Integrand size = 41, antiderivative size = 287 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a^4 (44 A+49 B+56 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac {a^4 (44 A+49 B+56 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 (988 A+1113 B+1232 C) \sec ^2(c+d x) \tan (c+d x)}{840 d}+\frac {(436 A+511 B+504 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{840 d}+\frac {(16 A+21 B+14 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{70 d}+\frac {a (4 A+7 B) (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{42 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d} \]
1/16*a^4*(44*A+49*B+56*C)*arctanh(sin(d*x+c))/d+1/105*a^4*(454*A+504*B+581 *C)*tan(d*x+c)/d+1/16*a^4*(44*A+49*B+56*C)*sec(d*x+c)*tan(d*x+c)/d+1/840*a ^4*(988*A+1113*B+1232*C)*sec(d*x+c)^2*tan(d*x+c)/d+1/840*(436*A+511*B+504* C)*(a^4+a^4*cos(d*x+c))*sec(d*x+c)^3*tan(d*x+c)/d+1/70*(16*A+21*B+14*C)*(a ^2+a^2*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d+1/42*a*(4*A+7*B)*(a+a*cos(d *x+c))^3*sec(d*x+c)^5*tan(d*x+c)/d+1/7*A*(a+a*cos(d*x+c))^4*sec(d*x+c)^6*t an(d*x+c)/d
Time = 6.97 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.52 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a^4 \left (105 (44 A+49 B+56 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (105 (44 A+49 B+56 C) \sec (c+d x)+70 (44 A+41 B+24 C) \sec ^3(c+d x)+280 (4 A+B) \sec ^5(c+d x)+16 \left (840 (A+B+C)+140 (4 A+3 B+2 C) \tan ^2(c+d x)+21 (9 A+4 B+C) \tan ^4(c+d x)+15 A \tan ^6(c+d x)\right )\right )\right )}{1680 d} \]
(a^4*(105*(44*A + 49*B + 56*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(105*( 44*A + 49*B + 56*C)*Sec[c + d*x] + 70*(44*A + 41*B + 24*C)*Sec[c + d*x]^3 + 280*(4*A + B)*Sec[c + d*x]^5 + 16*(840*(A + B + C) + 140*(4*A + 3*B + 2* C)*Tan[c + d*x]^2 + 21*(9*A + 4*B + C)*Tan[c + d*x]^4 + 15*A*Tan[c + d*x]^ 6))))/(1680*d)
Time = 2.36 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.06, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 3522, 3042, 3454, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 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.
| \(\displaystyle \int \sec ^8(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (a (4 A+7 B)+a (2 A+7 C) \cos (c+d x)) \sec ^7(c+d x)dx}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (4 A+7 B)+a (2 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{6} \int (\cos (c+d x) a+a)^3 \left (3 (16 A+21 B+14 C) a^2+2 (10 A+7 B+21 C) \cos (c+d x) a^2\right ) \sec ^6(c+d x)dx+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 (16 A+21 B+14 C) a^2+2 (10 A+7 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int (\cos (c+d x) a+a)^2 \left ((436 A+511 B+504 C) a^3+98 (2 A+2 B+3 C) \cos (c+d x) a^3\right ) \sec ^5(c+d x)dx+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((436 A+511 B+504 C) a^3+98 (2 A+2 B+3 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 (\cos (c+d x) a+a) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \cos (c+d x) a^4\right ) \sec ^4(c+d x)dx+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (\cos (c+d x) a+a) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \cos (c+d x) a^4\right ) \sec ^4(c+d x)dx+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \left (2 (276 A+301 B+364 C) \cos ^2(c+d x) a^5+(988 A+1113 B+1232 C) a^5+\left (2 (276 A+301 B+364 C) a^5+(988 A+1113 B+1232 C) a^5\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {2 (276 A+301 B+364 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(988 A+1113 B+1232 C) a^5+\left (2 (276 A+301 B+364 C) a^5+(988 A+1113 B+1232 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \left (105 (44 A+49 B+56 C) a^5+8 (454 A+504 B+581 C) \cos (c+d x) a^5\right ) \sec ^3(c+d x)dx+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \frac {105 (44 A+49 B+56 C) a^5+8 (454 A+504 B+581 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \int \sec ^3(c+d x)dx+8 a^5 (454 A+504 B+581 C) \int \sec ^2(c+d x)dx\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^5 (454 A+504 B+581 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+105 a^5 (44 A+49 B+56 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^5 (454 A+504 B+581 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 a^5 (454 A+504 B+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^5 (454 A+504 B+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^5 (454 A+504 B+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {a^2 (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^5 (454 A+504 B+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + ((a^2*(4*A + 7*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((3*a^3 *(16*A + 21*B + 14*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/ (5*d) + (((436*A + 511*B + 504*C)*(a^5 + a^5*Cos[c + d*x])*Sec[c + d*x]^3* Tan[c + d*x])/(4*d) + (3*((a^5*(988*A + 1113*B + 1232*C)*Sec[c + d*x]^2*Ta n[c + d*x])/(3*d) + ((8*a^5*(454*A + 504*B + 581*C)*Tan[c + d*x])/d + 105* a^5*(44*A + 49*B + 56*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[ c + d*x])/(2*d)))/3))/4)/5)/6)/(7*a)
3.4.38.3.1 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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
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_)])^(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/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
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 = 14.94 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.08
| method | result | size |
| parallelrisch | \(-\frac {11 a^{4} \left (\left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \left (A +\frac {49 B}{44}+\frac {14 C}{11}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) \left (A +\frac {49 B}{44}+\frac {14 C}{11}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {134 A}{33}-\frac {217 B}{66}-\frac {204 C}{77}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {248 A}{55}-\frac {1696 B}{385}-\frac {1604 C}{385}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {40 A}{21}-\frac {458 B}{231}-\frac {144 C}{77}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {1816 A}{1155}-\frac {2204 C}{1155}-\frac {96 B}{55}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {2 A}{7}-\frac {4 C}{11}-\frac {7 B}{22}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {332 C}{1155}-\frac {96 B}{385}-\frac {1816 A}{8085}\right ) \sin \left (7 d x +7 c \right )-\frac {40 \left (A +\frac {4 B}{5}+\frac {7 C}{10}\right ) \sin \left (d x +c \right )}{11}\right )}{4 \left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) d}\) | \(311\) |
| parts | \(-\frac {a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,a^{4} \tan \left (d x +c \right )}{d}\) | \(339\) |
| risch | \(-\frac {i a^{4} \left (-68320 A \,{\mathrm e}^{8 i \left (d x +c \right )}-9296 C -7264 A -8064 B -149184 A \,{\mathrm e}^{4 i \left (d x +c \right )}-155904 B \,{\mathrm e}^{4 i \left (d x +c \right )}+4620 A \,{\mathrm e}^{13 i \left (d x +c \right )}+5145 B \,{\mathrm e}^{13 i \left (d x +c \right )}-56448 B \,{\mathrm e}^{2 i \left (d x +c \right )}-164976 C \,{\mathrm e}^{4 i \left (d x +c \right )}-63392 C \,{\mathrm e}^{2 i \left (d x +c \right )}-13440 B \,{\mathrm e}^{10 i \left (d x +c \right )}-30800 A \,{\mathrm e}^{3 i \left (d x +c \right )}-65660 A \,{\mathrm e}^{5 i \left (d x +c \right )}-53165 B \,{\mathrm e}^{5 i \left (d x +c \right )}-50848 A \,{\mathrm e}^{2 i \left (d x +c \right )}-4620 A \,{\mathrm e}^{i \left (d x +c \right )}-5145 B \,{\mathrm e}^{i \left (d x +c \right )}-188160 B \,{\mathrm e}^{6 i \left (d x +c \right )}-32060 B \,{\mathrm e}^{3 i \left (d x +c \right )}-1680 C \,{\mathrm e}^{12 i \left (d x +c \right )}-3360 A \,{\mathrm e}^{10 i \left (d x +c \right )}+5880 C \,{\mathrm e}^{13 i \left (d x +c \right )}-185920 A \,{\mathrm e}^{6 i \left (d x +c \right )}-203840 C \,{\mathrm e}^{6 i \left (d x +c \right )}-5880 C \,{\mathrm e}^{i \left (d x +c \right )}-94080 B \,{\mathrm e}^{8 i \left (d x +c \right )}-42840 C \,{\mathrm e}^{5 i \left (d x +c \right )}-30240 C \,{\mathrm e}^{3 i \left (d x +c \right )}+32060 B \,{\mathrm e}^{11 i \left (d x +c \right )}+30800 A \,{\mathrm e}^{11 i \left (d x +c \right )}+30240 C \,{\mathrm e}^{11 i \left (d x +c \right )}-30240 C \,{\mathrm e}^{10 i \left (d x +c \right )}-121520 C \,{\mathrm e}^{8 i \left (d x +c \right )}+42840 C \,{\mathrm e}^{9 i \left (d x +c \right )}+65660 A \,{\mathrm e}^{9 i \left (d x +c \right )}+53165 B \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {11 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{16 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {11 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{16 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\) | \(574\) |
| derivativedivides | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \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 \,a^{4} \tan \left (d x +c \right )+4 a^{4} A \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 )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{4} \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 )-6 a^{4} A \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 )+6 B \,a^{4} \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 )-6 C \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \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 )-4 B \,a^{4} \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 )+4 C \,a^{4} \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 )-a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+B \,a^{4} \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^{4} \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}\) | \(577\) |
| default | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{4} \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 \,a^{4} \tan \left (d x +c \right )+4 a^{4} A \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 )-4 B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{4} \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 )-6 a^{4} A \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 )+6 B \,a^{4} \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 )-6 C \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \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 )-4 B \,a^{4} \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 )+4 C \,a^{4} \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 )-a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+B \,a^{4} \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^{4} \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}\) | \(577\) |
-11/4*a^4*((cos(5*d*x+5*c)+1/7*cos(7*d*x+7*c)+3*cos(3*d*x+3*c)+5*cos(d*x+c ))*(A+49/44*B+14/11*C)*ln(tan(1/2*d*x+1/2*c)-1)-(cos(5*d*x+5*c)+1/7*cos(7* d*x+7*c)+3*cos(3*d*x+3*c)+5*cos(d*x+c))*(A+49/44*B+14/11*C)*ln(tan(1/2*d*x +1/2*c)+1)+(-134/33*A-217/66*B-204/77*C)*sin(2*d*x+2*c)+(-248/55*A-1696/38 5*B-1604/385*C)*sin(3*d*x+3*c)+(-40/21*A-458/231*B-144/77*C)*sin(4*d*x+4*c )+(-1816/1155*A-2204/1155*C-96/55*B)*sin(5*d*x+5*c)+(-2/7*A-4/11*C-7/22*B) *sin(6*d*x+6*c)+(-332/1155*C-96/385*B-1816/8085*A)*sin(7*d*x+7*c)-40/11*(A +4/5*B+7/10*C)*sin(d*x+c))/(cos(5*d*x+5*c)+1/7*cos(7*d*x+7*c)+3*cos(3*d*x+ 3*c)+5*cos(d*x+c))/d
Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.79 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 16 \, {\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (44 \, A + 41 \, B + 24 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 48 \, {\left (48 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 240 \, A a^{4}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^8, x, algorithm="fricas")
1/3360*(105*(44*A + 49*B + 56*C)*a^4*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(44*A + 49*B + 56*C)*a^4*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*( 16*(454*A + 504*B + 581*C)*a^4*cos(d*x + c)^6 + 105*(44*A + 49*B + 56*C)*a ^4*cos(d*x + c)^5 + 16*(227*A + 252*B + 238*C)*a^4*cos(d*x + c)^4 + 70*(44 *A + 41*B + 24*C)*a^4*cos(d*x + c)^3 + 48*(48*A + 28*B + 7*C)*a^4*cos(d*x + c)^2 + 280*(4*A + B)*a^4*cos(d*x + c) + 240*A*a^4)*sin(d*x + c))/(d*cos( d*x + c)^7)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (271) = 542\).
Time = 0.22 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.55 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{4} + 1344 \, {\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^{4} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 896 \, {\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 )} B a^{4} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 224 \, {\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 )} C a^{4} + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 140 \, A a^{4} {\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 )} - 35 \, B a^{4} {\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 )} - 840 \, A a^{4} {\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 )} - 1260 \, B a^{4} {\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 )} - 840 \, C a^{4} {\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 )} - 840 \, B a^{4} {\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 )} - 3360 \, C a^{4} {\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 )} + 3360 \, C a^{4} \tan \left (d x + c\right )}{3360 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^8, x, algorithm="maxima")
1/3360*(96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35* tan(d*x + c))*A*a^4 + 1344*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan( d*x + c))*A*a^4 + 1120*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 896*(3*ta n(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 4480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 + 224*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 6720*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 140*A*a^4*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(s in(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)) - 35*B*a^4*(2*(15*sin(d*x + c)^5 - 4 0*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) ) - 840*A*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*s in(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 1260*B*a^4*(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)) - 840 *C*a^4*(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)) - 840*B*a ^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin( d*x + c) - 1)) - 3360*C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin (d*x + c) + 1) + log(sin(d*x + c) - 1)) + 3360*C*a^4*tan(d*x + c))/d
Time = 0.43 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.54 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (4620 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 5880 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 30800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 34300 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 39200 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 110936 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135168 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150528 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 172032 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 159656 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 58800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 73220 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 86240 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21735 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21000 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{1680 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^8, x, algorithm="giac")
1/1680*(105*(44*A*a^4 + 49*B*a^4 + 56*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(44*A*a^4 + 49*B*a^4 + 56*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(4620*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 5145*B*a^4*tan(1/2*d*x + 1 /2*c)^13 + 5880*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 30800*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 34300*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 39200*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 87164*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 97069*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 110936*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 135168*A*a^4*tan(1/2*d *x + 1/2*c)^7 - 150528*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 172032*C*a^4*tan(1/2 *d*x + 1/2*c)^7 + 126084*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 134099*B*a^4*tan(1 /2*d*x + 1/2*c)^5 + 159656*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 58800*A*a^4*tan( 1/2*d*x + 1/2*c)^3 - 73220*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 86240*C*a^4*tan( 1/2*d*x + 1/2*c)^3 + 22260*A*a^4*tan(1/2*d*x + 1/2*c) + 21735*B*a^4*tan(1/ 2*d*x + 1/2*c) + 21000*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d
Time = 5.28 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.33 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a^4\,\mathrm {atanh}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (44\,A+49\,B+56\,C\right )}{4\,\left (11\,A\,a^4+\frac {49\,B\,a^4}{4}+14\,C\,a^4\right )}\right )\,\left (44\,A+49\,B+56\,C\right )}{8\,d}-\frac {\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {110\,A\,a^4}{3}-\frac {245\,B\,a^4}{6}-\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {13867\,B\,a^4}{120}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {5632\,A\,a^4}{35}-\frac {896\,B\,a^4}{5}-\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {19157\,B\,a^4}{120}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-70\,A\,a^4-\frac {523\,B\,a^4}{6}-\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+\frac {207\,B\,a^4}{8}+25\,C\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(a^4*atanh((a^4*tan(c/2 + (d*x)/2)*(44*A + 49*B + 56*C))/(4*(11*A*a^4 + (4 9*B*a^4)/4 + 14*C*a^4)))*(44*A + 49*B + 56*C))/(8*d) - (tan(c/2 + (d*x)/2) ^13*((11*A*a^4)/2 + (49*B*a^4)/8 + 7*C*a^4) - tan(c/2 + (d*x)/2)^11*((110* A*a^4)/3 + (245*B*a^4)/6 + (140*C*a^4)/3) - tan(c/2 + (d*x)/2)^3*(70*A*a^4 + (523*B*a^4)/6 + (308*C*a^4)/3) - tan(c/2 + (d*x)/2)^7*((5632*A*a^4)/35 + (896*B*a^4)/5 + (1024*C*a^4)/5) + tan(c/2 + (d*x)/2)^9*((3113*A*a^4)/30 + (13867*B*a^4)/120 + (1981*C*a^4)/15) + tan(c/2 + (d*x)/2)^5*((1501*A*a^4 )/10 + (19157*B*a^4)/120 + (2851*C*a^4)/15) + tan(c/2 + (d*x)/2)*((53*A*a^ 4)/2 + (207*B*a^4)/8 + 25*C*a^4))/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan( c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))