Integrand size = 33, antiderivative size = 378 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=-\frac {b \left (12 A b^6-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (12 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 a^5 d}-\frac {b \left (12 A b^4+a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4+3 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
-b*(12*A*b^6-a^2*b^4*(29*A-2*C)+5*a^4*b^2*(4*A-C)+6*a^6*C)*arctan((a-b)^(1 /2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(5/2)/(a+b)^(5/2)/d+1/2*(12* A*b^2+a^2*(A+2*C))*arctanh(sin(d*x+c))/a^5/d-1/2*b*(12*A*b^4+a^4*(6*A-5*C) -a^2*b^2*(21*A-2*C))*tan(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(6*A*b^4+a^4*(A-4*C) -a^2*b^2*(10*A-C))*sec(d*x+c)*tan(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2+C*a^ 2)*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/2*(7*A*a^2*b^2 -4*A*b^4+3*C*a^4)*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(856\) vs. \(2(378)=756\).
Time = 8.55 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.26 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {2 b \left (20 a^4 A b^2-29 a^2 A b^4+12 A b^6+6 a^6 C-5 a^4 b^2 C+2 a^2 b^4 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right ) \cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right )}{a^5 \left (a^2-b^2\right )^2 \sqrt {-a^2+b^2} d (2 A+C+C \cos (2 c+2 d x))}+\frac {\left (-a^2 A-12 A b^2-2 a^2 C\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (C+A \sec ^2(c+d x)\right )}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac {\left (a^2 A+12 A b^2+2 a^2 C\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (C+A \sec ^2(c+d x)\right )}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac {A \cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right )}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {6 A b \cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {A \cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right )}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {6 A b \cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right ) \left (A b^4 \sin (c+d x)+a^2 b^2 C \sin (c+d x)\right )}{a^3 (a-b) (a+b) d (a+b \cos (c+d x))^2 (2 A+C+C \cos (2 c+2 d x))}+\frac {\cos ^2(c+d x) \left (C+A \sec ^2(c+d x)\right ) \left (9 a^2 A b^4 \sin (c+d x)-6 A b^6 \sin (c+d x)+5 a^4 b^2 C \sin (c+d x)-2 a^2 b^4 C \sin (c+d x)\right )}{a^4 (a-b)^2 (a+b)^2 d (a+b \cos (c+d x)) (2 A+C+C \cos (2 c+2 d x))} \]
(2*b*(20*a^4*A*b^2 - 29*a^2*A*b^4 + 12*A*b^6 + 6*a^6*C - 5*a^4*b^2*C + 2*a ^2*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x ]^2*(C + A*Sec[c + d*x]^2))/(a^5*(a^2 - b^2)^2*Sqrt[-a^2 + b^2]*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((-(a^2*A) - 12*A*b^2 - 2*a^2*C)*Cos[c + d*x]^2* Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d*(2 *A + C + C*Cos[2*c + 2*d*x])) + ((a^2*A + 12*A*b^2 + 2*a^2*C)*Cos[c + d*x] ^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d *(2*A + C + C*Cos[2*c + 2*d*x])) + (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2 ))/(2*a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d* x)/2])^2) - (6*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2]) /(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2 ])) - (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^3*d*(2*A + C + C*Cos[ 2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (6*A*b*Cos[c + d* x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (Cos[c + d*x]^2*(C + A* Sec[c + d*x]^2)*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a - b )*(a + b)*d*(a + b*Cos[c + d*x])^2*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[ c + d*x]^2*(C + A*Sec[c + d*x]^2)*(9*a^2*A*b^4*Sin[c + d*x] - 6*A*b^6*Sin[ c + d*x] + 5*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(a^4*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])*(2*A + C + C*Cos[2*c + 2*d*x]))
Time = 2.71 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3535, 25, 3042, 3534, 25, 3042, 3534, 27, 3042, 3534, 25, 3042, 3480, 3042, 3138, 218, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {\left (-3 \left (C a^2+A b^2\right ) \cos ^2(c+d x)+2 a b (A+C) \cos (c+d x)+2 \left (2 A b^2-a^2 (A-C)\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {\left (-3 \left (C a^2+A b^2\right ) \cos ^2(c+d x)+2 a b (A+C) \cos (c+d x)+2 \left (2 A b^2-a^2 (A-C)\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-3 \left (C a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (2 A b^2-a^2 (A-C)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {\left (2 \left (3 C a^4+7 A b^2 a^2-4 A b^4\right ) \cos ^2(c+d x)+a b \left (A b^2-a^2 (4 A+3 C)\right ) \cos (c+d x)+2 \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {\left (2 \left (3 C a^4+7 A b^2 a^2-4 A b^4\right ) \cos ^2(c+d x)+a b \left (A b^2-a^2 (4 A+3 C)\right ) \cos (c+d x)+2 \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {2 \left (3 C a^4+7 A b^2 a^2-4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a b \left (A b^2-a^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\int -\frac {2 \left (-b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \cos ^2(c+d x)+a \left (-\left ((A+2 C) a^4\right )-b^2 (4 A+C) a^2+2 A b^4\right ) \cos (c+d x)+b \left ((6 A-5 C) a^4-b^2 (21 A-2 C) a^2+12 A b^4\right )\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {\left (-b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \cos ^2(c+d x)+a \left (-\left ((A+2 C) a^4\right )-b^2 (4 A+C) a^2+2 A b^4\right ) \cos (c+d x)+b \left ((6 A-5 C) a^4-b^2 (21 A-2 C) a^2+12 A b^4\right )\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {-b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-\left ((A+2 C) a^4\right )-b^2 (4 A+C) a^2+2 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+b \left ((6 A-5 C) a^4-b^2 (21 A-2 C) a^2+12 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\int -\frac {\left (\left ((A+2 C) a^2+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\int \frac {\left (\left ((A+2 C) a^2+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\int \frac {\left ((A+2 C) a^2+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4-b^2 (10 A-C) a^2+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)+12 A b^2\right ) \int \sec (c+d x)dx}{a}-\frac {b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\left (3 a^4 C+7 a^2 A b^2-4 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {b \left (a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 (A+2 C)+12 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b \left (6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}\) |
((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[ c + d*x])^2) - (-(((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))) - (((6*A*b^4 + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Sec[c + d*x]*Tan[c + d*x])/(a*d) - (-(((-2*b*(12*A* b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + ((a^ 2 - b^2)^2*(12*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(a*d))/a) + ( b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Tan[c + d*x])/(a*d)) /a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2))
3.6.84.3.1 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_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 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[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 4.35 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}+C \,a^{3} b -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b^{2}-A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}-C \,a^{3} b -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (20 A \,a^{4} b^{2}-29 A \,a^{2} b^{4}+12 A \,b^{6}+6 C \,a^{6}-5 C \,a^{4} b^{2}+2 C \,a^{2} b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (A \,a^{2}+12 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}+\frac {A \left (a +6 b \right )}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-A \,a^{2}-12 A \,b^{2}-2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}+\frac {A \left (a +6 b \right )}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(458\) |
| default | \(\frac {-\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}+C \,a^{3} b -2 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b^{2}-A a \,b^{3}-6 A \,b^{4}+6 C \,a^{4}-C \,a^{3} b -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (20 A \,a^{4} b^{2}-29 A \,a^{2} b^{4}+12 A \,b^{6}+6 C \,a^{6}-5 C \,a^{4} b^{2}+2 C \,a^{2} b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (A \,a^{2}+12 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}+\frac {A \left (a +6 b \right )}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-A \,a^{2}-12 A \,b^{2}-2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}+\frac {A \left (a +6 b \right )}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(458\) |
| risch | \(\text {Expression too large to display}\) | \(2026\) |
1/d*(-2*b/a^5*((-1/2*(10*A*a^2*b^2+A*a*b^3-6*A*b^4+6*C*a^4+C*a^3*b-2*C*a^2 *b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*a*(10*A*a^2*b^2 -A*a*b^3-6*A*b^4+6*C*a^4-C*a^3*b-2*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/ 2*c))/(tan(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c)^2+a+b)^2+1/2*(20*A*a^4* b^2-29*A*a^2*b^4+12*A*b^6+6*C*a^6-5*C*a^4*b^2+2*C*a^2*b^4)/(a^4-2*a^2*b^2+ b^4)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/ 2)))-1/2*A/a^3/(tan(1/2*d*x+1/2*c)+1)^2+1/2*(A*a^2+12*A*b^2+2*C*a^2)/a^5*l n(tan(1/2*d*x+1/2*c)+1)+1/2*A*(a+6*b)/a^4/(tan(1/2*d*x+1/2*c)+1)+1/2*A/a^3 /(tan(1/2*d*x+1/2*c)-1)^2+1/2/a^5*(-A*a^2-12*A*b^2-2*C*a^2)*ln(tan(1/2*d*x +1/2*c)-1)+1/2*A*(a+6*b)/a^4/(tan(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (359) = 718\).
Time = 18.87 (sec) , antiderivative size = 2078, normalized size of antiderivative = 5.50 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
[-1/4*(((6*C*a^6*b^3 + 5*(4*A - C)*a^4*b^5 - (29*A - 2*C)*a^2*b^7 + 12*A*b ^9)*cos(d*x + c)^4 + 2*(6*C*a^7*b^2 + 5*(4*A - C)*a^5*b^4 - (29*A - 2*C)*a ^3*b^6 + 12*A*a*b^8)*cos(d*x + c)^3 + (6*C*a^8*b + 5*(4*A - C)*a^6*b^3 - ( 29*A - 2*C)*a^4*b^5 + 12*A*a^2*b^7)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log(( 2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a* cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b* cos(d*x + c) + a^2)) - (((A + 2*C)*a^8*b^2 + 3*(3*A - 2*C)*a^6*b^4 - 3*(11 *A - 2*C)*a^4*b^6 + (35*A - 2*C)*a^2*b^8 - 12*A*b^10)*cos(d*x + c)^4 + 2*( (A + 2*C)*a^9*b + 3*(3*A - 2*C)*a^7*b^3 - 3*(11*A - 2*C)*a^5*b^5 + (35*A - 2*C)*a^3*b^7 - 12*A*a*b^9)*cos(d*x + c)^3 + ((A + 2*C)*a^10 + 3*(3*A - 2* C)*a^8*b^2 - 3*(11*A - 2*C)*a^6*b^4 + (35*A - 2*C)*a^4*b^6 - 12*A*a^2*b^8) *cos(d*x + c)^2)*log(sin(d*x + c) + 1) + (((A + 2*C)*a^8*b^2 + 3*(3*A - 2* C)*a^6*b^4 - 3*(11*A - 2*C)*a^4*b^6 + (35*A - 2*C)*a^2*b^8 - 12*A*b^10)*co s(d*x + c)^4 + 2*((A + 2*C)*a^9*b + 3*(3*A - 2*C)*a^7*b^3 - 3*(11*A - 2*C) *a^5*b^5 + (35*A - 2*C)*a^3*b^7 - 12*A*a*b^9)*cos(d*x + c)^3 + ((A + 2*C)* a^10 + 3*(3*A - 2*C)*a^8*b^2 - 3*(11*A - 2*C)*a^6*b^4 + (35*A - 2*C)*a^4*b ^6 - 12*A*a^2*b^8)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(A*a^10 - 3* A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6 - ((6*A - 5*C)*a^7*b^3 - (27*A - 7*C)* a^5*b^5 + (33*A - 2*C)*a^3*b^7 - 12*A*a*b^9)*cos(d*x + c)^3 - ((11*A - 6*C )*a^8*b^2 - (43*A - 9*C)*a^6*b^4 + (50*A - 3*C)*a^4*b^6 - 18*A*a^2*b^8)...
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (359) = 718\).
Time = 0.37 (sec) , antiderivative size = 1191, normalized size of antiderivative = 3.15 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
1/2*(2*(6*C*a^6*b + 20*A*a^4*b^3 - 5*C*a^4*b^3 - 29*A*a^2*b^5 + 2*C*a^2*b^ 5 + 12*A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(- (a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(a^2 - b^2)) + 2*(A*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 6*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 2*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2*c)^7 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^5 + 4*A*a ^6*b*tan(1/2*d*x + 1/2*c)^5 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^5 *b^2*tan(1/2*d*x + 1/2*c)^5 - 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 15*C*a ^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*C* a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 - 6*C *a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 36*A *b^7*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 - 4*A*a^6*b*t an(1/2*d*x + 1/2*c)^3 + 5*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 6*C*a^5*b^2*t an(1/2*d*x + 1/2*c)^3 + 26*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*C*a^4*b^3 *tan(1/2*d*x + 1/2*c)^3 - 29*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 3*C*a^3*b^ 4*tan(1/2*d*x + 1/2*c)^3 - 67*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^...
Time = 15.59 (sec) , antiderivative size = 10422, normalized size of antiderivative = 27.57 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
- ((tan(c/2 + (d*x)/2)*(12*A*b^6 - A*a^6 - 23*A*a^2*b^4 - 10*A*a^3*b^3 + 8 *A*a^4*b^2 + 2*C*a^2*b^4 + C*a^3*b^3 - 6*C*a^4*b^2 + 6*A*a*b^5 + 5*A*a^5*b ))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^3*(3*A*a^7 + 36*A*b^7 - 67*A*a^2*b^5 - 29*A*a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 + 6*C* a^2*b^5 + 3*C*a^3*b^4 - 15*C*a^4*b^3 - 6*C*a^5*b^2 + 18*A*a*b^6 - 4*A*a^6* b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^5*(3*A*a^7 - 36*A*b^7 + 67*A*a^2*b^5 - 29*A*a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5*b^2 - 6 *C*a^2*b^5 + 3*C*a^3*b^4 + 15*C*a^4*b^3 - 6*C*a^5*b^2 + 18*A*a*b^6 + 4*A*a ^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^7*(A*a^ 6 - 12*A*b^6 + 23*A*a^2*b^4 - 10*A*a^3*b^3 - 8*A*a^4*b^2 - 2*C*a^2*b^4 + C *a^3*b^3 + 6*C*a^4*b^2 + 6*A*a*b^5 + 5*A*a^5*b))/((a^4*b - a^5)*(a + b)^2) )/(d*(2*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 - 6*b^2) - tan(c/2 + (d*x)/2)^2* (4*a*b + 4*b^2) + tan(c/2 + (d*x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2 )^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan((((6*A*b^2 + a^2*(A/2 + C))* ((8*tan(c/2 + (d*x)/2)*(A^2*a^14 + 288*A^2*b^14 + 4*C^2*a^14 - 288*A^2*a*b ^13 - 2*A^2*a^13*b - 8*C^2*a^13*b - 1104*A^2*a^2*b^12 + 1104*A^2*a^3*b^11 + 1538*A^2*a^4*b^10 - 1538*A^2*a^5*b^9 - 827*A^2*a^6*b^8 + 872*A^2*a^7*b^7 + 18*A^2*a^8*b^6 - 108*A^2*a^9*b^5 + 74*A^2*a^10*b^4 - 40*A^2*a^11*b^3 + 21*A^2*a^12*b^2 + 8*C^2*a^4*b^10 - 8*C^2*a^5*b^9 - 32*C^2*a^6*b^8 + 32*C^2 *a^7*b^7 + 57*C^2*a^8*b^6 - 48*C^2*a^9*b^5 - 52*C^2*a^10*b^4 + 32*C^2*a...