Integrand size = 33, antiderivative size = 246 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=4 a b^3 C x+\frac {\left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
4*a*b^3*C*x+1/8*(8*A*b^4+24*a^2*b^2*(A+2*C)+a^4*(3*A+4*C))*arctanh(sin(d*x +c))/d-1/24*b^2*(2*b^2*(13*A-12*C)+3*a^2*(3*A+4*C))*sin(d*x+c)/d+1/12*a*b* (12*A*b^2+a^2*(23*A+36*C))*tan(d*x+c)/d+1/8*(4*A*b^2+a^2*(3*A+4*C))*(a+b*c os(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*A*b*(a+b*cos(d*x+c))^3*sec(d*x+c) ^2*tan(d*x+c)/d+1/4*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^3*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(612\) vs. \(2(246)=492\).
Time = 9.18 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.49 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {4 a b^3 C (c+d x)}{d}+\frac {\left (-3 a^4 A-24 a^2 A b^2-8 A b^4-4 a^4 C-48 a^2 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\left (3 a^4 A+24 a^2 A b^2+8 A b^4+4 a^4 C+48 a^2 b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {a^4 A}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {9 a^4 A+16 a^3 A b+72 a^2 A b^2+12 a^4 C}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {a^4 A}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {-9 a^4 A-16 a^3 A b-72 a^2 A b^2-12 a^4 C}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b^4 C \sin (c+d x)}{d} \]
(4*a*b^3*C*(c + d*x))/d + ((-3*a^4*A - 24*a^2*A*b^2 - 8*A*b^4 - 4*a^4*C - 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(8*d) + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 4*a^4*C + 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(8*d) + (a^4*A)/(16*d*(Cos[(c + d*x)/2] - Sin[(c + d*x) /2])^4) + (9*a^4*A + 16*a^3*A*b + 72*a^2*A*b^2 + 12*a^4*C)/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*a^3*A*b*Sin[(c + d*x)/2])/(3*d*(Cos[( c + d*x)/2] - Sin[(c + d*x)/2])^3) - (a^4*A)/(16*d*(Cos[(c + d*x)/2] + Sin [(c + d*x)/2])^4) + (2*a^3*A*b*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-9*a^4*A - 16*a^3*A*b - 72*a^2*A*b^2 - 12*a^4*C)/( 48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(2*a^3*A*b*Sin[(c + d*x )/2] + 3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos [(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*(2*a^3*A*b*Sin[(c + d*x)/2] + 3*a* A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])) + (b^4*C*Sin[c + d*x])/d
Time = 1.99 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 3527, 3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3502, 27, 3042, 3214, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^3 \left (-b (A-4 C) \cos ^2(c+d x)+a (3 A+4 C) \cos (c+d x)+4 A b\right ) \sec ^4(c+d x)dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (3 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x))^2 \left (-b^2 (7 A-12 C) \cos ^2(c+d x)+2 a b (7 A+12 C) \cos (c+d x)+3 \left ((3 A+4 C) a^2+4 A b^2\right )\right ) \sec ^3(c+d x)dx+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-b^2 (7 A-12 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (7 A+12 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left ((3 A+4 C) a^2+4 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int (a+b \cos (c+d x)) \left (-b \left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x)+a \left (3 (3 A+4 C) a^2+2 b^2 (13 A+36 C)\right ) \cos (c+d x)+2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right ) \sec ^2(c+d x)dx+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (3 (3 A+4 C) a^2+2 b^2 (13 A+36 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 C b)\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}-\int -\left (\left (96 a C \cos (c+d x) b^3-\left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x) b^2+3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+8 A b^4\right )\right ) \sec (c+d x)\right )dx\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \left (96 a C \cos (c+d x) b^3-\left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \cos ^2(c+d x) b^2+3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+8 A b^4\right )\right ) \sec (c+d x)dx+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int \frac {96 a C \sin \left (c+d x+\frac {\pi }{2}\right ) b^3-\left (3 (3 A+4 C) a^2+2 b^2 (13 A-12 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+8 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+32 b^3 C \cos (c+d x) a+8 A b^4\right ) \sec (c+d x)dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left ((3 A+4 C) a^4+24 b^2 (A+2 C) a^2+32 b^3 C \cos (c+d x) a+8 A b^4\right ) \sec (c+d x)dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {(3 A+4 C) a^4+24 b^2 (A+2 C) a^2+32 b^3 C \sin \left (c+d x+\frac {\pi }{2}\right ) a+8 A b^4}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \int \sec (c+d x)dx+32 a b^3 C x\right )-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+32 a b^3 C x\right )-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}\right )+\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3 \left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac {1}{2} \left (-\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{d}+\frac {2 a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{d}+3 \left (\frac {\left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{d}+32 a b^3 C x\right )\right )\right )+\frac {4 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}\) |
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((4*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*(4*A*b^2 + a^ 2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ( 3*(32*a*b^3*C*x + ((8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcT anh[Sin[c + d*x]])/d) - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin [c + d*x])/d + (2*a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Tan[c + d*x])/d)/2)/3 )/4
3.6.55.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
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[((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/(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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 10.31 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) C \,b^{4}}{d}-\frac {4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {4 C a \,b^{3} \left (d x +c \right )}{d}\) | \(219\) |
| derivativedivides | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{3} b \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(247\) |
| default | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 C \,a^{3} b \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 A a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(247\) |
| parallelrisch | \(\frac {-36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{4}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {8 A \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{4}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {8 A \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+384 C a \,b^{3} d x \cos \left (2 d x +2 c \right )+96 C a \,b^{3} d x \cos \left (4 d x +4 c \right )+\left (\left (18 A +24 C \right ) a^{4}+144 A \,a^{2} b^{2}+36 C \,b^{4}\right ) \sin \left (3 d x +3 c \right )+256 b \left (a^{2} \left (A +\frac {3 C}{4}\right )+\frac {3 A \,b^{2}}{4}\right ) a \sin \left (2 d x +2 c \right )+64 \left (\left (A +\frac {3 C}{2}\right ) a^{2}+\frac {3 A \,b^{2}}{2}\right ) b a \sin \left (4 d x +4 c \right )+12 C \sin \left (5 d x +5 c \right ) b^{4}+\left (\left (66 A +24 C \right ) a^{4}+144 A \,a^{2} b^{2}+24 C \,b^{4}\right ) \sin \left (d x +c \right )+288 C a \,b^{3} d x}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(350\) |
| risch | \(4 a \,b^{3} C x -\frac {i C \,b^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i C \,b^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a \left (9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-96 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-96 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-288 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-33 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-256 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-288 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-64 A \,a^{2} b -96 A \,b^{3}-96 C \,a^{2} b \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}+\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}\) | \(629\) |
a^4*A/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ tan(d*x+c)))+(A*b^4+6*C*a^2*b^2)/d*ln(sec(d*x+c)+tan(d*x+c))+(4*A*a*b^3+4* C*a^3*b)/d*tan(d*x+c)+(6*A*a^2*b^2+C*a^4)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2 *ln(sec(d*x+c)+tan(d*x+c)))+1/d*sin(d*x+c)*C*b^4-4*A*a^3*b/d*(-2/3-1/3*sec (d*x+c)^2)*tan(d*x+c)+4*C*a*b^3/d*(d*x+c)
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {192 \, C a b^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{3} b \cos \left (d x + c\right ) + 6 \, A a^{4} + 32 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} b + 3 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
1/48*(192*C*a*b^3*d*x*cos(d*x + c)^4 + 3*((3*A + 4*C)*a^4 + 24*(A + 2*C)*a ^2*b^2 + 8*A*b^4)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*((3*A + 4*C)*a^ 4 + 24*(A + 2*C)*a^2*b^2 + 8*A*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(24*C*b^4*cos(d*x + c)^4 + 32*A*a^3*b*cos(d*x + c) + 6*A*a^4 + 32*((2* A + 3*C)*a^3*b + 3*A*a*b^3)*cos(d*x + c)^3 + 3*((3*A + 4*C)*a^4 + 24*A*a^2 *b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.24 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 192 \, {\left (d x + c\right )} C a b^{3} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
1/48*(64*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3*b + 192*(d*x + c)*C*a*b^3 - 3*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)) - 12* C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(s in(d*x + c) - 1)) - 72*A*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - lo g(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 144*C*a^2*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*A*b^4*(log(sin(d*x + c) + 1) - log (sin(d*x + c) - 1)) + 48*C*b^4*sin(d*x + c) + 192*C*a^3*b*tan(d*x + c) + 1 92*A*a*b^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (234) = 468\).
Time = 0.38 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.40 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} C a b^{3} + \frac {48 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a b^{3} \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 )}^{4}}}{24 \, d} \]
1/24*(96*(d*x + c)*C*a*b^3 + 48*C*b^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 3*(3*A*a^4 + 4*C*a^4 + 24*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b ^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(3*A*a^4 + 4*C*a^4 + 24*A*a^2*b ^2 + 48*C*a^2*b^2 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(15*A* a^4*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 96*A*a^3*b* tan(1/2*d*x + 1/2*c)^7 - 96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 72*A*a^2*b^2* tan(1/2*d*x + 1/2*c)^7 - 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^4*tan(1 /2*d*x + 1/2*c)^5 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^3*b*tan(1/2* d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 72*A*a^2*b^2*tan(1/2 *d*x + 1/2*c)^5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A*a^4*tan(1/2*d*x + 1/ 2*c) + 12*C*a^4*tan(1/2*d*x + 1/2*c) + 96*A*a^3*b*tan(1/2*d*x + 1/2*c) + 9 6*C*a^3*b*tan(1/2*d*x + 1/2*c) + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*A* a*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d
Time = 4.23 (sec) , antiderivative size = 1988, normalized size of antiderivative = 8.08 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
((27*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + 9*A*b^4*atanh (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + (9*C*a^4*atanh(sin(c/2 + (d*x)/2 )/cos(c/2 + (d*x)/2)))/2 + (9*A*a^4*sin(3*c + 3*d*x))/8 + (3*C*a^4*sin(3*c + 3*d*x))/2 + (9*C*b^4*sin(3*c + 3*d*x))/4 + (3*C*b^4*sin(5*c + 5*d*x))/4 + (33*A*a^4*sin(c + d*x))/8 + (3*C*a^4*sin(c + d*x))/2 + (3*C*b^4*sin(c + d*x))/2 + 12*A*a*b^3*sin(2*c + 2*d*x) + 16*A*a^3*b*sin(2*c + 2*d*x) + 6*A *a*b^3*sin(4*c + 4*d*x) + 4*A*a^3*b*sin(4*c + 4*d*x) + 9*A*a^2*b^2*sin(c + d*x) + 12*C*a^3*b*sin(2*c + 2*d*x) + 6*C*a^3*b*sin(4*c + 4*d*x) + 36*C*a* b^3*atan((9*A^2*a^8*sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(c/2 + (d*x)/2) + 1 6*C^2*a^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*A^ 2*a^4*b^4*sin(c/2 + (d*x)/2) + 144*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1024*C ^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*C^2*a^4*b^4*sin(c/2 + (d*x)/2) + 384* C^2*a^6*b^2*sin(c/2 + (d*x)/2) + 24*A*C*a^8*sin(c/2 + (d*x)/2) + 768*A*C*a ^2*b^6*sin(c/2 + (d*x)/2) + 2368*A*C*a^4*b^4*sin(c/2 + (d*x)/2) + 480*A*C* a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A^2*a^8 + 64*A^2*b^8 + 16*C^2*a^8 + 384*A^2*a^2*b^6 + 624*A^2*a^4*b^4 + 144*A^2*a^6*b^2 + 1024*C^ 2*a^2*b^6 + 2304*C^2*a^4*b^4 + 384*C^2*a^6*b^2 + 24*A*C*a^8 + 768*A*C*a^2* b^6 + 2368*A*C*a^4*b^4 + 480*A*C*a^6*b^2))) + (9*A*a^4*atanh(sin(c/2 + (d* x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x))/2 + (9*A*a^4*atanh(sin(c/2 + ( d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/8 + 27*A*a^2*b^2*atanh(si...