Integrand size = 31, antiderivative size = 367 \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=-\frac {4 A b x}{a^5}-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
-4*A*b*x/a^5-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8-2*a^8*C-a^6*b^2*(20*A+3*C) )*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b )^(7/2)/d+1/6*(68*a^2*A*b^4-24*A*b^6+a^6*(6*A-11*C)-a^4*b^2*(65*A+4*C))*si n(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2+C*a^2)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b *sec(d*x+c))^3-1/6*(4*A*b^4-3*a^4*C-a^2*b^2*(9*A+2*C))*sin(d*x+c)/a^2/(a^2 -b^2)^2/d/(a+b*sec(d*x+c))^2-1/2*(11*a^2*A*b^4-4*A*b^6-2*a^6*C-3*a^4*b^2*( 4*A+C))*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
Result contains complex when optimal does not.
Time = 9.27 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.46 \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-24 A b x (b+a \cos (c+d x))^3-\frac {6 i \left (-35 a^4 A b^4+28 a^2 A b^6-8 A b^8+2 a^8 C+a^6 b^2 (20 A+3 C)\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^3 (\cos (c)-i \sin (c))}{\left (a^2-b^2\right )^{7/2} d \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {2 b^3 \left (A b^2+a^2 C\right ) \sec (c) (b \sin (c)-a \sin (d x))}{\left (a^2-b^2\right ) d}+\frac {b^2 (b+a \cos (c+d x)) \sec (c) \left (\left (12 A b^5-11 a^4 b C+a^2 b^3 (-17 A+6 C)\right ) \sin (c)+a \left (-10 A b^4+a^2 b^2 (15 A-4 C)+9 a^4 C\right ) \sin (d x)\right )}{\left (a^2-b^2\right )^2 d}+\frac {b (b+a \cos (c+d x))^2 \sec (c) \left (3 b \left (12 A b^6+a^4 b^2 (25 A-6 C)+9 a^6 C+2 a^2 b^4 (-16 A+C)\right ) \sin (c)-a \left (26 A b^6+5 a^4 b^2 (12 A-C)+18 a^6 C+a^2 b^4 (-71 A+2 C)\right ) \sin (d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {6 a A (b+a \cos (c+d x))^3 \sin (c+d x)}{d}\right )}{3 a^5 (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^4} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-24*A*b*x*(b + a*Cos[c + d*x])^3 - ((6*I)*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 + 2*a ^8*C + a^6*b^2*(20*A + 3*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*( b + a*Cos[c + d*x])^3*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(7/2)*d*Sqrt[(Cos[ c] - I*Sin[c])^2]) + (2*b^3*(A*b^2 + a^2*C)*Sec[c]*(b*Sin[c] - a*Sin[d*x]) )/((a^2 - b^2)*d) + (b^2*(b + a*Cos[c + d*x])*Sec[c]*((12*A*b^5 - 11*a^4*b *C + a^2*b^3*(-17*A + 6*C))*Sin[c] + a*(-10*A*b^4 + a^2*b^2*(15*A - 4*C) + 9*a^4*C)*Sin[d*x]))/((a^2 - b^2)^2*d) + (b*(b + a*Cos[c + d*x])^2*Sec[c]* (3*b*(12*A*b^6 + a^4*b^2*(25*A - 6*C) + 9*a^6*C + 2*a^2*b^4*(-16*A + C))*S in[c] - a*(26*A*b^6 + 5*a^4*b^2*(12*A - C) + 18*a^6*C + a^2*b^4*(-71*A + 2 *C))*Sin[d*x]))/((a^2 - b^2)^3*d) + (6*a*A*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/d))/(3*a^5*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^4)
Time = 2.70 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4589, 3042, 4588, 25, 3042, 4588, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4589 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (-\left ((3 A-C) a^2\right )+3 b (A+C) \sec (c+d x) a+4 A b^2-3 \left (C a^2+A b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {-\left ((3 A-C) a^2\right )+3 b (A+C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+4 A b^2-3 \left (C a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {\cos (c+d x) \left (-\left ((6 A-5 C) a^4\right )+23 A b^2 a^2-2 b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x) a-12 A b^4+2 \left (-3 C a^4-b^2 (9 A+2 C) a^2+4 A b^4\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\int \frac {\cos (c+d x) \left (-\left ((6 A-5 C) a^4\right )+23 A b^2 a^2-2 b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x) a-12 A b^4+2 \left (-3 C a^4-b^2 (9 A+2 C) a^2+4 A b^4\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\int \frac {-\left ((6 A-5 C) a^4\right )+23 A b^2 a^2-2 b \left (A b^2-a^2 (6 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-12 A b^4+2 \left (-3 C a^4-b^2 (9 A+2 C) a^2+4 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left ((6 A-11 C) a^6-b^2 (65 A+4 C) a^4+68 A b^4 a^2-b \left ((18 A+11 C) a^4-b^2 (7 A-4 C) a^2+4 A b^4\right ) \sec (c+d x) a-24 A b^6-3 \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {(6 A-11 C) a^6-b^2 (65 A+4 C) a^4+68 A b^4 a^2-b \left ((18 A+11 C) a^4-b^2 (7 A-4 C) a^2+4 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-24 A b^6-3 \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {\int \frac {3 \left (8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {8 A b x \left (a^2-b^2\right )^3}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {8 A b x \left (a^2-b^2\right )^3}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a b}+\frac {8 A b x \left (a^2-b^2\right )^3}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a b}+\frac {8 A b x \left (a^2-b^2\right )^3}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}+\frac {8 A b x \left (a^2-b^2\right )^3}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {8 A b x \left (a^2-b^2\right )^3}{a}+\frac {2 \left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(2*a*(a^2 - b^ 2)*d*(a + b*Sec[c + d*x])^2) + ((3*(11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a ^4*b^2*(4*A + C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) - ( (-3*((8*A*b*(a^2 - b^2)^3*x)/a + (2*(35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/ Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + ((68*a^2*A*b^4 - 24*A*b^ 6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(a*d))/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.8.3.3.1 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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b ^2 + 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] + Simp[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^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)* (m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 = 1.18 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+4 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(503\) |
| default | \(\frac {-\frac {2 A \left (-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+4 b \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}-\frac {2 \left (\frac {-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 a^{5} C b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}}{d}\) | \(503\) |
| risch | \(\text {Expression too large to display}\) | \(1818\) |
1/d*(-2*A/a^5*(-a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+4*b*arctan(t an(1/2*d*x+1/2*c)))-2/a^5*((-1/2*(20*A*a^4*b^2+5*A*a^3*b^3-18*A*a^2*b^4-2* A*a*b^5+6*A*b^6+6*C*a^6+3*C*a^5*b+2*C*a^4*b^2)*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*(30*A*a^4*b^2-29*A*a^2*b^4+9*A*b^6+9*C*a ^6+C*a^4*b^2)*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 *(20*A*a^4*b^2-5*A*a^3*b^3-18*A*a^2*b^4+2*A*a*b^5+6*A*b^6+6*C*a^6-3*C*a^5* b+2*C*a^4*b^2)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*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)^3-1/2*(20*A*a^6*b^2-35*A* a^4*b^4+28*A*a^2*b^6-8*A*b^8+2*C*a^8+3*C*a^6*b^2)/(a^6-3*a^4*b^2+3*a^2*b^4 -b^6)/((a+b)*(a-b))^(1/2)*arctanh((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. 934 vs. \(2 (347) = 694\).
Time = 0.43 (sec) , antiderivative size = 1925, normalized size of antiderivative = 5.25 \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
[-1/12*(48*(A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + A*a^3*b^9 )*d*x*cos(d*x + c)^3 + 144*(A*a^10*b^2 - 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a ^4*b^8 + A*a^2*b^10)*d*x*cos(d*x + c)^2 + 144*(A*a^9*b^3 - 4*A*a^7*b^5 + 6 *A*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*d*x*cos(d*x + c) + 48*(A*a^8*b^4 - 4* A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*b^10 + A*b^12)*d*x - 3*(2*C*a^8*b^3 + (2 0*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b^9 - 8*A*b^11 + (2*C*a^11 + (20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(2*C*a^10*b + (20*A + 3*C)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^ 7 - 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*b^4 - 35*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*cos(d*x + c))*sqrt(a^2 - b^2)*lo g((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*((6*A - 11*C)*a^9*b^3 - (71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7 - 92*A*a^3*b^9 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10* b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^3 + (18*(A - C)* a^11*b - (132*A - 23*C)*a^9*b^3 + (239*A - 7*C)*a^7*b^5 - (169*A - 2*C)*a^ 5*b^7 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(3*(2*A - 3*C)*a^10*b^2 - (59*A - 8*C)*a^8*b^4 + (110*A + C)*a^6*b^6 - 77*A*a^4*b^8 + 20*A*a^2*b^10)*cos(d* x + c))*sin(d*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8* b^8)*d*cos(d*x + c)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*b^7...
\[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
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*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (347) = 694\).
Time = 0.39 (sec) , antiderivative size = 847, normalized size of antiderivative = 2.31 \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
1/3*(3*(2*C*a^8 + 20*A*a^6*b^2 + 3*C*a^6*b^2 - 35*A*a^4*b^4 + 28*A*a^2*b^6 - 8*A*b^8)*(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^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(-a^2 + b^2)) - 12*(d*x + c)*A*b/a^ 5 + (18*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c) ^5 + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c )^5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2 *c)^5 - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x + 1/ 2*c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2 *c)^5 - 36*C*a^8*b*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^6*b^3*tan(1/2*d*x + 1/ 2*c)^3 + 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^9*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1 /2*c) + 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2*d*x + 1/2 *c) + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c ) + 3*C*a^5*b^4*tan(1/2*d*x + 1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*C*a^4*b^5*tan(1/2*d*x + 1/2*c) - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) - 2 4*A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) + 18*A* b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*(a*...
Time = 30.72 (sec) , antiderivative size = 10120, normalized size of antiderivative = 27.57 \[ \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)*(8*A*b^7 - 2*A*a^7 - 24*A*a^2*b^5 - 11*A*a^3*b^4 + 26 *A*a^4*b^3 + 6*A*a^5*b^2 + 2*C*a^4*b^3 - 3*C*a^5*b^2 + 4*A*a*b^6 - 2*A*a^6 *b + 6*C*a^6*b))/((a + b)*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) + (tan(c/ 2 + (d*x)/2)^7*(2*A*a^7 + 8*A*b^7 - 24*A*a^2*b^5 + 11*A*a^3*b^4 + 26*A*a^4 *b^3 - 6*A*a^5*b^2 + 2*C*a^4*b^3 + 3*C*a^5*b^2 - 4*A*a*b^6 - 2*A*a^6*b + 6 *C*a^6*b))/((a^4*b - a^5)*(a + b)^3) + (tan(c/2 + (d*x)/2)^3*(18*A*a^8 + 7 2*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72 *A*a^6*b^2 + 10*C*a^4*b^4 - 7*C*a^5*b^3 + 45*C*a^6*b^2 - 12*A*a*b^7 - 18*C *a^7*b))/(3*(a + b)^2*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) - (tan(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 - 47*A*a^3*b^5 + 273*A*a^4 *b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 + 10*C*a^4*b^4 + 7*C*a^5*b^3 + 45*C*a^6 *b^2 + 12*A*a*b^7 + 18*C*a^7*b))/(3*(a^4*b - a^5)*(a + b)^3*(a - b)))/(d*( 3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) + tan(c/2 + (d* x)/2)^2*(6*a*b^2 - 2*a^3 + 4*b^3) + tan(c/2 + (d*x)/2)^6*(2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a^2*b + a^3 - b^3 ))) - (8*A*b*atan(((4*A*b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*C^2*a^1 6 - 128*A^2*a*b^15 - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^ 12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a ^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a ^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 9*C^2*a^12*b^4 + 12*C^2*...