Integrand size = 33, antiderivative size = 263 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a^4 (11 A+14 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac {a^4 (11 A+14 C) \sec (c+d x) \tan (c+d x)}{4 d}+\frac {a^4 (247 A+308 C) \sec ^2(c+d x) \tan (c+d x)}{210 d}+\frac {(109 A+126 C) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {(8 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d} \] Output:
1/4*a^4*(11*A+14*C)*arctanh(sin(d*x+c))/d+1/105*a^4*(454*A+581*C)*tan(d*x+ c)/d+1/4*a^4*(11*A+14*C)*sec(d*x+c)*tan(d*x+c)/d+1/210*a^4*(247*A+308*C)*s ec(d*x+c)^2*tan(d*x+c)/d+1/210*(109*A+126*C)*(a^4+a^4*cos(d*x+c))*sec(d*x+ c)^3*tan(d*x+c)/d+1/35*(8*A+7*C)*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^4*tan(d *x+c)/d+2/21*a*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^5*tan(d*x+c)/d+1/7*A*(a+a*c os(d*x+c))^4*sec(d*x+c)^6*tan(d*x+c)/d
Time = 5.18 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.06 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {11 a^4 A \text {arctanh}(\sin (c+d x))}{4 d}+\frac {7 a^4 C \text {arctanh}(\sin (c+d x))}{2 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 C \tan (c+d x)}{d}+\frac {11 a^4 A \sec (c+d x) \tan (c+d x)}{4 d}+\frac {7 a^4 C \sec (c+d x) \tan (c+d x)}{2 d}+\frac {11 a^4 A \sec ^3(c+d x) \tan (c+d x)}{6 d}+\frac {a^4 C \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {2 a^4 A \sec ^5(c+d x) \tan (c+d x)}{3 d}+\frac {16 a^4 A \tan ^3(c+d x)}{3 d}+\frac {8 a^4 C \tan ^3(c+d x)}{3 d}+\frac {9 a^4 A \tan ^5(c+d x)}{5 d}+\frac {a^4 C \tan ^5(c+d x)}{5 d}+\frac {a^4 A \tan ^7(c+d x)}{7 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]
Output:
(11*a^4*A*ArcTanh[Sin[c + d*x]])/(4*d) + (7*a^4*C*ArcTanh[Sin[c + d*x]])/( 2*d) + (8*a^4*A*Tan[c + d*x])/d + (8*a^4*C*Tan[c + d*x])/d + (11*a^4*A*Sec [c + d*x]*Tan[c + d*x])/(4*d) + (7*a^4*C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (11*a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/(6*d) + (a^4*C*Sec[c + d*x]^3*Tan [c + d*x])/d + (2*a^4*A*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) + (16*a^4*A*Tan [c + d*x]^3)/(3*d) + (8*a^4*C*Tan[c + d*x]^3)/(3*d) + (9*a^4*A*Tan[c + d*x ]^5)/(5*d) + (a^4*C*Tan[c + d*x]^5)/(5*d) + (a^4*A*Tan[c + d*x]^7)/(7*d)
Time = 2.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.08, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3523, 3042, 3454, 27, 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.
| \(\displaystyle \int \sec ^8(c+d x) (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (4 a A+a (2 A+7 C) \cos (c+d x)) \sec ^7(c+d x)dx}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A+a (2 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{6} \int 2 (\cos (c+d x) a+a)^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \cos (c+d x) a^2\right ) \sec ^6(c+d x)dx+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \int (\cos (c+d x) a+a)^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \cos (c+d x) a^2\right ) \sec ^6(c+d x)dx+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \int (\cos (c+d x) a+a)^2 \left (2 (109 A+126 C) a^3+49 (2 A+3 C) \cos (c+d x) a^3\right ) \sec ^5(c+d x)dx+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 (109 A+126 C) a^3+49 (2 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{4} \int 6 (\cos (c+d x) a+a) \left ((247 A+308 C) a^4+2 (69 A+91 C) \cos (c+d x) a^4\right ) \sec ^4(c+d x)dx+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left ((247 A+308 C) a^4+2 (69 A+91 C) \cos (c+d x) a^4\right ) \sec ^4(c+d x)dx+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((247 A+308 C) a^4+2 (69 A+91 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int \left (2 (69 A+91 C) \cos ^2(c+d x) a^5+(247 A+308 C) a^5+\left (2 (69 A+91 C) a^5+(247 A+308 C) a^5\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int \frac {2 (69 A+91 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(247 A+308 C) a^5+\left (2 (69 A+91 C) a^5+(247 A+308 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \int \left (105 (11 A+14 C) a^5+2 (454 A+581 C) \cos (c+d x) a^5\right ) \sec ^3(c+d x)dx+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \int \frac {105 (11 A+14 C) a^5+2 (454 A+581 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \int \sec ^3(c+d x)dx+2 a^5 (454 A+581 C) \int \sec ^2(c+d x)dx\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (2 a^5 (454 A+581 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+105 a^5 (11 A+14 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {2 a^5 (454 A+581 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {2 a^5 (454 A+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 a^5 (454 A+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 a^5 (454 A+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {2 a^2 A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 a^5 (454 A+581 C) \tan (c+d x)}{d}\right )+\frac {a^5 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}\right )}{7 a}+\frac {A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}\) |
Input:
Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]
Output:
(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + ((2*a^2*A*( a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) + ((3*a^3*(8*A + 7*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (((109*A + 126*C)*(a^5 + a^5*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (3* ((a^5*(247*A + 308*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((2*a^5*(454*A + 581*C)*Tan[c + d*x])/d + 105*a^5*(11*A + 14*C)*(ArcTanh[Sin[c + d*x]]/(2 *d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/3))/2)/5)/3)/(7*a)
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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(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*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(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, 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 = 12.95 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.08
| method | result | size |
| parallelrisch | \(-\frac {11 \left (\left (A +\frac {14 C}{11}\right ) \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (A +\frac {14 C}{11}\right ) \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {204 C}{11}-\frac {938 A}{33}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {1736 A}{55}-\frac {1604 C}{55}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {144 C}{11}-\frac {40 A}{3}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {1816 A}{165}-\frac {2204 C}{165}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {28 C}{11}-2 A \right ) \sin \left (6 d x +6 c \right )+\left (-\frac {332 C}{165}-\frac {1816 A}{1155}\right ) \sin \left (7 d x +7 c \right )-\frac {280 \left (A +\frac {7 C}{10}\right ) \sin \left (d x +c \right )}{11}\right ) a^{4}}{4 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\) | \(284\) |
| parts | \(-\frac {a^{4} A \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} C \tan \left (d x +c \right )}{d}+\frac {4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {2 C \tan \left (d x +c \right ) \sec \left (d x +c \right ) a^{4}}{d}+\frac {2 C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}\) | \(311\) |
| derivativedivides | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} A \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(374\) |
| default | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} A \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(374\) |
| risch | \(-\frac {i a^{4} \left (-2324 C -1816 A +7700 A \,{\mathrm e}^{11 i \left (d x +c \right )}+7560 C \,{\mathrm e}^{11 i \left (d x +c \right )}-7560 C \,{\mathrm e}^{10 i \left (d x +c \right )}-46480 A \,{\mathrm e}^{6 i \left (d x +c \right )}-50960 C \,{\mathrm e}^{6 i \left (d x +c \right )}-37296 A \,{\mathrm e}^{4 i \left (d x +c \right )}-41244 C \,{\mathrm e}^{4 i \left (d x +c \right )}-12712 A \,{\mathrm e}^{2 i \left (d x +c \right )}-15848 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1155 A \,{\mathrm e}^{13 i \left (d x +c \right )}-420 C \,{\mathrm e}^{12 i \left (d x +c \right )}+1470 C \,{\mathrm e}^{13 i \left (d x +c \right )}-840 A \,{\mathrm e}^{10 i \left (d x +c \right )}-16415 A \,{\mathrm e}^{5 i \left (d x +c \right )}-10710 C \,{\mathrm e}^{5 i \left (d x +c \right )}-7700 A \,{\mathrm e}^{3 i \left (d x +c \right )}-7560 C \,{\mathrm e}^{3 i \left (d x +c \right )}-1155 A \,{\mathrm e}^{i \left (d x +c \right )}-17080 A \,{\mathrm e}^{8 i \left (d x +c \right )}-1470 C \,{\mathrm e}^{i \left (d x +c \right )}+16415 A \,{\mathrm e}^{9 i \left (d x +c \right )}+10710 C \,{\mathrm e}^{9 i \left (d x +c \right )}-30380 C \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{210 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\frac {11 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{4 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}+\frac {11 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{4 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}\) | \(395\) |
Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^8,x,method=_RETURNVER BOSE)
Output:
-11/4*((A+14/11*C)*(cos(7*d*x+7*c)+7*cos(5*d*x+5*c)+21*cos(3*d*x+3*c)+35*c os(d*x+c))*ln(tan(1/2*d*x+1/2*c)-1)-(A+14/11*C)*(cos(7*d*x+7*c)+7*cos(5*d* x+5*c)+21*cos(3*d*x+3*c)+35*cos(d*x+c))*ln(tan(1/2*d*x+1/2*c)+1)+(-204/11* C-938/33*A)*sin(2*d*x+2*c)+(-1736/55*A-1604/55*C)*sin(3*d*x+3*c)+(-144/11* C-40/3*A)*sin(4*d*x+4*c)+(-1816/165*A-2204/165*C)*sin(5*d*x+5*c)+(-28/11*C -2*A)*sin(6*d*x+6*c)+(-332/165*C-1816/1155*A)*sin(7*d*x+7*c)-280/11*(A+7/1 0*C)*sin(d*x+c))*a^4/d/(cos(7*d*x+7*c)+7*cos(5*d*x+5*c)+21*cos(3*d*x+3*c)+ 35*cos(d*x+c))
Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.76 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (454 \, A + 581 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 4 \, {\left (227 \, A + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (11 \, A + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 12 \, {\left (48 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, A a^{4} \cos \left (d x + c\right ) + 60 \, A a^{4}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^8,x, algorithm= "fricas")
Output:
1/840*(105*(11*A + 14*C)*a^4*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(1 1*A + 14*C)*a^4*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*(4*(454*A + 581* C)*a^4*cos(d*x + c)^6 + 105*(11*A + 14*C)*a^4*cos(d*x + c)^5 + 4*(227*A + 238*C)*a^4*cos(d*x + c)^4 + 70*(11*A + 6*C)*a^4*cos(d*x + c)^3 + 12*(48*A + 7*C)*a^4*cos(d*x + c)^2 + 280*A*a^4*cos(d*x + c) + 60*A*a^4)*sin(d*x + c ))/(d*cos(d*x + c)^7)
Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**8,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.76 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {24 \, {\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} + 336 \, {\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} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 56 \, {\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} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, 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 )} - 210 \, 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 )} - 210 \, 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 \, 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 )} + 840 \, C a^{4} \tan \left (d x + c\right )}{840 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^8,x, algorithm= "maxima")
Output:
1/840*(24*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*t an(d*x + c))*A*a^4 + 336*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d* x + c))*A*a^4 + 280*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 56*(3*tan(d* x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 1680*(tan(d*x + c) ^3 + 3*tan(d*x + c))*C*a^4 - 35*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*log(sin(d*x + c) - 1)) - 210*A*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)) - 210*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*C*a^4*(2*si n(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 840*C*a^4*tan(d*x + c))/d
Time = 0.42 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.19 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left (11 \, A a^{4} + 14 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (11 \, A a^{4} + 14 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1155 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1470 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 7700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 9800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 21791 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 27734 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 33792 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 43008 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 31521 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 39914 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21560 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5565 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5250 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{420 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^8,x, algorithm= "giac")
Output:
1/420*(105*(11*A*a^4 + 14*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105* (11*A*a^4 + 14*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(1155*A*a^4*t an(1/2*d*x + 1/2*c)^13 + 1470*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 7700*A*a^4*t an(1/2*d*x + 1/2*c)^11 - 9800*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 21791*A*a^4* tan(1/2*d*x + 1/2*c)^9 + 27734*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 33792*A*a^4* tan(1/2*d*x + 1/2*c)^7 - 43008*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 31521*A*a^4* tan(1/2*d*x + 1/2*c)^5 + 39914*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 14700*A*a^4* tan(1/2*d*x + 1/2*c)^3 - 21560*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 5565*A*a^4*t an(1/2*d*x + 1/2*c) + 5250*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2* c)^2 - 1)^7)/d
Time = 1.81 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.14 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (11\,A+14\,C\right )}{2\,d}-\frac {\left (\frac {11\,A\,a^4}{2}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {110\,A\,a^4}{3}-\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {5632\,A\,a^4}{35}-\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-70\,A\,a^4-\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+25\,C\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4)/cos(c + d*x)^8,x)
Output:
(a^4*atanh(tan(c/2 + (d*x)/2))*(11*A + 14*C))/(2*d) - (tan(c/2 + (d*x)/2)* ((53*A*a^4)/2 + 25*C*a^4) + tan(c/2 + (d*x)/2)^13*((11*A*a^4)/2 + 7*C*a^4) - tan(c/2 + (d*x)/2)^11*((110*A*a^4)/3 + (140*C*a^4)/3) - tan(c/2 + (d*x) /2)^3*(70*A*a^4 + (308*C*a^4)/3) + tan(c/2 + (d*x)/2)^5*((1501*A*a^4)/10 + (2851*C*a^4)/15) + tan(c/2 + (d*x)/2)^9*((3113*A*a^4)/30 + (1981*C*a^4)/1 5) - tan(c/2 + (d*x)/2)^7*((5632*A*a^4)/35 + (1024*C*a^4)/5))/(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*ta n(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))
Time = 0.20 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.52 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^8,x)
Output:
(a**4*( - 1155*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a - 1470*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*c + 3465*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a + 4410*cos(c + d*x)*lo g(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c - 3465*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a - 4410*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*c + 1155*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a + 14 70*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*c + 1155*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a + 1470*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*c - 3465*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin( c + d*x)**4*a - 4410*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)** 4*c + 3465*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a + 4410 *cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*c - 1155*cos(c + d *x)*log(tan((c + d*x)/2) + 1)*a - 1470*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*c - 1155*cos(c + d*x)*sin(c + d*x)**5*a - 1470*cos(c + d*x)*sin(c + d* x)**5*c + 3080*cos(c + d*x)*sin(c + d*x)**3*a + 3360*cos(c + d*x)*sin(c + d*x)**3*c - 2205*cos(c + d*x)*sin(c + d*x)*a - 1890*cos(c + d*x)*sin(c + d *x)*c + 1816*sin(c + d*x)**7*a + 2324*sin(c + d*x)**7*c - 6356*sin(c + d*x )**5*a - 7924*sin(c + d*x)**5*c + 7840*sin(c + d*x)**3*a + 8960*sin(c + d* x)**3*c - 3360*sin(c + d*x)*a - 3360*sin(c + d*x)*c))/(420*cos(c + d*x)*d* (sin(c + d*x)**6 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**2 - 1))