Integrand size = 41, antiderivative size = 453 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) x}{2 a^5}-\frac {b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B-a^2 b^4 (29 A-2 C)+5 a^4 b^2 (4 A-C)+6 a^6 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B+a^4 b (6 A-5 C)-a^2 b^3 (21 A-2 C)\right ) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (6 A b^4+6 a^3 b B-3 a b^3 B+a^4 (A-4 C)-a^2 b^2 (10 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (7 a^2 A b^2-4 A b^4-5 a^3 b B+2 a b^3 B+3 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
1/2*(12*A*b^2-6*B*a*b+a^2*(A+2*C))*x/a^5-b*(12*A*b^6-12*a^5*b*B+15*a^3*b^3 *B-6*a*b^5*B-a^2*b^4*(29*A-2*C)+5*a^4*b^2*(4*A-C)+6*a^6*C)*arctanh((a-b)^( 1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(5/2)/(a+b)^(5/2)/d-1/2*(12 *A*b^5-2*a^5*B+11*a^3*b^2*B-6*a*b^4*B+a^4*b*(6*A-5*C)-a^2*b^3*(21*A-2*C))* sin(d*x+c)/a^4/(a^2-b^2)^2/d+1/2*(6*A*b^4+6*B*a^3*b-3*B*a*b^3+a^4*(A-4*C)- a^2*b^2*(10*A-C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*(A*b^2-a*(B* b-C*a))*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*(7*A*a^ 2*b^2-4*A*b^4-5*B*a^3*b+2*B*a*b^3+3*C*a^4)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2- b^2)^2/d/(a+b*sec(d*x+c))
Time = 10.03 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.94 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {16 b \left (12 A b^6-12 a^5 b B+15 a^3 b^3 B-6 a b^5 B+5 a^4 b^2 (4 A-C)+6 a^6 C+a^2 b^4 (-29 A+2 C)\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {4 a^8 A c+48 a^6 A b^2 c-12 a^4 A b^4 c-136 a^2 A b^6 c+96 A b^8 c-24 a^7 b B c+72 a^3 b^5 B c-48 a b^7 B c+8 a^8 c C-24 a^4 b^4 c C+16 a^2 b^6 c C+4 a^8 A d x+48 a^6 A b^2 d x-12 a^4 A b^4 d x-136 a^2 A b^6 d x+96 A b^8 d x-24 a^7 b B d x+72 a^3 b^5 B d x-48 a b^7 B d x+8 a^8 C d x-24 a^4 b^4 C d x+16 a^2 b^6 C d x+16 a b \left (a^2-b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) (c+d x) \cos (c+d x)+4 \left (a^3-a b^2\right )^2 \left (12 A b^2-6 a b B+a^2 (A+2 C)\right ) (c+d x) \cos (2 (c+d x))-8 a^7 A b \sin (c+d x)-32 a^5 A b^3 \sin (c+d x)+160 a^3 A b^5 \sin (c+d x)-96 a A b^7 \sin (c+d x)+4 a^8 B \sin (c+d x)+8 a^6 b^2 B \sin (c+d x)-84 a^4 b^4 B \sin (c+d x)+48 a^2 b^6 B \sin (c+d x)+40 a^5 b^3 C \sin (c+d x)-16 a^3 b^5 C \sin (c+d x)+2 a^8 A \sin (2 (c+d x))-48 a^6 A b^2 \sin (2 (c+d x))+130 a^4 A b^4 \sin (2 (c+d x))-72 a^2 A b^6 \sin (2 (c+d x))+16 a^7 b B \sin (2 (c+d x))-64 a^5 b^3 B \sin (2 (c+d x))+36 a^3 b^5 B \sin (2 (c+d x))+24 a^6 b^2 C \sin (2 (c+d x))-12 a^4 b^4 C \sin (2 (c+d x))-8 a^7 A b \sin (3 (c+d x))+16 a^5 A b^3 \sin (3 (c+d x))-8 a^3 A b^5 \sin (3 (c+d x))+4 a^8 B \sin (3 (c+d x))-8 a^6 b^2 B \sin (3 (c+d x))+4 a^4 b^4 B \sin (3 (c+d x))+a^8 A \sin (4 (c+d x))-2 a^6 A b^2 \sin (4 (c+d x))+a^4 A b^4 \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{16 a^5 d} \]
((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2] )/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12*a ^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 8*a^8*c*C - 24*a^4*b^4*c*C + 16*a^2*b^6*c*C + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 8*a^8*C*d*x - 24*a^ 4*b^4*C*d*x + 16*a^2*b^6*C*d*x + 16*a*b*(a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(12*A*b^2 - 6* a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] + 160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin [c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B *Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 40*a^5*b^3*C*Sin[c + d*x] - 16 *a^3*b^5*C*Sin[c + d*x] + 2*a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5 *B*Sin[2*(c + d*x)] + 24*a^6*b^2*C*Sin[2*(c + d*x)] - 12*a^4*b^4*C*Sin[2*( c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c + d*x)] - 8 *a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3 *(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2...
Time = 3.16 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.06, 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, 4588, 3042, 4592, 27, 3042, 4592, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)+2 a (A b+C b-a B) \sec (c+d x)+2 \left (-\left ((A-C) a^2\right )-b B a+2 A b^2\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {-3 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a (A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (-\left ((A-C) a^2\right )-b B a+2 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int \frac {\cos ^2(c+d x) \left (2 \left (3 C a^4-5 b B a^3+7 A b^2 a^2+2 b^3 B a-4 A b^4\right ) \sec ^2(c+d x)+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sec (c+d x)+2 \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right )\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int \frac {2 \left (3 C a^4-5 b B a^3+7 A b^2 a^2+2 b^3 B a-4 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\int \frac {2 \cos (c+d x) \left (-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \sec (c+d x) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{2 a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\int \frac {\cos (c+d x) \left (-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \sec (c+d x) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\int \frac {-2 B a^5+b (6 A-5 C) a^4+11 b^2 B a^3-b^3 (21 A-2 C) a^2-6 b^4 B a+\left (-\left ((A+2 C) a^4\right )+4 b B a^3-b^2 (4 A+C) a^2-b^3 B a+2 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+12 A b^5-b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\int \frac {\left ((A+2 C) a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\int \frac {\left ((A+2 C) a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left ((A-4 C) a^4+6 b B a^3-b^2 (10 A-C) a^2-3 b^3 B a+6 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {\left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {\left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {2 \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{a}}{a \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\sin (c+d x) \cos (c+d x) \left (3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right )}{a d}-\frac {\frac {\sin (c+d x) \left (-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right )}{a d}-\frac {\frac {x \left (a^2-b^2\right )^2 \left (a^2 (A+2 C)-6 a b B+12 A b^2\right )}{a}-\frac {2 b \left (6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (-(((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x] ))) - (((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(a*d) - (-((((a^2 - b^2)^2*(12*A*b^2 - 6*a *b*B + a^2*(A + 2*C))*x)/a - (2*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[ (Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) )/a) + ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(21*A - 2*C))*Sin[c + d*x])/(a*d))/a)/(a*(a^2 - b^2)))/(2*a*(a^ 2 - b^2))
3.10.22.3.1 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 = 1.04 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b -B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b +B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 a^{5} b B +15 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -3 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -3 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +12 A \,b^{2}-6 B a b +2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(487\) |
| default | \(\frac {\frac {2 b \left (\frac {-\frac {\left (10 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b -B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C +a^{3} b C -2 C \,a^{2} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (10 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-8 B \,a^{3} b +B \,a^{2} b^{2}+4 B a \,b^{3}+6 a^{4} C -a^{3} b C -2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (20 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+12 A \,b^{6}-12 a^{5} b B +15 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -5 a^{4} b^{2} C +2 C \,a^{2} b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{5}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -3 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -3 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +12 A \,b^{2}-6 B a b +2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(487\) |
| risch | \(\text {Expression too large to display}\) | \(2178\) |
1/d*(2*b/a^5*((-1/2*(10*A*a^2*b^2+A*a*b^3-6*A*b^4-8*B*a^3*b-B*a^2*b^2+4*B* a*b^3+6*C*a^4+C*a^3*b-2*C*a^2*b^2)*a*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1 /2*c)^3+1/2*b*a*(10*A*a^2*b^2-A*a*b^3-6*A*b^4-8*B*a^3*b+B*a^2*b^2+4*B*a*b^ 3+6*C*a^4-C*a^3*b-2*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(tan(1/2* d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-1/2*(20*A*a^4*b^2-29*A*a^2*b^ 4+12*A*b^6-12*B*a^5*b+15*B*a^3*b^3-6*B*a*b^5+6*C*a^6-5*C*a^4*b^2+2*C*a^2*b ^4)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2* c)/((a+b)*(a-b))^(1/2)))+2/a^5*(((-1/2*a^2*A-3*a*A*b+B*a^2)*tan(1/2*d*x+1/ 2*c)^3+(1/2*a^2*A-3*a*A*b+B*a^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c) ^2)^2+1/2*(A*a^2+12*A*b^2-6*B*a*b+2*C*a^2)*arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (433) = 866\).
Time = 0.46 (sec) , antiderivative size = 2127, normalized size of antiderivative = 4.70 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3, x, algorithm="fricas")
[1/4*(2*((A + 2*C)*a^10 - 6*B*a^9*b + 3*(3*A - 2*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(11*A - 2*C)*a^6*b^4 - 18*B*a^5*b^5 + (35*A - 2*C)*a^4*b^6 + 6*B*a^3* b^7 - 12*A*a^2*b^8)*d*x*cos(d*x + c)^2 + 4*((A + 2*C)*a^9*b - 6*B*a^8*b^2 + 3*(3*A - 2*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(11*A - 2*C)*a^5*b^5 - 18*B*a^4 *b^6 + (35*A - 2*C)*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*d*x*cos(d*x + c) + 2*((A + 2*C)*a^8*b^2 - 6*B*a^7*b^3 + 3*(3*A - 2*C)*a^6*b^4 + 18*B*a^5*b^5 - 3*(11*A - 2*C)*a^4*b^6 - 18*B*a^3*b^7 + (35*A - 2*C)*a^2*b^8 + 6*B*a*b^ 9 - 12*A*b^10)*d*x + (6*C*a^6*b^3 - 12*B*a^5*b^4 + 5*(4*A - C)*a^4*b^5 + 1 5*B*a^3*b^6 - (29*A - 2*C)*a^2*b^7 - 6*B*a*b^8 + 12*A*b^9 + (6*C*a^8*b - 1 2*B*a^7*b^2 + 5*(4*A - C)*a^6*b^3 + 15*B*a^5*b^4 - (29*A - 2*C)*a^4*b^5 - 6*B*a^3*b^6 + 12*A*a^2*b^7)*cos(d*x + c)^2 + 2*(6*C*a^7*b^2 - 12*B*a^6*b^3 + 5*(4*A - C)*a^5*b^4 + 15*B*a^4*b^5 - (29*A - 2*C)*a^3*b^6 - 6*B*a^2*b^7 + 12*A*a*b^8)*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*B*a^8*b^2 - (6*A - 5*C)*a^7*b^3 - 13*B*a^6*b^4 + (27*A - 7*C)*a^5*b^5 + 17*B*a^4*b^6 - (33*A - 2*C)*a^3*b^7 - 6*B*a^2*b^8 + 12*A*a*b^9 + (A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6)*cos(d*x + c)^3 + 2*(B*a^10 - 2*A* a^9*b - 3*B*a^8*b^2 + 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 - B*a^4*b^6 + 2*A*a^3*b^7)*cos(d*x + c)^2 + (4*B*a^9*b - (11*A - 6*C)*a^8*b^2 - 20*...
\[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**2/(a + b*s ec(c + d*x))**3, x)
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3, 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
Leaf count of result is larger than twice the leaf count of optimal. 3408 vs. \(2 (433) = 866\).
Time = 0.73 (sec) , antiderivative size = 3408, normalized size of antiderivative = 7.52 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3, x, algorithm="giac")
-1/2*(((a^6 - a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 23*a^2*b^4 - 6*a*b^5 + 12* b^6)*sqrt(-a^2 + b^2)*A*abs(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) - 3*(2* a^5*b + 2*a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5)*sqrt(-a^2 + b^2)*B*abs( a^9 - 2*a^7*b^2 + a^5*b^4)*abs(-a + b) + (2*a^6 + 4*a^5*b - 4*a^4*b^2 - a^ 3*b^3 + 2*a^2*b^4)*sqrt(-a^2 + b^2)*C*abs(a^9 - 2*a^7*b^2 + a^5*b^4)*abs(- a + b) - (a^15 - a^14*b + 8*a^13*b^2 - 28*a^12*b^3 - 42*a^11*b^4 + 111*a^1 0*b^5 + 68*a^9*b^6 - 158*a^8*b^7 - 47*a^7*b^8 + 100*a^6*b^9 + 12*a^5*b^10 - 24*a^4*b^11)*sqrt(-a^2 + b^2)*A*abs(-a + b) + 3*(2*a^14*b - 6*a^13*b^2 - 8*a^12*b^3 + 21*a^11*b^4 + 12*a^10*b^5 - 28*a^9*b^6 - 8*a^8*b^7 + 17*a^7* b^8 + 2*a^6*b^9 - 4*a^5*b^10)*sqrt(-a^2 + b^2)*B*abs(-a + b) - (2*a^15 - 8 *a^14*b - 8*a^13*b^2 + 25*a^12*b^3 + 12*a^11*b^4 - 30*a^10*b^5 - 8*a^9*b^6 + 17*a^8*b^7 + 2*a^7*b^8 - 4*a^6*b^9)*sqrt(-a^2 + b^2)*C*abs(-a + b))*(pi *floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^8*b - 2*a^6*b^3 + a^4*b^5 + sqrt((a^9 + a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + a^5*b^ 4 + a^4*b^5)*(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^5*b^4 - a^4*b^5) + ( a^8*b - 2*a^6*b^3 + a^4*b^5)^2))/(a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + a^ 5*b^4 - a^4*b^5))))/((a^9 - 2*a^7*b^2 + a^5*b^4)^2*(a^2 - 2*a*b + b^2) + ( a^10*b - 2*a^9*b^2 - a^8*b^3 + 4*a^7*b^4 - a^6*b^5 - 2*a^5*b^6 + a^4*b^7)* abs(a^9 - 2*a^7*b^2 + a^5*b^4)) + (A*a^15 + 2*C*a^15 - A*a^14*b - 6*B*a^14 *b - 8*C*a^14*b + 8*A*a^13*b^2 + 18*B*a^13*b^2 - 8*C*a^13*b^2 - 28*A*a^...
Time = 35.00 (sec) , antiderivative size = 16016, normalized size of antiderivative = 35.36 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^5*(3*A*a^7 - 36*A*b^7 - 2*B*a^7 + 67*A*a^2*b^5 - 29*A *a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2*b^5 - 35*B*a^3*b^4 + 16*B* a^4*b^3 + 10*B*a^5*b^2 - 6*C*a^2*b^5 + 3*C*a^3*b^4 + 15*C*a^4*b^3 - 6*C*a^ 5*b^2 + 18*A*a*b^6 + 4*A*a^6*b + 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) - (tan(c/2 + (d*x)/2)^3*(3*A*a^7 + 36*A*b^7 + 2*B*a^ 7 - 67*A*a^2*b^5 - 29*A*a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2*b^5 + 35*B*a^3*b^4 + 16*B*a^4*b^3 - 10*B*a^5*b^2 + 6*C*a^2*b^5 + 3*C*a^3*b^4 - 15*C*a^4*b^3 - 6*C*a^5*b^2 + 18*A*a*b^6 - 4*A*a^6*b - 18*B*a*b^6 - 4*B*a ^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^7*(A*a^ 6 - 12*A*b^6 - 2*B*a^6 + 23*A*a^2*b^4 - 10*A*a^3*b^3 - 8*A*a^4*b^2 - 3*B*a ^2*b^4 - 12*B*a^3*b^3 + 4*B*a^4*b^2 - 2*C*a^2*b^4 + C*a^3*b^3 + 6*C*a^4*b^ 2 + 6*A*a*b^5 + 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a^4*b - a^5)*(a + b) ^2) + (tan(c/2 + (d*x)/2)*(A*a^6 - 12*A*b^6 + 2*B*a^6 + 23*A*a^2*b^4 + 10* A*a^3*b^3 - 8*A*a^4*b^2 + 3*B*a^2*b^4 - 12*B*a^3*b^3 - 4*B*a^4*b^2 - 2*C*a ^2*b^4 - C*a^3*b^3 + 6*C*a^4*b^2 - 6*A*a*b^5 - 5*A*a^5*b + 6*B*a*b^5 + 2*B *a^5*b))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/(d*(2*a*b - tan(c/2 + (d*x)/ 2)^4*(2*a^2 - 6*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*b^2) - tan(c/2 + (d *x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((((4*(4*A*a^21 + 8*C*a^21 - 48*A*a^10*b^11 + 24*A*a^11* b^10 + 212*A*a^12*b^9 - 100*A*a^13*b^8 - 360*A*a^14*b^7 + 164*A*a^15*b^...