 - 20*C*b^3 + (3*A*a*b^2)/2 - (156*A*a^2*b)/5 - 36*B*a*b^2 + (3*B*a^2*b)/2 + 6*C*a*b^2 - 36*C*a^2*b))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2...
Time = 0.21 (sec) , antiderivative size = 1767, normalized size of antiderivative = 5.26 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7,x)
Output:
( - 75*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**4 - 90*co s(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**3*c - 540*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**2*b**2 - 360*cos(c + d*x )*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**2*c - 120*cos(c + d*x)*lo g(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b**4 + 225*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**4 + 270*cos(c + d*x)*log(tan((c + d*x)/ 2) - 1)*sin(c + d*x)**4*a**3*c + 1620*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b**2 + 1080*cos(c + d*x)*log(tan((c + d*x)/2) - 1) *sin(c + d*x)**4*a*b**2*c + 360*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin (c + d*x)**4*b**4 - 225*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x )**2*a**4 - 270*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a** 3*c - 1620*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b** 2 - 1080*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**2*c - 360*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**4 + 75*cos( c + d*x)*log(tan((c + d*x)/2) - 1)*a**4 + 90*cos(c + d*x)*log(tan((c + d*x )/2) - 1)*a**3*c + 540*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**2*b**2 + 360*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**2*c + 120*cos(c + d*x)*log (tan((c + d*x)/2) - 1)*b**4 + 75*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*si n(c + d*x)**6*a**4 + 90*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x )**6*a**3*c + 540*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**...