Integrand size = 33, antiderivative size = 273 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {b \left (5 b^2 (2 A+3 C)+6 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {A b (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]
1/16*a*(6*b^2*(3*A+4*C)+a^2*(5*A+6*C))*arctanh(sin(d*x+c))/d+1/15*b*(5*b^2 *(2*A+3*C)+6*a^2*(4*A+5*C))*tan(d*x+c)/d+1/16*a*(6*b^2*(3*A+4*C)+a^2*(5*A+ 6*C))*sec(d*x+c)*tan(d*x+c)/d+1/15*b*(A*b^2+3*a^2*(4*A+5*C))*sec(d*x+c)^2* tan(d*x+c)/d+1/120*a*(6*A*b^2+5*a^2*(5*A+6*C))*sec(d*x+c)^3*tan(d*x+c)/d+1 /10*A*b*(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 = 1.72 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.67 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x)+10 a \left (18 A b^2+a^2 (5 A+6 C)\right ) \sec ^3(c+d x)+40 a^3 A \sec ^5(c+d x)+16 b \left (15 \left (3 a^2+b^2\right ) (A+C)+5 \left (A b^2+3 a^2 (2 A+C)\right ) \tan ^2(c+d x)+9 a^2 A \tan ^4(c+d x)\right )\right )}{240 d} \]
(15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x] + 10*a*(18 *A*b^2 + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 40*a^3*A*Sec[c + d*x]^5 + 16*b* (15*(3*a^2 + b^2)*(A + C) + 5*(A*b^2 + 3*a^2*(2*A + C))*Tan[c + d*x]^2 + 9 *a^2*A*Tan[c + d*x]^4)))/(240*d)
Time = 1.82 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3527, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{6} \int (a+b \cos (c+d x))^2 \left (2 b (A+3 C) \cos ^2(c+d x)+a (5 A+6 C) \cos (c+d x)+3 A b\right ) \sec ^6(c+d x)dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (5 A+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int (a+b \cos (c+d x)) \left (5 (5 A+6 C) a^2+b (47 A+60 C) \cos (c+d x) a+6 A b^2+2 b^2 (8 A+15 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x)dx+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+6 C) a^2+b (47 A+60 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2+2 b^2 (8 A+15 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {1}{4} \int -\left (\left (8 (8 A+15 C) \cos ^2(c+d x) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \cos (c+d x)\right ) \sec ^4(c+d x)\right )dx\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \left (8 (8 A+15 C) \cos ^2(c+d x) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 (8 A+15 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 \left (15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right )+8 b \left (6 (4 A+5 C) a^2+5 b^2 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \left (15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right )+8 b \left (6 (4 A+5 C) a^2+5 b^2 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right )+8 b \left (6 (4 A+5 C) a^2+5 b^2 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \sec ^3(c+d x)dx+8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \int \sec ^2(c+d x)dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (-\frac {8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \int 1d(-\tan (c+d x))}{d}+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 b \left (6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{4 d}\right )+\frac {3 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}\) |
(A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((3*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 + 5* a^2*(5*A + 6*C))*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((8*b*(5*b^2*(2*A + 3*C) + 6*a^2*(4*A + 5*C))*Tan[c + d*x])/d + (8*b*(A*b^2 + 3*a^2*(4*A + 5*C ))*Sec[c + d*x]^2*Tan[c + d*x])/d + 15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6 *C))*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4) /5)/6
3.6.48.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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 = 12.62 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.93
| method | result | size |
| parts | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (A \,b^{3}+3 C \,a^{2} b \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A a \,b^{2}+C \,a^{3}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,b^{3} \tan \left (d x +c \right )}{d}-\frac {3 A \,a^{2} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(254\) |
| derivativedivides | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 A \,a^{2} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-3 C \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 A a \,b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \tan \left (d x +c \right ) b^{3}}{d}\) | \(298\) |
| default | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 A \,a^{2} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-3 C \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 A a \,b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 C a \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \tan \left (d x +c \right ) b^{3}}{d}\) | \(298\) |
| parallelrisch | \(\frac {-75 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {18 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+75 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {18 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+5760 \left (\frac {\left (A +\frac {5 C}{6}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) b \sin \left (2 d x +2 c \right )+850 \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {18 b^{2} \left (A +\frac {12 C}{17}\right )}{5}\right ) a \sin \left (3 d x +3 c \right )+2304 b \left (a^{2} \left (A +\frac {5 C}{4}\right )+\frac {5 b^{2} \left (A +C \right )}{12}\right ) \sin \left (4 d x +4 c \right )+150 \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {18 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) a \sin \left (5 d x +5 c \right )+384 b \left (a^{2} \left (A +\frac {5 C}{4}\right )+\frac {5 b^{2} \left (A +\frac {3 C}{2}\right )}{12}\right ) \sin \left (6 d x +6 c \right )+1980 \left (\left (A +\frac {14 C}{33}\right ) a^{2}+\frac {14 \left (A +\frac {4 C}{7}\right ) b^{2}}{11}\right ) \sin \left (d x +c \right ) a}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(365\) |
| risch | \(-\frac {i \left (-1260 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-270 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+1260 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-5760 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+1080 C a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-1440 C \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-720 C a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-1080 C a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+720 C a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-3840 A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-360 C a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+270 A a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+360 C a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+1530 A a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-240 C \,b^{3}-384 A \,a^{2} b -160 A \,b^{3}-480 C \,a^{2} b -990 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-960 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-425 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-75 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+990 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-1600 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-1920 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-420 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-510 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-90 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+425 A \,a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-2400 C \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+510 C \,a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-480 A \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+75 A \,a^{3} {\mathrm e}^{11 i \left (d x +c \right )}-1200 C \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-1200 C \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-2400 C \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+90 C \,a^{3} {\mathrm e}^{11 i \left (d x +c \right )}-240 C \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-4800 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-5760 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-2880 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+420 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-2304 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-1530 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {5 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{8 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b^{2} C}{2 d}-\frac {5 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {9 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{8 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b^{2} C}{2 d}\) | \(847\) |
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)))-(A*b^3+3*C*a^2*b)/d*(-2/3-1/3*sec(d*x+c)^ 2)*tan(d*x+c)+(3*A*a*b^2+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)-3*A*a^2*b/d*(-8/ 15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+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)))
Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (6 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, {\left (2 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} + 144 \, A a^{2} b \cos \left (d x + c\right ) + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} + 40 \, A a^{3} + 16 \, {\left (3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
1/480*(15*((5*A + 6*C)*a^3 + 6*(3*A + 4*C)*a*b^2)*cos(d*x + c)^6*log(sin(d *x + c) + 1) - 15*((5*A + 6*C)*a^3 + 6*(3*A + 4*C)*a*b^2)*cos(d*x + c)^6*l og(-sin(d*x + c) + 1) + 2*(16*(6*(4*A + 5*C)*a^2*b + 5*(2*A + 3*C)*b^3)*co s(d*x + c)^5 + 144*A*a^2*b*cos(d*x + c) + 15*((5*A + 6*C)*a^3 + 6*(3*A + 4 *C)*a*b^2)*cos(d*x + c)^4 + 40*A*a^3 + 16*(3*(4*A + 5*C)*a^2*b + 5*A*b^3)* cos(d*x + c)^3 + 10*((5*A + 6*C)*a^3 + 18*A*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.41 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{2} b + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{3} - 5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, C a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C b^{3} \tan \left (d x + c\right )}{480 \, d} \]
1/480*(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 + 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*log(sin(d*x + c) - 1)) - 90*A*a*b^2*(2*(3*si n(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(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. 932 vs. \(2 (259) = 518\).
Time = 0.36 (sec) , antiderivative size = 932, normalized size of antiderivative = 3.41 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Too large to display} \]
1/240*(15*(5*A*a^3 + 6*C*a^3 + 18*A*a*b^2 + 24*C*a*b^2)*log(abs(tan(1/2*d* x + 1/2*c) + 1)) - 15*(5*A*a^3 + 6*C*a^3 + 18*A*a*b^2 + 24*C*a*b^2)*log(ab s(tan(1/2*d*x + 1/2*c) - 1)) + 2*(165*A*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 - 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 + 36 0*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 - 24 0*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 - 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 + 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 - 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 + 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 + 60*C*a^3*t an(1/2*d*x + 1/2*c)^7 - 3744*A*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 + 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 - 2400*C*b^3*t an(1/2*d*x + 1/2*c)^7 + 450*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 60*C*a^3*tan(1/ 2*d*x + 1/2*c)^5 + 3744*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 4320*C*a^2*b*tan( 1/2*d*x + 1/2*c)^5 + 180*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 720*C*a*b^2*tan( 1/2*d*x + 1/2*c)^5 + 1440*A*b^3*tan(1/2*d*x + 1/2*c)^5 + 2400*C*b^3*tan(1/ 2*d*x + 1/2*c)^5 + 25*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 210*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 2640*C*a^2*b*tan(1/...
Time = 5.98 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.10 \[ \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (\frac {11\,A\,a^3}{8}-2\,A\,b^3+\frac {5\,C\,a^3}{4}-2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}-6\,A\,a^2\,b+3\,C\,a\,b^2-6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^3}{24}+\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}+10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b-9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,A\,a^3}{4}-12\,A\,b^3+\frac {C\,a^3}{2}-20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}-\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2-36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^3}{4}+12\,A\,b^3+\frac {C\,a^3}{2}+20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,A\,a^3}{24}-\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}-10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}-14\,A\,a^2\,b-9\,C\,a\,b^2-22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^3}{8}+2\,A\,b^3+\frac {5\,C\,a^3}{4}+2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b+3\,C\,a\,b^2+6\,C\,a^2\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atanh}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{4\,\left (\frac {5\,A\,a^3}{4}+\frac {3\,C\,a^3}{2}+\frac {9\,A\,a\,b^2}{2}+6\,C\,a\,b^2\right )}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((11*A*a^3)/8 + 2*A*b^3 + (5*C*a^3)/4 + 2*C*b^3 + (15* A*a*b^2)/4 + 6*A*a^2*b + 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 + (5*C*a^3)/4 - 2*C*b^3 + (15*A*a*b^2)/4 - 6*A*a^2*b + 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 + (7*C*a^3)/4 + 10*C*b^3 + (21*A*a*b^2)/4 + 14*A*a^2*b + 9*C*a*b^2 + 2 2*C*a^2*b) + tan(c/2 + (d*x)/2)^9*((5*A*a^3)/24 + (22*A*b^3)/3 - (7*C*a^3) /4 + 10*C*b^3 - (21*A*a*b^2)/4 + 14*A*a^2*b - 9*C*a*b^2 + 22*C*a^2*b) + ta n(c/2 + (d*x)/2)^5*((15*A*a^3)/4 + 12*A*b^3 + (C*a^3)/2 + 20*C*b^3 + (3*A* a*b^2)/2 + (156*A*a^2*b)/5 + 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 + (C*a^3)/2 - 20*C*b^3 + (3*A*a*b^2)/2 - (156*A *a^2*b)/5 + 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)^8 - 6*ta n(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a*atanh((a*tan(c/2 + (d*x)/2)*(5*A*a^2 + 18*A*b^2 + 6*C*a^2 + 24*C*b^2))/(4*((5*A*a^3)/4 + (3*C *a^3)/2 + (9*A*a*b^2)/2 + 6*C*a*b^2)))*(5*A*a^2 + 18*A*b^2 + 6*C*a^2 + 24* C*b^2))/(8*d)