Integrand size = 41, antiderivative size = 253 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac {a^4 (417 A+488 B+550 C) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {(43 A+52 B+50 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(37 A+48 B+30 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {a (2 A+3 B) (a+a \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]
7/16*a^4*(7*A+8*B+10*C)*arctanh(sin(d*x+c))/d+1/15*a^4*(72*A+83*B+100*C)*t an(d*x+c)/d+1/240*a^4*(417*A+488*B+550*C)*sec(d*x+c)*tan(d*x+c)/d+1/60*(43 *A+52*B+50*C)*(a^4+a^4*cos(d*x+c))*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(37*A+4 8*B+30*C)*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^3*tan(d*x+c)/d+1/15*a*(2*A+3*B )*(a+a*cos(d*x+c))^3*sec(d*x+c)^4*tan(d*x+c)/d+1/6*A*(a+a*cos(d*x+c))^4*se c(d*x+c)^5*tan(d*x+c)/d
Time = 6.49 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.51 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^4 \left (105 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 (49 A+56 B+54 C) \sec (c+d x)+10 (41 A+6 (4 B+C)) \sec ^3(c+d x)+40 A \sec ^5(c+d x)+16 \left (120 (A+B+C)+20 (3 A+2 B+C) \tan ^2(c+d x)+3 (4 A+B) \tan ^4(c+d x)\right )\right )\right )}{240 d} \]
(a^4*(105*(7*A + 8*B + 10*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(49* A + 56*B + 54*C)*Sec[c + d*x] + 10*(41*A + 6*(4*B + C))*Sec[c + d*x]^3 + 4 0*A*Sec[c + d*x]^5 + 16*(120*(A + B + C) + 20*(3*A + 2*B + C)*Tan[c + d*x] ^2 + 3*(4*A + B)*Tan[c + d*x]^4))))/(240*d)
Time = 2.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.463, Rules used = {3042, 3522, 3042, 3454, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 24, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (2 a (2 A+3 B)+a (A+6 C) \cos (c+d x)) \sec ^6(c+d x)dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (2 a (2 A+3 B)+a (A+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \int (\cos (c+d x) a+a)^3 \left ((37 A+48 B+30 C) a^2+3 (3 A+2 B+10 C) \cos (c+d x) a^2\right ) \sec ^5(c+d x)dx+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((37 A+48 B+30 C) a^2+3 (3 A+2 B+10 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int (\cos (c+d x) a+a)^2 \left (6 (43 A+52 B+50 C) a^3+(73 A+72 B+150 C) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (6 (43 A+52 B+50 C) a^3+(73 A+72 B+150 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 (\cos (c+d x) a+a) \left ((417 A+488 B+550 C) a^4+(159 A+176 B+250 C) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int (\cos (c+d x) a+a) \left ((417 A+488 B+550 C) a^4+(159 A+176 B+250 C) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((417 A+488 B+550 C) a^4+(159 A+176 B+250 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \left ((159 A+176 B+250 C) \cos ^2(c+d x) a^5+(417 A+488 B+550 C) a^5+\left ((159 A+176 B+250 C) a^5+(417 A+488 B+550 C) a^5\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {(159 A+176 B+250 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(417 A+488 B+550 C) a^5+\left ((159 A+176 B+250 C) a^5+(417 A+488 B+550 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \left (16 (72 A+83 B+100 C) a^5+105 (7 A+8 B+10 C) \cos (c+d x) a^5\right ) \sec ^2(c+d x)dx+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {16 (72 A+83 B+100 C) a^5+105 (7 A+8 B+10 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (16 a^5 (72 A+83 B+100 C) \int \sec ^2(c+d x)dx+105 a^5 (7 A+8 B+10 C) \int \sec (c+d x)dx\right )+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+8 B+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+16 a^5 (72 A+83 B+100 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+8 B+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {16 a^5 (72 A+83 B+100 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (105 a^5 (7 A+8 B+10 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {16 a^5 (72 A+83 B+100 C) \tan (c+d x)}{d}\right )+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )+\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {2 a^2 (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (\frac {105 a^5 (7 A+8 B+10 C) \text {arctanh}(\sin (c+d x))}{d}+\frac {16 a^5 (72 A+83 B+100 C) \tan (c+d x)}{d}\right )+\frac {a^5 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {2 (43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {a^3 (37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}\right )}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}\) |
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((2*a^2*(2* A + 3*B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((a^3 *(37*A + 48*B + 30*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/ (4*d) + ((a^5*(417*A + 488*B + 550*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ( 2*(43*A + 52*B + 50*C)*(a^5 + a^5*Cos[c + d*x])*Sec[c + d*x]^2*Tan[c + d*x ])/d + ((105*a^5*(7*A + 8*B + 10*C)*ArcTanh[Sin[c + d*x]])/d + (16*a^5*(72 *A + 83*B + 100*C)*Tan[c + d*x])/d)/2)/4)/5)/(6*a)
3.4.37.3.1 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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 13.07 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.07
| method | result | size |
| parallelrisch | \(-\frac {49 a^{4} \left (\left (A +\frac {8 B}{7}+\frac {10 C}{7}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (A +\frac {8 B}{7}+\frac {10 C}{7}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {704 C}{49}-\frac {128 A}{7}-16 B \right ) \sin \left (2 d x +2 c \right )+\left (-\frac {1538 A}{147}-\frac {464 B}{49}-\frac {356 C}{49}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {2304 A}{245}-\frac {2496 B}{245}-\frac {512 C}{49}\right ) \sin \left (4 d x +4 c \right )+\left (-2 A -\frac {16 B}{7}-\frac {108 C}{49}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {384 A}{245}-\frac {320 C}{147}-\frac {1328 B}{735}\right ) \sin \left (6 d x +6 c \right )-\frac {500 \left (\frac {88 B}{125}+\frac {62 C}{125}+A \right ) \sin \left (d x +c \right )}{49}\right )}{16 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(270\) |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}\) | \(314\) |
| derivativedivides | \(\frac {a^{4} A \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 )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 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 )+4 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 )+4 C \,a^{4} \tan \left (d x +c \right )+6 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 )-6 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 )+6 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 )-4 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 )+4 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 )-4 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 )+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 )-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 )+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 )}{d}\) | \(506\) |
| default | \(\frac {a^{4} A \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 )+B \,a^{4} \tan \left (d x +c \right )+C \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 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 )+4 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 )+4 C \,a^{4} \tan \left (d x +c \right )+6 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 )-6 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 )+6 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 )-4 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 )+4 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 )-4 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 )+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 )-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 )+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 )}{d}\) | \(506\) |
| risch | \(-\frac {i a^{4} \left (-1920 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1600 C -1152 A -1328 B -15360 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15840 B \,{\mathrm e}^{4 i \left (d x +c \right )}-7728 B \,{\mathrm e}^{2 i \left (d x +c \right )}-17280 C \,{\mathrm e}^{4 i \left (d x +c \right )}-8640 C \,{\mathrm e}^{2 i \left (d x +c \right )}-240 B \,{\mathrm e}^{10 i \left (d x +c \right )}+2640 B \,{\mathrm e}^{7 i \left (d x +c \right )}-3845 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3750 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2640 B \,{\mathrm e}^{5 i \left (d x +c \right )}-6912 A \,{\mathrm e}^{2 i \left (d x +c \right )}-735 A \,{\mathrm e}^{i \left (d x +c \right )}-840 B \,{\mathrm e}^{i \left (d x +c \right )}-13280 B \,{\mathrm e}^{6 i \left (d x +c \right )}-3480 B \,{\mathrm e}^{3 i \left (d x +c \right )}-11520 A \,{\mathrm e}^{6 i \left (d x +c \right )}-16000 C \,{\mathrm e}^{6 i \left (d x +c \right )}+3750 A \,{\mathrm e}^{7 i \left (d x +c \right )}-810 C \,{\mathrm e}^{i \left (d x +c \right )}-4080 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1860 C \,{\mathrm e}^{7 i \left (d x +c \right )}-1860 C \,{\mathrm e}^{5 i \left (d x +c \right )}-2670 C \,{\mathrm e}^{3 i \left (d x +c \right )}+840 B \,{\mathrm e}^{11 i \left (d x +c \right )}+735 A \,{\mathrm e}^{11 i \left (d x +c \right )}+810 C \,{\mathrm e}^{11 i \left (d x +c \right )}-960 C \,{\mathrm e}^{10 i \left (d x +c \right )}-6720 C \,{\mathrm e}^{8 i \left (d x +c \right )}+2670 C \,{\mathrm e}^{9 i \left (d x +c \right )}+3845 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3480 B \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}+\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}-\frac {35 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(550\) |
-49/16*a^4*((A+8/7*B+10/7*C)*(cos(6*d*x+6*c)+6*cos(4*d*x+4*c)+15*cos(2*d*x +2*c)+10)*ln(tan(1/2*d*x+1/2*c)-1)-(A+8/7*B+10/7*C)*(cos(6*d*x+6*c)+6*cos( 4*d*x+4*c)+15*cos(2*d*x+2*c)+10)*ln(tan(1/2*d*x+1/2*c)+1)+(-704/49*C-128/7 *A-16*B)*sin(2*d*x+2*c)+(-1538/147*A-464/49*B-356/49*C)*sin(3*d*x+3*c)+(-2 304/245*A-2496/245*B-512/49*C)*sin(4*d*x+4*c)+(-2*A-16/7*B-108/49*C)*sin(5 *d*x+5*c)+(-384/245*A-320/147*C-1328/735*B)*sin(6*d*x+6*c)-500/49*(88/125* B+62/125*C+A)*sin(d*x+c))/d/(cos(6*d*x+6*c)+6*cos(4*d*x+4*c)+15*cos(2*d*x+ 2*c)+10)
Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.80 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (72 \, A + 83 \, B + 100 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (41 \, A + 24 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="fricas")
1/480*(105*(7*A + 8*B + 10*C)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 1 05*(7*A + 8*B + 10*C)*a^4*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(7 2*A + 83*B + 100*C)*a^4*cos(d*x + c)^5 + 15*(49*A + 56*B + 54*C)*a^4*cos(d *x + c)^4 + 32*(18*A + 17*B + 10*C)*a^4*cos(d*x + c)^3 + 10*(41*A + 24*B + 6*C)*a^4*cos(d*x + c)^2 + 48*(4*A + B)*a^4*cos(d*x + c) + 40*A*a^4)*sin(d *x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 645 vs. \(2 (239) = 478\).
Time = 0.23 (sec) , antiderivative size = 645, 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 ^7(c+d x) \, dx=\frac {128 \, {\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} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 32 \, {\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} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 5 \, 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 )} - 180 \, 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 )} - 120 \, 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 )} - 30 \, 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 )} - 120 \, A 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 )} - 480 \, 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 )} - 720 \, 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 )} + 240 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, B a^{4} \tan \left (d x + c\right ) + 1920 \, C a^{4} \tan \left (d x + c\right )}{480 \, 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)^7, x, algorithm="maxima")
1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 32*(3*tan(d*x + c)^5 + 10* tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 960*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 5*A*a^4*(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*lo g(sin(d*x + c) - 1)) - 180*A*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(s in(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(si n(d*x + c) - 1)) - 120*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)) - 30*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)) - 120*A*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)) - 480*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)) - 720*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)) + 240*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 480*B* a^4*tan(d*x + c) + 1920*C*a^4*tan(d*x + c))/d
Time = 0.40 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.55 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, 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 (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2790 \, 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 )}^{6}}}{240 \, 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)^7, x, algorithm="giac")
1/240*(105*(7*A*a^4 + 8*B*a^4 + 10*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1 )) - 105*(7*A*a^4 + 8*B*a^4 + 10*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(735*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 840*B*a^4*tan(1/2*d*x + 1/2*c)^1 1 + 1050*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 4165*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 4760*B*a^4*tan(1/2*d*x + 1/2*c)^9 - 5950*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 11088*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 13860*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 11802*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 13488*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 16860*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 7355*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 9320*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 10 690*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 3105*A*a^4*tan(1/2*d*x + 1/2*c) - 3000* B*a^4*tan(1/2*d*x + 1/2*c) - 2790*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6)/d
Time = 5.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.34 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {7\,a^4\,\mathrm {atanh}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+8\,B+10\,C\right )}{4\,\left (\frac {49\,A\,a^4}{4}+14\,B\,a^4+\frac {35\,C\,a^4}{2}\right )}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,d}-\frac {\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {833\,A\,a^4}{24}-\frac {119\,B\,a^4}{3}-\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {462\,B\,a^4}{5}+\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {1967\,A\,a^4}{20}-\frac {562\,B\,a^4}{5}-\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {233\,B\,a^4}{3}+\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {207\,A\,a^4}{8}-25\,B\,a^4-\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(7*a^4*atanh((7*a^4*tan(c/2 + (d*x)/2)*(7*A + 8*B + 10*C))/(4*((49*A*a^4)/ 4 + 14*B*a^4 + (35*C*a^4)/2)))*(7*A + 8*B + 10*C))/(8*d) - (tan(c/2 + (d*x )/2)^11*((49*A*a^4)/8 + 7*B*a^4 + (35*C*a^4)/4) - tan(c/2 + (d*x)/2)^9*((8 33*A*a^4)/24 + (119*B*a^4)/3 + (595*C*a^4)/12) + tan(c/2 + (d*x)/2)^7*((16 17*A*a^4)/20 + (462*B*a^4)/5 + (231*C*a^4)/2) + tan(c/2 + (d*x)/2)^3*((147 1*A*a^4)/24 + (233*B*a^4)/3 + (1069*C*a^4)/12) - tan(c/2 + (d*x)/2)^5*((19 67*A*a^4)/20 + (562*B*a^4)/5 + (281*C*a^4)/2) - tan(c/2 + (d*x)/2)*((207*A *a^4)/8 + 25*B*a^4 + (93*C*a^4)/4))/(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*tan (c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))