Integrand size = 41, antiderivative size = 381 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {\left (24 a^3 b B+32 a b^3 B+8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (8 a^4 B+60 a^2 b^2 B+15 b^4 B+20 a b^3 (2 A+3 C)+8 a^3 b (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {\left (24 A b^4+360 a^3 b B+336 a b^3 B+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (4 A b^3+16 a^3 B+36 a b^2 B+a^2 b (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {\left (12 A b^2+48 a b B+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(2 A b+3 a B) (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]
1/16*(24*B*a^3*b+32*B*a*b^3+8*b^4*(A+2*C)+12*a^2*b^2*(3*A+4*C)+a^4*(5*A+6* C))*arctanh(sin(d*x+c))/d+1/15*(8*B*a^4+60*B*a^2*b^2+15*B*b^4+20*a*b^3*(2* A+3*C)+8*a^3*b*(4*A+5*C))*tan(d*x+c)/d+1/240*(24*A*b^4+360*B*a^3*b+336*B*a *b^3+15*a^4*(5*A+6*C)+10*a^2*b^2*(49*A+66*C))*sec(d*x+c)*tan(d*x+c)/d+1/60 *a*(4*A*b^3+16*B*a^3+36*B*a*b^2+a^2*b*(39*A+50*C))*sec(d*x+c)^2*tan(d*x+c) /d+1/120*(12*A*b^2+48*B*a*b+5*a^2*(5*A+6*C))*(a+b*cos(d*x+c))^2*sec(d*x+c) ^3*tan(d*x+c)/d+1/15*(2*A*b+3*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^4*tan(d*x +c)/d+1/6*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^5*tan(d*x+c)/d
Time = 6.33 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.07 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {5 a^4 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {b^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {b^2 \left (A b^2+4 a b B+6 a^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {3 a^2 \left (6 A b^2+a (4 b B+a C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b^3 (b B+4 a C) \tan (c+d x)}{d}+\frac {5 a^4 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b^2 \left (A b^2+4 a b B+6 a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {3 a^2 \left (6 A b^2+a (4 b B+a C)\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a^4 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^2 \left (6 A b^2+a (4 b B+a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {a^4 A \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {2 a b \left (2 A b^2+a (3 b B+2 a C)\right ) \left (3 \tan (c+d x)+\tan ^3(c+d x)\right )}{3 d}+\frac {a^3 (4 A b+a B) \left (15 \tan (c+d x)+10 \tan ^3(c+d x)+3 \tan ^5(c+d x)\right )}{15 d} \]
(5*a^4*A*ArcTanh[Sin[c + d*x]])/(16*d) + (b^4*C*ArcTanh[Sin[c + d*x]])/d + (b^2*(A*b^2 + 4*a*b*B + 6*a^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a^2*(6 *A*b^2 + a*(4*b*B + a*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (b^3*(b*B + 4*a*C )*Tan[c + d*x])/d + (5*a^4*A*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (b^2*(A*b ^2 + 4*a*b*B + 6*a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*a^2*(6*A*b^2 + a*(4*b*B + a*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (5*a^4*A*Sec[c + d* x]^3*Tan[c + d*x])/(24*d) + (a^2*(6*A*b^2 + a*(4*b*B + a*C))*Sec[c + d*x]^ 3*Tan[c + d*x])/(4*d) + (a^4*A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (2*a*b *(2*A*b^2 + a*(3*b*B + 2*a*C))*(3*Tan[c + d*x] + Tan[c + d*x]^3))/(3*d) + (a^3*(4*A*b + a*B)*(15*Tan[c + d*x] + 10*Tan[c + d*x]^3 + 3*Tan[c + d*x]^5 ))/(15*d)
Time = 2.62 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3526, 3042, 3526, 3042, 3526, 3042, 3510, 27, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \int (a+b \cos (c+d x))^3 \left (b (A+6 C) \cos ^2(c+d x)+(5 a A+6 b B+6 a C) \cos (c+d x)+2 (2 A b+3 a B)\right ) \sec ^6(c+d x)dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+6 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 (2 A b+3 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int (a+b \cos (c+d x))^2 \left (5 (5 A+6 C) a^2+48 b B a+12 A b^2+3 b (3 A b+10 C b+2 a B) \cos ^2(c+d x)+2 \left (12 B a^2+b (23 A+30 C) a+15 b^2 B\right ) \cos (c+d x)\right ) \sec ^5(c+d x)dx+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+6 C) a^2+48 b B a+12 A b^2+3 b (3 A b+10 C b+2 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (12 B a^2+b (23 A+30 C) a+15 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int (a+b \cos (c+d x)) \left (b \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \cos ^2(c+d x)+\left (15 (5 A+6 C) a^3+264 b B a^2+8 b^2 (32 A+45 C) a+120 b^3 B\right ) \cos (c+d x)+6 \left (16 B a^3+b (39 A+50 C) a^2+36 b^2 B a+4 A b^3\right )\right ) \sec ^4(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (15 (5 A+6 C) a^3+264 b B a^2+8 b^2 (32 A+45 C) a+120 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (16 B a^3+b (39 A+50 C) a^2+36 b^2 B a+4 A b^3\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}-\frac {1}{3} \int -3 \left (15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \cos ^2(c+d x)+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \left (15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \cos ^2(c+d x)+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \left (16 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right )+15 \left ((5 A+6 C) a^4+24 b B a^3+12 b^2 (3 A+4 C) a^2+32 b^3 B a+8 b^4 (A+2 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {16 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right )+15 \left ((5 A+6 C) a^4+24 b B a^3+12 b^2 (3 A+4 C) a^2+32 b^3 B a+8 b^4 (A+2 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (16 \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right ) \int \sec ^2(c+d x)dx+15 \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \int \sec (c+d x)dx\right )+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (15 \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+16 \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (15 \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {16 \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right ) \int 1d(-\tan (c+d x))}{d}\right )+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{2} \left (15 \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {16 \tan (c+d x) \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {\tan (c+d x) \sec ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{4 d}+\frac {1}{4} \left (\frac {2 a \tan (c+d x) \sec ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {1}{2} \left (\frac {15 \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {16 \tan (c+d x) \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right )}{d}\right )+\frac {\tan (c+d x) \sec (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )\right )+\frac {2 (3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}\right )+\frac {A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}\) |
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((2*(2*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (((12 *A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x] ^3*Tan[c + d*x])/(4*d) + (((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4* (5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (2*a*(4*A*b^3 + 16*a^3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Sec[c + d*x] ^2*Tan[c + d*x])/d + ((15*(24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12* a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]])/d + (16*(8*a ^4*B + 60*a^2*b^2*B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C ))*Tan[c + d*x])/d)/2)/4)/5)/6
3.10.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[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 = 1.00 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.86
| method | result | size |
| parts | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(329\) |
| derivativedivides | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-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 )+a^{4} C \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 A \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \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 a^{3} b C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{2} b^{2} \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^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a A \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B a \,b^{3} \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 \,b^{3} \tan \left (d x +c \right )+A \,b^{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 )+B \,b^{4} \tan \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(521\) |
| default | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-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 )+a^{4} C \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 A \,a^{3} b \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \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 a^{3} b C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{2} b^{2} \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^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a A \,b^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B a \,b^{3} \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 \,b^{3} \tan \left (d x +c \right )+A \,b^{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 )+B \,b^{4} \tan \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(521\) |
| parallelrisch | \(\frac {-450 \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {24 B \,a^{3} b}{5}+\frac {36 \left (A +\frac {4 C}{3}\right ) b^{2} a^{2}}{5}+\frac {32 B a \,b^{3}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \left (\frac {5}{3}+\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+450 \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {24 B \,a^{3} b}{5}+\frac {36 \left (A +\frac {4 C}{3}\right ) b^{2} a^{2}}{5}+\frac {32 B a \,b^{3}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \left (\frac {5}{3}+\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1920 B \,a^{4}+7680 a^{3} \left (A +\frac {3 C}{4}\right ) b +8640 B \,a^{2} b^{2}+5760 \left (A +\frac {5 C}{6}\right ) a \,b^{3}+1200 B \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (\left (850 A +1020 C \right ) a^{4}+4080 B \,a^{3} b +6120 \left (A +\frac {12 C}{17}\right ) b^{2} a^{2}+2880 B a \,b^{3}+720 A \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (768 B \,a^{4}+3072 \left (A +\frac {5 C}{4}\right ) b \,a^{3}+5760 B \,a^{2} b^{2}+3840 b^{3} \left (A +C \right ) a +960 B \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (\left (150 A +180 C \right ) a^{4}+720 B \,a^{3} b +1080 \left (A +\frac {4 C}{3}\right ) b^{2} a^{2}+960 B a \,b^{3}+240 A \,b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (128 B \,a^{4}+512 \left (A +\frac {5 C}{4}\right ) b \,a^{3}+960 B \,a^{2} b^{2}+640 \left (A +\frac {3 C}{2}\right ) b^{3} a +240 B \,b^{4}\right ) \sin \left (6 d x +6 c \right )+1980 \left (\left (A +\frac {14 C}{33}\right ) a^{4}+\frac {56 B \,a^{3} b}{33}+\frac {28 \left (A +\frac {4 C}{7}\right ) b^{2} a^{2}}{11}+\frac {32 B a \,b^{3}}{33}+\frac {8 A \,b^{4}}{33}\right ) \sin \left (d x +c \right )}{240 d \left (10+\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 )\right )}\) | \(551\) |
| risch | \(\text {Expression too large to display}\) | \(1564\) |
A*a^4/d*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c) +5/16*ln(sec(d*x+c)+tan(d*x+c)))-(4*A*a^3*b+B*a^4)/d*(-8/15-1/5*sec(d*x+c) ^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+(B*b^4+4*C*a*b^3)/d*tan(d*x+c)+(A*b^4+4*B *a*b^3+6*C*a^2*b^2)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x +c)))-(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+ c)+(6*A*a^2*b^2+4*B*a^3*b+C*a^4)/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*ta n(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))+C*b^4/d*ln(sec(d*x+c)+tan(d*x+c))
Time = 0.32 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.03 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 60 \, B a^{2} b^{2} + 20 \, {\left (2 \, A + 3 \, C\right )} a b^{3} + 15 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, A a^{4} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 32 \, {\left (2 \, B a^{4} + 2 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 15 \, B a^{2} b^{2} + 10 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
integrate((a+b*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*(15*((5*A + 6*C)*a^4 + 24*B*a^3*b + 12*(3*A + 4*C)*a^2*b^2 + 32*B*a* b^3 + 8*(A + 2*C)*b^4)*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*((5*A + 6 *C)*a^4 + 24*B*a^3*b + 12*(3*A + 4*C)*a^2*b^2 + 32*B*a*b^3 + 8*(A + 2*C)*b ^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(8*B*a^4 + 8*(4*A + 5*C) *a^3*b + 60*B*a^2*b^2 + 20*(2*A + 3*C)*a*b^3 + 15*B*b^4)*cos(d*x + c)^5 + 40*A*a^4 + 15*((5*A + 6*C)*a^4 + 24*B*a^3*b + 12*(3*A + 4*C)*a^2*b^2 + 32* B*a*b^3 + 8*A*b^4)*cos(d*x + c)^4 + 32*(2*B*a^4 + 2*(4*A + 5*C)*a^3*b + 15 *B*a^2*b^2 + 10*A*a*b^3)*cos(d*x + c)^3 + 10*((5*A + 6*C)*a^4 + 24*B*a^3*b + 36*A*a^2*b^2)*cos(d*x + c)^2 + 48*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin (d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.73 \[ \int (a+b \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 {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} + 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^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} b + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} b^{2} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} - 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 )} - 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 \, B a^{3} b {\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 )} - 180 \, A a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 720 \, C a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 480 \, B a b^{3} {\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 )} - 120 \, A b^{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 b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1920 \, C a b^{3} \tan \left (d x + c\right ) + 480 \, B b^{4} \tan \left (d x + c\right )}{480 \, d} \]
integrate((a+b*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*(32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3*b + 64 0*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3*b + 960*(tan(d*x + c)^3 + 3*tan( d*x + c))*B*a^2*b^2 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a*b^3 - 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*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*B*a^3*b*(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)) - 180*A*a^2*b^2*(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)) - 720*C*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 480*B*a*b^3*(2*sin(d *x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 120*A*b^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*b^4*(log(sin(d*x + c) + 1) - log(sin(d *x + c) - 1)) + 1920*C*a*b^3*tan(d*x + c) + 480*B*b^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (367) = 734\).
Time = 0.42 (sec) , antiderivative size = 1658, normalized size of antiderivative = 4.35 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Too large to display} \]
integrate((a+b*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*(15*(5*A*a^4 + 6*C*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4 + 16*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*( 5*A*a^4 + 6*C*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4 + 16*C*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(165*A*a^4*ta n(1/2*d*x + 1/2*c)^11 - 240*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 150*C*a^4*tan( 1/2*d*x + 1/2*c)^11 - 960*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 600*B*a^3*b*ta n(1/2*d*x + 1/2*c)^11 - 960*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 900*A*a^2*b^ 2*tan(1/2*d*x + 1/2*c)^11 - 1440*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 720*C *a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 960*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 4 80*B*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 960*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 240*B*b^4*tan(1/2*d*x + 1/2*c)^11 + 2 5*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 560*B*a^4*tan(1/2*d*x + 1/2*c)^9 - 210*C* a^4*tan(1/2*d*x + 1/2*c)^9 + 2240*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 840*B*a ^3*b*tan(1/2*d*x + 1/2*c)^9 + 3520*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 1260*A *a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 5280*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 2160*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^ 9 - 1440*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 4800*C*a*b^3*tan(1/2*d*x + 1/2*c )^9 - 360*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 1200*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 1248*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 +...
Time = 4.20 (sec) , antiderivative size = 942, normalized size of antiderivative = 2.47 \[ \int (a+b \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 {\left (\frac {11\,A\,a^4}{8}+A\,b^4-2\,B\,a^4-2\,B\,b^4+\frac {5\,C\,a^4}{4}+\frac {15\,A\,a^2\,b^2}{2}-12\,B\,a^2\,b^2+6\,C\,a^2\,b^2-8\,A\,a\,b^3-8\,A\,a^3\,b+4\,B\,a\,b^3+5\,B\,a^3\,b-8\,C\,a\,b^3-8\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^4}{24}-3\,A\,b^4+\frac {14\,B\,a^4}{3}+10\,B\,b^4-\frac {7\,C\,a^4}{4}-\frac {21\,A\,a^2\,b^2}{2}+44\,B\,a^2\,b^2-18\,C\,a^2\,b^2+\frac {88\,A\,a\,b^3}{3}+\frac {56\,A\,a^3\,b}{3}-12\,B\,a\,b^3-7\,B\,a^3\,b+40\,C\,a\,b^3+\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,A\,a^4}{4}+2\,A\,b^4-\frac {52\,B\,a^4}{5}-20\,B\,b^4+\frac {C\,a^4}{2}+3\,A\,a^2\,b^2-72\,B\,a^2\,b^2+12\,C\,a^2\,b^2-48\,A\,a\,b^3-\frac {208\,A\,a^3\,b}{5}+8\,B\,a\,b^3+2\,B\,a^3\,b-80\,C\,a\,b^3-48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^4}{4}+2\,A\,b^4+\frac {52\,B\,a^4}{5}+20\,B\,b^4+\frac {C\,a^4}{2}+3\,A\,a^2\,b^2+72\,B\,a^2\,b^2+12\,C\,a^2\,b^2+48\,A\,a\,b^3+\frac {208\,A\,a^3\,b}{5}+8\,B\,a\,b^3+2\,B\,a^3\,b+80\,C\,a\,b^3+48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,A\,a^4}{24}-3\,A\,b^4-\frac {14\,B\,a^4}{3}-10\,B\,b^4-\frac {7\,C\,a^4}{4}-\frac {21\,A\,a^2\,b^2}{2}-44\,B\,a^2\,b^2-18\,C\,a^2\,b^2-\frac {88\,A\,a\,b^3}{3}-\frac {56\,A\,a^3\,b}{3}-12\,B\,a\,b^3-7\,B\,a^3\,b-40\,C\,a\,b^3-\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^4}{8}+A\,b^4+2\,B\,a^4+2\,B\,b^4+\frac {5\,C\,a^4}{4}+\frac {15\,A\,a^2\,b^2}{2}+12\,B\,a^2\,b^2+6\,C\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b+4\,B\,a\,b^3+5\,B\,a^3\,b+8\,C\,a\,b^3+8\,C\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,A\,a^4}{16}+\frac {A\,b^4}{2}+\frac {3\,C\,a^4}{8}+C\,b^4+\frac {9\,A\,a^2\,b^2}{4}+3\,C\,a^2\,b^2+2\,B\,a\,b^3+\frac {3\,B\,a^3\,b}{2}\right )}{\frac {5\,A\,a^4}{4}+2\,A\,b^4+\frac {3\,C\,a^4}{2}+4\,C\,b^4+9\,A\,a^2\,b^2+12\,C\,a^2\,b^2+8\,B\,a\,b^3+6\,B\,a^3\,b}\right )\,\left (\frac {5\,A\,a^4}{8}+A\,b^4+\frac {3\,C\,a^4}{4}+2\,C\,b^4+\frac {9\,A\,a^2\,b^2}{2}+6\,C\,a^2\,b^2+4\,B\,a\,b^3+3\,B\,a^3\,b\right )}{d} \]
(tan(c/2 + (d*x)/2)*((11*A*a^4)/8 + A*b^4 + 2*B*a^4 + 2*B*b^4 + (5*C*a^4)/ 4 + (15*A*a^2*b^2)/2 + 12*B*a^2*b^2 + 6*C*a^2*b^2 + 8*A*a*b^3 + 8*A*a^3*b + 4*B*a*b^3 + 5*B*a^3*b + 8*C*a*b^3 + 8*C*a^3*b) + tan(c/2 + (d*x)/2)^11*( (11*A*a^4)/8 + A*b^4 - 2*B*a^4 - 2*B*b^4 + (5*C*a^4)/4 + (15*A*a^2*b^2)/2 - 12*B*a^2*b^2 + 6*C*a^2*b^2 - 8*A*a*b^3 - 8*A*a^3*b + 4*B*a*b^3 + 5*B*a^3 *b - 8*C*a*b^3 - 8*C*a^3*b) - tan(c/2 + (d*x)/2)^3*(3*A*b^4 - (5*A*a^4)/24 + (14*B*a^4)/3 + 10*B*b^4 + (7*C*a^4)/4 + (21*A*a^2*b^2)/2 + 44*B*a^2*b^2 + 18*C*a^2*b^2 + (88*A*a*b^3)/3 + (56*A*a^3*b)/3 + 12*B*a*b^3 + 7*B*a^3*b + 40*C*a*b^3 + (88*C*a^3*b)/3) + tan(c/2 + (d*x)/2)^9*((5*A*a^4)/24 - 3*A *b^4 + (14*B*a^4)/3 + 10*B*b^4 - (7*C*a^4)/4 - (21*A*a^2*b^2)/2 + 44*B*a^2 *b^2 - 18*C*a^2*b^2 + (88*A*a*b^3)/3 + (56*A*a^3*b)/3 - 12*B*a*b^3 - 7*B*a ^3*b + 40*C*a*b^3 + (88*C*a^3*b)/3) + tan(c/2 + (d*x)/2)^5*((15*A*a^4)/4 + 2*A*b^4 + (52*B*a^4)/5 + 20*B*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 + 72*B*a^2*b^ 2 + 12*C*a^2*b^2 + 48*A*a*b^3 + (208*A*a^3*b)/5 + 8*B*a*b^3 + 2*B*a^3*b + 80*C*a*b^3 + 48*C*a^3*b) + tan(c/2 + (d*x)/2)^7*((15*A*a^4)/4 + 2*A*b^4 - (52*B*a^4)/5 - 20*B*b^4 + (C*a^4)/2 + 3*A*a^2*b^2 - 72*B*a^2*b^2 + 12*C*a^ 2*b^2 - 48*A*a*b^3 - (208*A*a^3*b)/5 + 8*B*a*b^3 + 2*B*a^3*b - 80*C*a*b^3 - 48*C*a^3*b))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 20*t an(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)) + (atanh((4*tan(c/2 + (d*x)/2)*((5*A*a^4)/1...