Integrand size = 41, antiderivative size = 657 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 a^6 d}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]
1/2*(20*A*b^2-8*B*a*b+a^2*(A+2*C))*arctanh(sin(d*x+c))/a^6/d+b*(20*A*b^8+2 0*a^7*b*B-35*a^5*b^3*B+28*a^3*b^5*B-8*a*b^7*B-a^2*b^6*(69*A-2*C)-8*a^6*b^2 *(5*A-C)+7*a^4*b^4*(12*A-C)-8*a^8*C)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c) /(a+b)^(1/2))/a^6/(a^2-b^2)^3/d/(a-b)^(1/2)/(a+b)^(1/2)+1/6*(60*A*b^7+6*a^ 7*B-65*a^5*b^2*B+68*a^3*b^4*B-24*a*b^6*B+a^4*b^3*(146*A-17*C)-a^2*b^5*(167 *A-6*C)-a^6*(24*A*b-26*C*b))*tan(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^6-12 *B*a^5*b+11*B*a^3*b^3-4*B*a*b^5-a^6*(A-6*C)+a^4*b^2*(23*A-2*C)-a^2*b^4*(27 *A-C))*sec(d*x+c)*tan(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2-a*(B*b-C*a))*sec (d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(5*A*b^4+7*B*a^3*b -2*B*a*b^3-4*a^4*C-a^2*b^2*(10*A+C))*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2 /d/(a+b*cos(d*x+c))^2+1/6*(20*A*b^6-27*B*a^5*b+20*B*a^3*b^3-8*B*a*b^5-a^2* b^4*(53*A-2*C)+12*a^6*C+a^4*b^2*(48*A+C))*sec(d*x+c)*tan(d*x+c)/a^3/(a^2-b ^2)^3/d/(a+b*cos(d*x+c))
Time = 9.07 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.04 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {b \left (40 a^6 A b^2-84 a^4 A b^4+69 a^2 A b^6-20 A b^8-20 a^7 b B+35 a^5 b^3 B-28 a^3 b^5 B+8 a b^7 B+8 a^8 C-8 a^6 b^2 C+7 a^4 b^4 C-2 a^2 b^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{a^6 \left (a^2-b^2\right )^3 \sqrt {-a^2+b^2} d}+\frac {\left (-a^2 A-20 A b^2+8 a b B-2 a^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (a^2 A+20 A b^2-8 a b B+2 a^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\sec (c+d x) (-4 A b \sin (c+d x)+a B \sin (c+d x))}{a^5 d}+\frac {A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)+a^2 b^2 C \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {14 a^2 A b^4 \sin (c+d x)-9 A b^6 \sin (c+d x)-11 a^3 b^3 B \sin (c+d x)+6 a b^5 B \sin (c+d x)+8 a^4 b^2 C \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {74 a^4 A b^4 \sin (c+d x)-95 a^2 A b^6 \sin (c+d x)+36 A b^8 \sin (c+d x)-47 a^5 b^3 B \sin (c+d x)+50 a^3 b^5 B \sin (c+d x)-18 a b^7 B \sin (c+d x)+26 a^6 b^2 C \sin (c+d x)-17 a^4 b^4 C \sin (c+d x)+6 a^2 b^6 C \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^4 d} \]
(b*(40*a^6*A*b^2 - 84*a^4*A*b^4 + 69*a^2*A*b^6 - 20*A*b^8 - 20*a^7*b*B + 3 5*a^5*b^3*B - 28*a^3*b^5*B + 8*a*b^7*B + 8*a^8*C - 8*a^6*b^2*C + 7*a^4*b^4 *C - 2*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a ^6*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(a^2*A) - 20*A*b^2 + 8*a*b*B - 2 *a^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^6*d) + ((a^2*A + 20 *A*b^2 - 8*a*b*B + 2*a^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a ^6*d) + (Sec[c + d*x]*(-4*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(a^5*d) + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(3*a^ 3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (14*a^2*A*b^4*Sin[c + d*x] - 9*A *b^6*Sin[c + d*x] - 11*a^3*b^3*B*Sin[c + d*x] + 6*a*b^5*B*Sin[c + d*x] + 8 *a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^2*d *(a + b*Cos[c + d*x])^2) + (74*a^4*A*b^4*Sin[c + d*x] - 95*a^2*A*b^6*Sin[c + d*x] + 36*A*b^8*Sin[c + d*x] - 47*a^5*b^3*B*Sin[c + d*x] + 50*a^3*b^5*B *Sin[c + d*x] - 18*a*b^7*B*Sin[c + d*x] + 26*a^6*b^2*C*Sin[c + d*x] - 17*a ^4*b^4*C*Sin[c + d*x] + 6*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*( a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d)
Time = 4.89 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.05, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 3534, 25, 3042, 3534, 25, 3042, 3534, 25, 3042, 3534, 27, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\int -\frac {\left (-\left ((3 A-2 C) a^2\right )-2 b B a+3 (A b+C b-a B) \cos (c+d x) a+5 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {\left (-\left ((3 A-2 C) a^2\right )-2 b B a+3 (A b+C b-a B) \cos (c+d x) a+5 A b^2-4 \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {-\left ((3 A-2 C) a^2\right )-2 b B a+3 (A b+C b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a+5 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int -\frac {\left (-3 \left (-4 C a^4+7 b B a^3-b^2 (10 A+C) a^2-2 b^3 B a+5 A b^4\right ) \cos ^2(c+d x)+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \cos (c+d x)+2 \left (3 (A-2 C) a^4+9 b B a^3-b^2 (18 A-C) a^2-4 b^3 B a+10 A b^4\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {\left (-3 \left (-4 C a^4+7 b B a^3-b^2 (10 A+C) a^2-2 b^3 B a+5 A b^4\right ) \cos ^2(c+d x)+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \cos (c+d x)+2 \left (3 (A-2 C) a^4+9 b B a^3-b^2 (18 A-C) a^2-4 b^3 B a+10 A b^4\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-3 \left (-4 C a^4+7 b B a^3-b^2 (10 A+C) a^2-2 b^3 B a+5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 (A-2 C) a^4+9 b B a^3-b^2 (18 A-C) a^2-4 b^3 B a+10 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\int -\frac {\left (-2 \left (12 C a^6-27 b B a^5+b^2 (48 A+C) a^4+20 b^3 B a^3-b^4 (53 A-2 C) a^2-8 b^5 B a+20 A b^6\right ) \cos ^2(c+d x)+a \left (-6 B a^5+2 b (9 A+5 C) a^4-7 b^2 B a^3-b^3 (8 A-5 C) a^2-2 b^4 B a+5 A b^5\right ) \cos (c+d x)+6 \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {\left (-2 \left (12 C a^6-27 b B a^5+b^2 (48 A+C) a^4+20 b^3 B a^3-b^4 (53 A-2 C) a^2-8 b^5 B a+20 A b^6\right ) \cos ^2(c+d x)+a \left (-6 B a^5+2 b (9 A+5 C) a^4-7 b^2 B a^3-b^3 (8 A-5 C) a^2-2 b^4 B a+5 A b^5\right ) \cos (c+d x)+6 \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {-2 \left (12 C a^6-27 b B a^5+b^2 (48 A+C) a^4+20 b^3 B a^3-b^4 (53 A-2 C) a^2-8 b^5 B a+20 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-6 B a^5+2 b (9 A+5 C) a^4-7 b^2 B a^3-b^3 (8 A-5 C) a^2-2 b^4 B a+5 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int -\frac {2 \left (6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\int \frac {\left (6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\int \frac {6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {\int \frac {3 \left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \int \frac {\left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \int \sec (c+d x)dx}{a}+\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}+\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\tan (c+d x) \sec (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \tan (c+d x) \sec (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {\tan (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}+\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}+\frac {2 b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - (((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2* b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^ 4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Sec[c + d*x]*Tan[c + d*x]) /(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((3*(10*A*b^6 - 12*a^5*b*B + 11* a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27 *A - C))*Sec[c + d*x]*Tan[c + d*x])/(a*d) - ((3*((2*b*(20*A*b^8 + 20*a^7*b *B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^ 6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*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)^3 *(20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(a*d)))/a + ( (60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*( 146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Tan[c + d*x ])/(a*d))/a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.11.8.3.1 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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.90 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.27
| method | result | size |
| derivativedivides | \(\frac {-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+20 A \,b^{2}-8 B a b +2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-20 A \,b^{2}+8 B a b -2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}-\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 B \,a^{5} b -5 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}+2 B \,a^{2} b^{4}-6 B a \,b^{5}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 a^{2} C \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 B \,a^{5} b +29 B \,a^{3} b^{3}-9 B a \,b^{5}+18 a^{6} C -11 a^{4} b^{2} C +3 a^{2} C \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 B \,a^{5} b +5 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}-2 B \,a^{2} b^{4}-6 B a \,b^{5}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{3}}+\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}-20 a^{7} b B +35 a^{5} b^{3} B -28 a^{3} b^{5} B +8 a \,b^{7} B +8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\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^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{6}}}{d}\) | \(834\) |
| default | \(\frac {-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+20 A \,b^{2}-8 B a b +2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-20 A \,b^{2}+8 B a b -2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}-\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 B \,a^{5} b -5 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}+2 B \,a^{2} b^{4}-6 B a \,b^{5}+12 a^{6} C +4 C \,a^{5} b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 a^{2} C \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 B \,a^{5} b +29 B \,a^{3} b^{3}-9 B a \,b^{5}+18 a^{6} C -11 a^{4} b^{2} C +3 a^{2} C \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 B \,a^{5} b +5 B \,a^{4} b^{2}+18 B \,a^{3} b^{3}-2 B \,a^{2} b^{4}-6 B a \,b^{5}+12 a^{6} C -4 C \,a^{5} b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 a^{2} C \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{3}}+\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}-20 a^{7} b B +35 a^{5} b^{3} B -28 a^{3} b^{5} B +8 a \,b^{7} B +8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\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^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{6}}}{d}\) | \(834\) |
| risch | \(\text {Expression too large to display}\) | \(4513\) |
1/d*(-1/2*A/a^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(tan(1 /2*d*x+1/2*c)+1)+1/2*(A*a^2+20*A*b^2-8*B*a*b+2*C*a^2)/a^6*ln(tan(1/2*d*x+1 /2*c)+1)+1/2*A/a^4/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(ta n(1/2*d*x+1/2*c)-1)+1/2/a^6*(-A*a^2-20*A*b^2+8*B*a*b-2*C*a^2)*ln(tan(1/2*d *x+1/2*c)-1)-2*b/a^6*((-1/2*(30*A*a^4*b^2+6*A*a^3*b^3-34*A*a^2*b^4-3*A*a*b ^5+12*A*b^6-20*B*a^5*b-5*B*a^4*b^2+18*B*a^3*b^3+2*B*a^2*b^4-6*B*a*b^5+12*C *a^6+4*C*a^5*b-6*C*a^4*b^2-C*a^3*b^3+2*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3 *a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(45*A*a^4*b^2-53*A*a^2*b^4+18*A*b^6-3 0*B*a^5*b+29*B*a^3*b^3-9*B*a*b^5+18*C*a^6-11*C*a^4*b^2+3*C*a^2*b^4)*a*b/(a ^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(30*A*a^4*b^2-6*A*a ^3*b^3-34*A*a^2*b^4+3*A*a*b^5+12*A*b^6-20*B*a^5*b+5*B*a^4*b^2+18*B*a^3*b^3 -2*B*a^2*b^4-6*B*a*b^5+12*C*a^6-4*C*a^5*b-6*C*a^4*b^2+C*a^3*b^3+2*C*a^2*b^ 4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/ 2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3+1/2*(40*A*a^6*b^2-84*A*a^4*b^4+69*A *a^2*b^6-20*A*b^8-20*B*a^7*b+35*B*a^5*b^3-28*B*a^3*b^5+8*B*a*b^7+8*C*a^8-8 *C*a^6*b^2+7*C*a^4*b^4-2*C*a^2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*( a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 2269 vs. \(2 (633) = 1266\).
Time = 276.48 (sec) , antiderivative size = 4607, normalized size of antiderivative = 7.01 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4, x, algorithm="fricas")
[-1/12*(3*((8*C*a^8*b^4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5*b^ 7 - 7*(12*A - C)*a^4*b^8 - 28*B*a^3*b^9 + (69*A - 2*C)*a^2*b^10 + 8*B*a*b^ 11 - 20*A*b^12)*cos(d*x + c)^5 + 3*(8*C*a^9*b^3 - 20*B*a^8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*B*a^6*b^6 - 7*(12*A - C)*a^5*b^7 - 28*B*a^4*b^8 + (69*A - 2*C)*a^3*b^9 + 8*B*a^2*b^10 - 20*A*a*b^11)*cos(d*x + c)^4 + 3*(8*C*a^10*b^ 2 - 20*B*a^9*b^3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b ^6 - 28*B*a^5*b^7 + (69*A - 2*C)*a^4*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*co s(d*x + c)^3 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^9*b^3 + 35*B*a^ 8*b^4 - 7*(12*A - C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a ^4*b^8 - 20*A*a^3*b^9)*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)) - 3*(((A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)*a^8*b^5 + 32* B*a^7*b^6 - 2*(37*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8 + 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12 + 20*A*b^13)*cos(d*x + c)^5 + 3*((A + 2*C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9*b^4 + 32* B*a^8*b^5 - 2*(37*A - 6*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(29*A - 2*C)*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b^12)*cos( d*x + c)^4 + 3*((A + 2*C)*a^12*b - 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 3 2*B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5 - 48*B*a^7*b^6 + 4*(29*A - 2*C)*a^...
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4, x, algorithm="maxima")
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. 1482 vs. \(2 (633) = 1266\).
Time = 0.43 (sec) , antiderivative size = 1482, normalized size of antiderivative = 2.26 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4, x, algorithm="giac")
1/6*(6*(8*C*a^8*b - 20*B*a^7*b^2 + 40*A*a^6*b^3 - 8*C*a^6*b^3 + 35*B*a^5*b ^4 - 84*A*a^4*b^5 + 7*C*a^4*b^5 - 28*B*a^3*b^6 + 69*A*a^2*b^7 - 2*C*a^2*b^ 7 + 8*B*a*b^8 - 20*A*b^9)*(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^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(a^2 - b^2)) + 2*(36* C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 6 0*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c )^5 + 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2 *c)^5 - 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan(1/2*d*x + 1 /2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2*d* x + 1/2*c)^5 + 42*B*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8*tan(1/2*d *x + 1/2*c)^5 - 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 + 180*A*a^6*b^4*tan(1/2* d*x + 1/2*c)^3 - 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*B*a^5*b^5*tan( 1/2*d*x + 1/2*c)^3 - 392*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 + 56*C*a^4*b^6*t an(1/2*d*x + 1/2*c)^3 - 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 + 284*A*a^...
Time = 24.30 (sec) , antiderivative size = 21844, normalized size of antiderivative = 33.25 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 2*B*a^8 - 59*A*a^2*b^6 + 27*A*a ^3*b^5 + 57*A*a^4*b^4 - 21*A*a^5*b^3 - 11*A*a^6*b^2 + 4*B*a^2*b^6 + 24*B*a ^3*b^5 - 11*B*a^4*b^4 - 26*B*a^5*b^3 + 6*B*a^6*b^2 + 2*C*a^2*b^6 - C*a^3*b ^5 - 6*C*a^4*b^4 + 4*C*a^5*b^3 + 12*C*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b - 8 *B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)^3*(a - b)) + (2*tan(c/2 + (d*x)/2)^3*( 6*A*a^9 - 120*A*b^9 + 6*B*a^9 + 364*A*a^2*b^7 + 71*A*a^3*b^6 - 369*A*a^4*b ^5 - 45*A*a^5*b^4 + 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 - 148*B*a^3 *b^6 - 29*B*a^4*b^5 + 159*B*a^5*b^4 + 18*B*a^6*b^3 - 30*B*a^7*b^2 - 12*C*a ^2*b^7 - 3*C*a^3*b^6 + 37*C*a^4*b^5 + 8*C*a^5*b^4 - 60*C*a^6*b^3 - 30*A*a* b^8 - 21*A*a^8*b + 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) + (2*tan(c/2 + (d*x)/2)^7*(6*A*a^9 + 120*A*b^9 - 6*B*a^9 - 364*A*a^2*b^7 + 7 1*A*a^3*b^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 + 148*B*a^3*b^6 - 29*B*a^4*b^5 - 159*B*a^5*b^4 + 18*B*a^6*b^ 3 + 30*B*a^7*b^2 + 12*C*a^2*b^7 - 3*C*a^3*b^6 - 37*C*a^4*b^5 + 8*C*a^5*b^4 + 60*C*a^6*b^3 - 30*A*a*b^8 + 21*A*a^8*b - 48*B*a*b^8 - 6*B*a^8*b))/(3*a^ 5*(a + b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9*A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*b^4 + 36*A*a^8*b^2 + 248*B*a^3*b ^7 - 320*B*a^5*b^5 + 132*B*a^7*b^3 + 18*C*a^2*b^8 - 62*C*a^4*b^6 + 110*C*a ^6*b^4 - 36*C*a^8*b^2 - 72*B*a*b^9 - 18*B*a^9*b))/(3*a^5*(a + b)^3*(a - b) ^3) + (tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 + 2*B*a^8 - 59*A*a^2*b^6 - ...