Integrand size = 41, antiderivative size = 278 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a^4 (44 A+49 B+56 C) x+\frac {a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (44 A+49 B+56 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (988 A+1113 B+1232 C) \cos ^2(c+d x) \sin (c+d x)}{840 d}+\frac {a (4 A+7 B) \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(16 A+21 B+14 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{70 d}+\frac {(436 A+511 B+504 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{840 d} \]
1/16*a^4*(44*A+49*B+56*C)*x+1/105*a^4*(454*A+504*B+581*C)*sin(d*x+c)/d+1/1 6*a^4*(44*A+49*B+56*C)*cos(d*x+c)*sin(d*x+c)/d+1/840*a^4*(988*A+1113*B+123 2*C)*cos(d*x+c)^2*sin(d*x+c)/d+1/42*a*(4*A+7*B)*cos(d*x+c)^5*(a+a*sec(d*x+ c))^3*sin(d*x+c)/d+1/7*A*cos(d*x+c)^6*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/70 *(16*A+21*B+14*C)*cos(d*x+c)^4*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/d+1/840*( 436*A+511*B+504*C)*cos(d*x+c)^3*(a^4+a^4*sec(d*x+c))*sin(d*x+c)/d
Time = 0.62 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.73 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (11760 A c+20580 B c+18480 A d x+20580 B d x+23520 C d x+105 (323 A+352 B+392 C) \sin (c+d x)+105 (124 A+127 B+128 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+5040 B \sin (3 (c+d x))+4060 C \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+1575 B \sin (4 (c+d x))+840 C \sin (4 (c+d x))+651 A \sin (5 (c+d x))+336 B \sin (5 (c+d x))+84 C \sin (5 (c+d x))+140 A \sin (6 (c+d x))+35 B \sin (6 (c+d x))+15 A \sin (7 (c+d x)))}{6720 d} \]
(a^4*(11760*A*c + 20580*B*c + 18480*A*d*x + 20580*B*d*x + 23520*C*d*x + 10 5*(323*A + 352*B + 392*C)*Sin[c + d*x] + 105*(124*A + 127*B + 128*C)*Sin[2 *(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 4060*C*S in[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 840* C*Sin[4*(c + d*x)] + 651*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 84* C*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A *Sin[7*(c + d*x)]))/(6720*d)
Time = 1.86 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 4574, 3042, 4505, 3042, 4505, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 3115, 24, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^7(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) (\sec (c+d x) a+a)^4 (a (4 A+7 B)+a (2 A+7 C) \sec (c+d x))dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (4 A+7 B)+a (2 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{6} \int \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left (3 (16 A+21 B+14 C) a^2+2 (10 A+7 B+21 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 (16 A+21 B+14 C) a^2+2 (10 A+7 B+21 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left ((436 A+511 B+504 C) a^3+98 (2 A+2 B+3 C) \sec (c+d x) a^3\right )dx+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((436 A+511 B+504 C) a^3+98 (2 A+2 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \sec (c+d x) a^4\right )dx+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \sec (c+d x) a^4\right )dx+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((988 A+1113 B+1232 C) a^4+2 (276 A+301 B+364 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (105 (44 A+49 B+56 C) a^5+8 (454 A+504 B+581 C) \sec (c+d x) a^5\right )dx\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (105 (44 A+49 B+56 C) a^5+8 (454 A+504 B+581 C) \sec (c+d x) a^5\right )dx+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \frac {105 (44 A+49 B+56 C) a^5+8 (454 A+504 B+581 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^5}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^5 (44 A+49 B+56 C) \int \cos ^2(c+d x)dx+8 a^5 (454 A+504 B+581 C) \int \cos (c+d x)dx\right )+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^5 (454 A+504 B+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (44 A+49 B+56 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^5 (454 A+504 B+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (44 A+49 B+56 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^5 (454 A+504 B+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (44 A+49 B+56 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {a^2 (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {a^5 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {8 a^5 (454 A+504 B+581 C) \sin (c+d x)}{d}+105 a^5 (44 A+49 B+56 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{4 d}\right )+\frac {3 a^3 (16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\) |
(A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((a^2*(4*A + 7*B)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*a^3 *(16*A + 21*B + 14*C)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/ (5*d) + (((436*A + 511*B + 504*C)*Cos[c + d*x]^3*(a^5 + a^5*Sec[c + d*x])* Sin[c + d*x])/(4*d) + (3*((a^5*(988*A + 1113*B + 1232*C)*Cos[c + d*x]^2*Si n[c + d*x])/(3*d) + ((8*a^5*(454*A + 504*B + 581*C)*Sin[c + d*x])/d + 105* a^5*(44*A + 49*B + 56*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/3))/4) /5)/6)/(7*a)
3.5.47.3.1 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[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.61 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.54
| method | result | size |
| parallelrisch | \(\frac {157 \left (\frac {3 \left (124 A +127 B +128 C \right ) \sin \left (2 d x +2 c \right )}{157}+\left (A +\frac {144 B}{157}+\frac {116 C}{157}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (20 A +15 B +8 C \right ) \sin \left (4 d x +4 c \right )}{157}+\frac {3 \left (31 A +16 B +4 C \right ) \sin \left (5 d x +5 c \right )}{785}+\frac {\left (4 A +B \right ) \sin \left (6 d x +6 c \right )}{157}+\frac {3 A \sin \left (7 d x +7 c \right )}{1099}+\frac {3 \left (323 A +352 B +392 C \right ) \sin \left (d x +c \right )}{157}+\frac {528 x d \left (A +\frac {49 B}{44}+\frac {14 C}{11}\right )}{157}\right ) a^{4}}{192 d}\) | \(149\) |
| risch | \(\frac {11 a^{4} A x}{4}+\frac {49 a^{4} x B}{16}+\frac {7 a^{4} x C}{2}+\frac {323 \sin \left (d x +c \right ) a^{4} A}{64 d}+\frac {11 \sin \left (d x +c \right ) B \,a^{4}}{2 d}+\frac {49 \sin \left (d x +c \right ) a^{4} C}{8 d}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{48 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{192 d}+\frac {31 a^{4} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{20 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} C}{80 d}+\frac {5 a^{4} A \sin \left (4 d x +4 c \right )}{16 d}+\frac {15 \sin \left (4 d x +4 c \right ) B \,a^{4}}{64 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{8 d}+\frac {157 a^{4} A \sin \left (3 d x +3 c \right )}{192 d}+\frac {3 \sin \left (3 d x +3 c \right ) B \,a^{4}}{4 d}+\frac {29 \sin \left (3 d x +3 c \right ) a^{4} C}{48 d}+\frac {31 \sin \left (2 d x +2 c \right ) a^{4} A}{16 d}+\frac {127 \sin \left (2 d x +2 c \right ) B \,a^{4}}{64 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} C}{d}\) | \(338\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(490\) |
| default | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(490\) |
157/192*(3/157*(124*A+127*B+128*C)*sin(2*d*x+2*c)+(A+144/157*B+116/157*C)* sin(3*d*x+3*c)+3/157*(20*A+15*B+8*C)*sin(4*d*x+4*c)+3/785*(31*A+16*B+4*C)* sin(5*d*x+5*c)+1/157*(4*A+B)*sin(6*d*x+6*c)+3/1099*A*sin(7*d*x+7*c)+3/157* (323*A+352*B+392*C)*sin(d*x+c)+528/157*x*d*(A+49/44*B+14/11*C))*a^4/d
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.60 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} d x + {\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (48 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (44 \, A + 41 \, B + 24 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (227 \, A + 252 \, B + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (44 \, A + 49 \, B + 56 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (454 \, A + 504 \, B + 581 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/1680*(105*(44*A + 49*B + 56*C)*a^4*d*x + (240*A*a^4*cos(d*x + c)^6 + 280 *(4*A + B)*a^4*cos(d*x + c)^5 + 48*(48*A + 28*B + 7*C)*a^4*cos(d*x + c)^4 + 70*(44*A + 41*B + 24*C)*a^4*cos(d*x + c)^3 + 16*(227*A + 252*B + 238*C)* a^4*cos(d*x + c)^2 + 105*(44*A + 49*B + 56*C)*a^4*cos(d*x + c) + 16*(454*A + 504*B + 581*C)*a^4)*sin(d*x + c))/d
Timed out. \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.74 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 6720 \, C a^{4} \sin \left (d x + c\right )}{6720 \, d} \]
integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
-1/6720*(192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 3 5*sin(d*x + c))*A*a^4 - 2688*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*si n(d*x + c))*A*a^4 + 140*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d* x + 4*c) - 48*sin(2*d*x + 2*c))*A*a^4 + 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A *a^4 - 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin( 2*d*x + 2*c))*B*a^4 + 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 1260* (12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 1680*(2*d* x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 448*(3*sin(d*x + c)^5 - 10*sin(d*x + c )^3 + 15*sin(d*x + c))*C*a^4 + 13440*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a ^4 - 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 - 6 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 - 6720*C*a^4*sin(d*x + c))/d
Time = 0.38 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.44 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (44 \, A a^{4} + 49 \, B a^{4} + 56 \, C a^{4}\right )} {\left (d x + c\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(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/1680*(105*(44*A*a^4 + 49*B*a^4 + 56*C*a^4)*(d*x + c) + 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*t an(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 + 15965 6*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 + 7322 0*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 + 2226 0*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 = 18.76 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.36 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {110\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}+\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {13867\,B\,a^4}{120}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {5632\,A\,a^4}{35}+\frac {896\,B\,a^4}{5}+\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {19157\,B\,a^4}{120}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (70\,A\,a^4+\frac {523\,B\,a^4}{6}+\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+\frac {207\,B\,a^4}{8}+25\,C\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (44\,A+49\,B+56\,C\right )}{8\,\left (\frac {11\,A\,a^4}{2}+\frac {49\,B\,a^4}{8}+7\,C\,a^4\right )}\right )\,\left (44\,A+49\,B+56\,C\right )}{8\,d} \]
(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)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(44*A + 49*B + 56*C ))/(8*((11*A*a^4)/2 + (49*B*a^4)/8 + 7*C*a^4)))*(44*A + 49*B + 56*C))/(8*d )