Integrand size = 41, antiderivative size = 396 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (8 A b^3-a^3 B-6 a b^2 B+2 a^2 b (A+2 C)\right ) x}{2 a^5}+\frac {2 b^2 \left (5 a^2 A b^2-4 A b^4-4 a^3 b B+3 a b^3 B+3 a^4 C-2 a^2 b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (12 A b^4+6 a^3 b B-9 a b^3 B-a^2 b^2 (7 A-6 C)-a^4 (2 A+3 C)\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}+\frac {\left (4 A b^3+a^3 B-3 a b^2 B-2 a^2 b (A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}-\frac {\left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
-1/2*(8*A*b^3-B*a^3-6*B*a*b^2+2*a^2*b*(A+2*C))*x/a^5+2*b^2*(5*A*a^2*b^2-4* A*b^4-4*B*a^3*b+3*B*a*b^3+3*C*a^4-2*C*a^2*b^2)*arctanh((a-b)^(1/2)*tan(1/2 *d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(3/2)/(a+b)^(3/2)/d-1/3*(12*A*b^4+6*B*a ^3*b-9*B*a*b^3-a^2*b^2*(7*A-6*C)-a^4*(2*A+3*C))*sin(d*x+c)/a^4/(a^2-b^2)/d +1/2*(4*A*b^3+B*a^3-3*B*a*b^2-2*a^2*b*(A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^ 2-b^2)/d-1/3*(4*A*b^2-3*B*a*b-a^2*(A-3*C))*cos(d*x+c)^2*sin(d*x+c)/a^2/(a^ 2-b^2)/d+(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*se c(d*x+c))
Time = 5.93 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.64 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {6 \left (-8 A b^3+a^3 B+6 a b^2 B-2 a^2 b (A+2 C)\right ) (c+d x)+\frac {24 b^2 \left (4 A b^4+4 a^3 b B-3 a b^3 B-3 a^4 C+a^2 b^2 (-5 A+2 C)\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+3 a \left (12 A b^2-8 a b B+a^2 (3 A+4 C)\right ) \sin (c+d x)-\frac {12 a b^3 \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}+3 a^2 (-2 A b+a B) \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 a^5 d} \]
(6*(-8*A*b^3 + a^3*B + 6*a*b^2*B - 2*a^2*b*(A + 2*C))*(c + d*x) + (24*b^2* (4*A*b^4 + 4*a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh [((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(12 *A*b^2 - 8*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] - (12*a*b^3*(A*b^2 + a*(- (b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + 3*a^2 *(-2*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)
Time = 2.68 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 4588, 3042, 4592, 3042, 4592, 3042, 4592, 27, 3042, 4407, 3042, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (-\left ((A-3 C) a^2\right )-3 b B a+(A b+C b-a B) \sec (c+d x) a+4 A b^2-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {-\left ((A-3 C) a^2\right )-3 b B a+(A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+4 A b^2-3 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\int \frac {\cos ^2(c+d x) \left (-2 b \left (-\left ((A-3 C) a^2\right )-3 b B a+4 A b^2\right ) \sec ^2(c+d x)+a \left ((2 A+3 C) a^2-3 b B a+A b^2\right ) \sec (c+d x)+3 \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right )\right )}{a+b \sec (c+d x)}dx}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\int \frac {-2 b \left (-\left ((A-3 C) a^2\right )-3 b B a+4 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left ((2 A+3 C) a^2-3 b B a+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-3 b \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right ) \sec ^2(c+d x)+a \left (-3 B a^3+2 b (A+3 C) a^2-3 b^2 B a+4 A b^3\right ) \sec (c+d x)+2 \left (-\left ((2 A+3 C) a^4\right )+6 b B a^3-b^2 (7 A-6 C) a^2-9 b^3 B a+12 A b^4\right )\right )}{a+b \sec (c+d x)}dx}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\int \frac {-3 b \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-3 B a^3+2 b (A+3 C) a^2-3 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (-\left ((2 A+3 C) a^4\right )+6 b B a^3-b^2 (7 A-6 C) a^2-9 b^3 B a+12 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}-\frac {\int -\frac {3 \left (\left (a^2-b^2\right ) \left (-B a^3+2 b (A+2 C) a^2-6 b^2 B a+8 A b^3\right )-a b \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right ) \left (-B a^3+2 b (A+2 C) a^2-6 b^2 B a+8 A b^3\right )-a b \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right ) \left (-B a^3+2 b (A+2 C) a^2-6 b^2 B a+8 A b^3\right )-a b \left (B a^3-2 b (A-C) a^2-3 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {2 b^2 \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {2 b^2 \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {2 b \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}+\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {2 b \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}+\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {4 b \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos ^2(c+d x) \left (-\left (a^2 (A-3 C)\right )-3 a b B+4 A b^2\right )}{3 a d}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right )}{2 a d}-\frac {\frac {3 \left (\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right )}{a}+\frac {4 b^2 \left (-3 a^4 C+4 a^3 b B-a^2 b^2 (5 A-2 C)-3 a b^3 B+4 A b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}+\frac {2 \sin (c+d x) \left (-\left (a^4 (2 A+3 C)\right )+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right )}{a d}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) - (((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Cos[c + d*x]^2* Sin[c + d*x])/(3*a*d) - ((3*(4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C) )*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((3*(((a^2 - b^2)*(8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*x)/a + (4*b^2*(4*A*b^4 + 4*a^3*b*B - 3*a* b^3*B - a^2*b^2*(5*A - 2*C) - 3*a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/ 2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + (2*(12*A*b^4 + 6*a^3 *b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(a *d))/(2*a))/(3*a))/(a*(a^2 - b^2))
3.10.15.3.1 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Time = 0.80 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}-4 B \,a^{3} b +3 B a \,b^{3}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b -a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}+4 B \,a^{2} b -2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 a A \,b^{2}+2 B \,a^{2} b -a^{3} C +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}+4 a^{2} b C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\) | \(397\) |
| default | \(\frac {-\frac {2 b^{2} \left (-\frac {a b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b^{2}-4 A \,b^{4}-4 B \,a^{3} b +3 B a \,b^{3}+3 a^{4} C -2 C \,a^{2} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 a A \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b -a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 a A \,b^{2}+4 B \,a^{2} b -2 a^{3} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 a A \,b^{2}+2 B \,a^{2} b -a^{3} C +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}+4 a^{2} b C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\) | \(397\) |
| risch | \(\text {Expression too large to display}\) | \(1436\) |
1/d*(-2*b^2/a^5*(-a*b*(A*b^2-B*a*b+C*a^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(ta n(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)-(5*A*a^2*b^2-4*A*b^4-4*B* a^3*b+3*B*a*b^3+3*C*a^4-2*C*a^2*b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arcta nh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))-2/a^5*(((-a^3*A-A*a^2*b- 3*a*A*b^2+1/2*B*a^3+2*B*a^2*b-a^3*C)*tan(1/2*d*x+1/2*c)^5+(-2/3*a^3*A-6*a* A*b^2+4*B*a^2*b-2*a^3*C)*tan(1/2*d*x+1/2*c)^3+(-a^3*A-3*a*A*b^2+2*B*a^2*b- a^3*C+A*a^2*b-1/2*B*a^3)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^3+1/ 2*(2*A*a^2*b+8*A*b^3-B*a^3-6*B*a*b^2+4*C*a^2*b)*arctan(tan(1/2*d*x+1/2*c)) ))
Time = 0.38 (sec) , antiderivative size = 1347, normalized size of antiderivative = 3.40 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2, x, algorithm="fricas")
[1/6*(3*(B*a^8 - 2*(A + 2*C)*a^7*b + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 1 1*B*a^4*b^4 + 2*(7*A - 2*C)*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4*b ^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*d*x + 3*( 3*C*a^4*b^3 - 4*B*a^3*b^4 + (5*A - 2*C)*a^2*b^5 + 3*B*a*b^6 - 4*A*b^7 + (3 *C*a^5*b^2 - 4*B*a^4*b^3 + (5*A - 2*C)*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6)* cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos( d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (2*(2*A + 3*C)*a^ 7*b - 12*B*a^6*b^2 + 2*(5*A - 9*C)*a^5*b^3 + 30*B*a^4*b^4 - 2*(19*A - 6*C) *a^3*b^5 - 18*B*a^2*b^6 + 24*A*a*b^7 + 2*(A*a^8 - 2*A*a^6*b^2 + A*a^4*b^4) *cos(d*x + c)^3 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6*b^2 + 8*A*a^5*b^3 + 3*B*a ^4*b^4 - 4*A*a^3*b^5)*cos(d*x + c)^2 + (2*(2*A + 3*C)*a^8 - 9*B*a^7*b + 4* (A - 3*C)*a^6*b^2 + 18*B*a^5*b^3 - 2*(10*A - 3*C)*a^4*b^4 - 9*B*a^3*b^5 + 12*A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 2*a^8*b^2 + a^6*b^4)*d* cos(d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d), 1/6*(3*(B*a^8 - 2*(A + 2* C)*a^7*b + 4*B*a^6*b^2 - 4*(A - 2*C)*a^5*b^3 - 11*B*a^4*b^4 + 2*(7*A - 2*C )*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a^7*b - 2*(A + 2*C)*a^6*b^2 + 4*B*a^5*b^3 - 4*(A - 2*C)*a^4*b^4 - 11*B*a^3*b^5 + 2*(7*A - 2*C)*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*d*x + 6*(3*C*a^4*b^3 - 4*B*a^3*b...
\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**3/(a + b*s ec(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2, 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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.33 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {12 \, {\left (3 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + 5 \, A a^{2} b^{4} - 2 \, C a^{2} b^{4} + 3 \, B a b^{5} - 4 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {12 \, {\left (C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b - 4 \, C a^{2} b + 6 \, B a b^{2} - 8 \, A b^{3}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A b^{2} \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 )}^{3} a^{4}}}{6 \, d} \]
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^2, x, algorithm="giac")
1/6*(12*(3*C*a^4*b^2 - 4*B*a^3*b^3 + 5*A*a^2*b^4 - 2*C*a^2*b^4 + 3*B*a*b^5 - 4*A*b^6)*(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^7 - a^5*b^2)*sqrt(-a^2 + b^2)) + 12*(C*a^2*b^3*tan(1/2*d*x + 1/2*c) - B*a*b^4 *tan(1/2*d*x + 1/2*c) + A*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - a^4*b^2)*(a*ta n(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) + 3*(B*a^3 - 2*A *a^2*b - 4*C*a^2*b + 6*B*a*b^2 - 8*A*b^3)*(d*x + c)/a^5 + 2*(6*A*a^2*tan(1 /2*d*x + 1/2*c)^5 - 3*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a*b*tan(1/2*d*x + 1/2*c)^5 - 12*B*a*b*tan(1/2*d*x + 1/2*c) ^5 + 18*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 4*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 12 *C*a^2*tan(1/2*d*x + 1/2*c)^3 - 24*B*a*b*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^2 *tan(1/2*d*x + 1/2*c)^3 + 6*A*a^2*tan(1/2*d*x + 1/2*c) + 3*B*a^2*tan(1/2*d *x + 1/2*c) + 6*C*a^2*tan(1/2*d*x + 1/2*c) - 6*A*a*b*tan(1/2*d*x + 1/2*c) - 12*B*a*b*tan(1/2*d*x + 1/2*c) + 18*A*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2 *d*x + 1/2*c)^2 + 1)^3*a^4))/d
Time = 29.66 (sec) , antiderivative size = 11743, normalized size of antiderivative = 29.65 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)*(2*A*a^5 - 8*A*b^5 + B*a^5 + 2*C*a^5 + 6*A*a^2*b^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 - 5*B*a^3*b^2 - 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A* a*b^4 + 6*B*a*b^4 - 3*B*a^4*b + 2*C*a^4*b))/(a^4*(a + b)*(a - b)) - (tan(c /2 + (d*x)/2)^3*(2*A*a^5 + 72*A*b^5 + 3*B*a^5 - 6*C*a^5 - 38*A*a^2*b^3 - 1 4*A*a^3*b^2 - 9*B*a^2*b^3 + 33*B*a^3*b^2 + 36*C*a^2*b^3 + 6*C*a^3*b^2 + 12 *A*a*b^4 - 16*A*a^4*b - 54*B*a*b^4 + 9*B*a^4*b - 18*C*a^4*b))/(3*a^4*(a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^5*(2*A*a^5 - 72*A*b^5 - 3*B*a^5 - 6*C*a^ 5 + 38*A*a^2*b^3 - 14*A*a^3*b^2 - 9*B*a^2*b^3 - 33*B*a^3*b^2 - 36*C*a^2*b^ 3 + 6*C*a^3*b^2 + 12*A*a*b^4 + 16*A*a^4*b + 54*B*a*b^4 + 9*B*a^4*b + 18*C* a^4*b))/(3*a^4*(a + b)*(a - b)) - (tan(c/2 + (d*x)/2)^7*(2*A*a^5 + 8*A*b^5 - B*a^5 + 2*C*a^5 - 6*A*a^2*b^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 + 5*B*a^3*b^2 + 4*C*a^2*b^3 - 2*C*a^3*b^2 - 4*A*a*b^4 - 6*B*a*b^4 - 3*B*a^4*b - 2*C*a^4 *b))/(a^4*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)/2)^8*(a - b) + tan (c/2 + (d*x)/2)^2*(2*a + 4*b) - tan(c/2 + (d*x)/2)^6*(2*a - 4*b) + 6*b*tan (c/2 + (d*x)/2)^4)) - (atan(((((((8*(2*B*a^18 + 16*A*a^10*b^8 - 8*A*a^11*b ^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4*A*a^15*b^3 - 12*B*a ^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 - 16*B*a^15*b^3 + 6 *B*a^16*b^2 + 8*C*a^12*b^6 - 4*C*a^13*b^5 - 20*C*a^14*b^4 + 12*C*a^15*b^3 + 12*C*a^16*b^2 - 4*A*a^17*b - 8*C*a^17*b))/(a^14*b + a^15 - a^12*b^3 - a^ 13*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^3*4i - (B*a^3*1i)/2 + a^2*(A*b*1i ...