Integrand size = 41, antiderivative size = 456 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) x}{2 b^5}-\frac {a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
1/2*(2*A*b^2-6*B*a*b+12*C*a^2+C*b^2)*x/b^5-a*(6*A*b^6-6*B*a^5*b+15*B*a^3*b ^3-12*B*a*b^5+a^4*b^2*(2*A-29*C)-5*a^2*b^4*(A-4*C)+12*a^6*C)*arctan((a-b)^ (1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^5/(a+b)^(5/2)/d+1/2*(6 *B*a^4*b-11*B*a^2*b^3+2*B*b^5-a^3*b^2*(2*A-21*C)+a*b^4*(5*A-6*C)-12*C*a^5) *sin(d*x+c)/b^4/(a^2-b^2)^2/d-1/2*(3*B*a^3*b-6*B*a*b^3-a^2*b^2*(A-10*C)+b^ 4*(4*A-C)-6*a^4*C)*cos(d*x+c)*sin(d*x+c)/b^3/(a^2-b^2)^2/d-1/2*(A*b^2-a*(B *b-C*a))*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/2*(3*A *b^4+a*(2*B*a^2*b-5*B*b^3-4*C*a^3+7*C*a*b^2))*cos(d*x+c)^2*sin(d*x+c)/b^2/ (a^2-b^2)^2/d/(a+b*cos(d*x+c))
Time = 8.08 (sec) , antiderivative size = 883, normalized size of antiderivative = 1.94 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {16 a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {16 a^6 A b^2 c-24 a^4 A b^4 c+8 A b^8 c-48 a^7 b B c+72 a^5 b^3 B c-24 a b^7 B c+96 a^8 c C-136 a^6 b^2 c C-12 a^4 b^4 c C+48 a^2 b^6 c C+4 b^8 c C+16 a^6 A b^2 d x-24 a^4 A b^4 d x+8 A b^8 d x-48 a^7 b B d x+72 a^5 b^3 B d x-24 a b^7 B d x+96 a^8 C d x-136 a^6 b^2 C d x-12 a^4 b^4 C d x+48 a^2 b^6 C d x+4 b^8 C d x+16 a b \left (a^2-b^2\right )^2 \left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) (c+d x) \cos (c+d x)+4 \left (-a^2 b+b^3\right )^2 \left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) (c+d x) \cos (2 (c+d x))-16 a^5 A b^3 \sin (c+d x)+40 a^3 A b^5 \sin (c+d x)+48 a^6 b^2 B \sin (c+d x)-84 a^4 b^4 B \sin (c+d x)+8 a^2 b^6 B \sin (c+d x)+4 b^8 B \sin (c+d x)-96 a^7 b C \sin (c+d x)+160 a^5 b^3 C \sin (c+d x)-32 a^3 b^5 C \sin (c+d x)-8 a b^7 C \sin (c+d x)-12 a^4 A b^4 \sin (2 (c+d x))+24 a^2 A b^6 \sin (2 (c+d x))+36 a^5 b^3 B \sin (2 (c+d x))-64 a^3 b^5 B \sin (2 (c+d x))+16 a b^7 B \sin (2 (c+d x))-72 a^6 b^2 C \sin (2 (c+d x))+130 a^4 b^4 C \sin (2 (c+d x))-48 a^2 b^6 C \sin (2 (c+d x))+2 b^8 C \sin (2 (c+d x))+4 a^4 b^4 B \sin (3 (c+d x))-8 a^2 b^6 B \sin (3 (c+d x))+4 b^8 B \sin (3 (c+d x))-8 a^5 b^3 C \sin (3 (c+d x))+16 a^3 b^5 C \sin (3 (c+d x))-8 a b^7 C \sin (3 (c+d x))+a^4 b^4 C \sin (4 (c+d x))-2 a^2 b^6 C \sin (4 (c+d x))+b^8 C \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{16 b^5 d} \]
((16*a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 2 9*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/ Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (16*a^6*A*b^2*c - 24*a^4*A*b^4*c + 8*A*b^8*c - 48*a^7*b*B*c + 72*a^5*b^3*B*c - 24*a*b^7*B*c + 96*a^8*c*C - 1 36*a^6*b^2*c*C - 12*a^4*b^4*c*C + 48*a^2*b^6*c*C + 4*b^8*c*C + 16*a^6*A*b^ 2*d*x - 24*a^4*A*b^4*d*x + 8*A*b^8*d*x - 48*a^7*b*B*d*x + 72*a^5*b^3*B*d*x - 24*a*b^7*B*d*x + 96*a^8*C*d*x - 136*a^6*b^2*C*d*x - 12*a^4*b^4*C*d*x + 48*a^2*b^6*C*d*x + 4*b^8*C*d*x + 16*a*b*(a^2 - b^2)^2*(2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*(2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[2*(c + d*x)] - 16*a^5*A*b^3*Si n[c + d*x] + 40*a^3*A*b^5*Sin[c + d*x] + 48*a^6*b^2*B*Sin[c + d*x] - 84*a^ 4*b^4*B*Sin[c + d*x] + 8*a^2*b^6*B*Sin[c + d*x] + 4*b^8*B*Sin[c + d*x] - 9 6*a^7*b*C*Sin[c + d*x] + 160*a^5*b^3*C*Sin[c + d*x] - 32*a^3*b^5*C*Sin[c + d*x] - 8*a*b^7*C*Sin[c + d*x] - 12*a^4*A*b^4*Sin[2*(c + d*x)] + 24*a^2*A* b^6*Sin[2*(c + d*x)] + 36*a^5*b^3*B*Sin[2*(c + d*x)] - 64*a^3*b^5*B*Sin[2* (c + d*x)] + 16*a*b^7*B*Sin[2*(c + d*x)] - 72*a^6*b^2*C*Sin[2*(c + d*x)] + 130*a^4*b^4*C*Sin[2*(c + d*x)] - 48*a^2*b^6*C*Sin[2*(c + d*x)] + 2*b^8*C* Sin[2*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] - 8*a^2*b^6*B*Sin[3*(c + d *x)] + 4*b^8*B*Sin[3*(c + d*x)] - 8*a^5*b^3*C*Sin[3*(c + d*x)] + 16*a^3*b^ 5*C*Sin[3*(c + d*x)] - 8*a*b^7*C*Sin[3*(c + d*x)] + a^4*b^4*C*Sin[4*(c ...
Time = 2.53 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 3526, 3042, 3526, 3042, 3528, 27, 3042, 3502, 3042, 3214, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (2 C a^2-b B a+A b^2-b^2 C\right ) \cos ^2(c+d x)+2 b (b B-a (A+C)) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-2 \left (2 C a^2-b B a+A b^2-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (-2 \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right ) \cos ^2(c+d x)+b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)+2 \left (3 A b^4+a \left (-4 C a^3+2 b B a^2+7 b^2 C a-5 b^3 B\right )\right )\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-2 \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 A b^4+a \left (-4 C a^3+2 b B a^2+7 b^2 C a-5 b^3 B\right )\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {2 \left (-\left (\left (-12 C a^5+6 b B a^4-b^2 (2 A-21 C) a^3-11 b^3 B a^2+b^4 (5 A-6 C) a+2 b^5 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^4+b B a^3+b^2 (A+4 C) a^2-4 b^3 B a+b^4 (2 A+C)\right ) \cos (c+d x)+a \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right )\right )}{a+b \cos (c+d x)}dx}{2 b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {-\left (\left (-12 C a^5+6 b B a^4-b^2 (2 A-21 C) a^3-11 b^3 B a^2+b^4 (5 A-6 C) a+2 b^5 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^4+b B a^3+b^2 (A+4 C) a^2-4 b^3 B a+b^4 (2 A+C)\right ) \cos (c+d x)+a \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right )}{a+b \cos (c+d x)}dx}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (12 C a^5-6 b B a^4+b^2 (2 A-21 C) a^3+11 b^3 B a^2-b^4 (5 A-6 C) a-2 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-2 C a^4+b B a^3+b^2 (A+4 C) a^2-4 b^3 B a+b^4 (2 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {a b \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right )-\left (a^2-b^2\right )^2 \left (12 C a^2-6 b B a+2 A b^2+b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)}dx}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {a b \left (-6 C a^4+3 b B a^3-b^2 (A-10 C) a^2-6 b^3 B a+b^4 (4 A-C)\right )-\left (a^2-b^2\right )^2 \left (12 C a^2-6 b B a+2 A b^2+b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}-\frac {x \left (a^2-b^2\right )^2 \left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right )}{b}}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {x \left (a^2-b^2\right )^2 \left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right )}{b}}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {x \left (a^2-b^2\right )^2 \left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right )}{b}}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\sin (c+d x) \cos ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {-\frac {\sin (c+d x) \cos (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{b d}-\frac {\frac {\frac {2 a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (a^2-b^2\right )^2 \left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right )}{b}}{b}-\frac {\sin (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\) |
-1/2*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^2) - (-(((3*A*b^4 + a*(2*a^2*b*B - 5*b^3*B - 4*a^3* C + 7*a*b^2*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))) - (-(((3*a^3*b*B - 6*a*b^3*B - a^2*b^2*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(b*d)) - ((-(((a^2 - b^2)^2*(2*A* b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*x)/b) + (2*a*(6*A*b^6 - 6*a^5*b*B + 15*a ^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^ 6*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sq rt[a + b]*d))/b - ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 21 *C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Sin[c + d*x])/(b*d))/b)/(b*(a^2 - b^2) ))/(2*b*(a^2 - b^2))
3.10.94.3.1 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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 0.93 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-4 B \,a^{3} b +B \,a^{2} b^{2}+8 B a \,b^{3}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-4 B \,a^{3} b -B \,a^{2} b^{2}+8 B a \,b^{3}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}-6 B \,a^{5} b +15 B \,a^{3} b^{3}-12 B a \,b^{5}+12 a^{6} C -29 a^{4} b^{2} C +20 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 \left (\left (B \,b^{2}-3 a b C -\frac {1}{2} C \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B \,b^{2}-3 a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}-6 B a b +12 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(483\) |
| default | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b^{2}-a A \,b^{3}-6 A \,b^{4}-4 B \,a^{3} b +B \,a^{2} b^{2}+8 B a \,b^{3}+6 a^{4} C -a^{3} b C -10 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b^{2}+a A \,b^{3}-6 A \,b^{4}-4 B \,a^{3} b -B \,a^{2} b^{2}+8 B a \,b^{3}+6 a^{4} C +a^{3} b C -10 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}^{2}}+\frac {\left (2 A \,a^{4} b^{2}-5 a^{2} A \,b^{4}+6 A \,b^{6}-6 B \,a^{5} b +15 B \,a^{3} b^{3}-12 B a \,b^{5}+12 a^{6} C -29 a^{4} b^{2} C +20 a^{2} C \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}+\frac {\frac {2 \left (\left (B \,b^{2}-3 a b C -\frac {1}{2} C \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B \,b^{2}-3 a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}-6 B a b +12 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\) | \(483\) |
| risch | \(\text {Expression too large to display}\) | \(2178\) |
1/d*(-2*a/b^5*((1/2*(2*A*a^2*b^2-A*a*b^3-6*A*b^4-4*B*a^3*b+B*a^2*b^2+8*B*a *b^3+6*C*a^4-C*a^3*b-10*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*(2*A*a^2*b^2+A*a*b^3-6*A*b^4-4*B*a^3*b-B*a^2*b^2+8*B*a*b^3 +6*C*a^4+C*a^3*b-10*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*(2*A*a^4*b^2-5*A*a^2*b^4+ 6*A*b^6-6*B*a^5*b+15*B*a^3*b^3-12*B*a*b^5+12*C*a^6-29*C*a^4*b^2+20*C*a^2*b ^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c )/((a-b)*(a+b))^(1/2)))+2/b^5*(((B*b^2-3*a*b*C-1/2*C*b^2)*tan(1/2*d*x+1/2* c)^3+(B*b^2-3*a*b*C+1/2*C*b^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2 )^2+1/2*(2*A*b^2-6*B*a*b+12*C*a^2+C*b^2)*arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 1018 vs. \(2 (437) = 874\).
Time = 0.47 (sec) , antiderivative size = 2107, normalized size of antiderivative = 4.62 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3, x, algorithm="fricas")
[1/4*(2*(12*C*a^8*b^2 - 6*B*a^7*b^3 + (2*A - 35*C)*a^6*b^4 + 18*B*a^5*b^5 - 3*(2*A - 11*C)*a^4*b^6 - 18*B*a^3*b^7 + 3*(2*A - 3*C)*a^2*b^8 + 6*B*a*b^ 9 - (2*A + C)*b^10)*d*x*cos(d*x + c)^2 + 4*(12*C*a^9*b - 6*B*a^8*b^2 + (2* A - 35*C)*a^7*b^3 + 18*B*a^6*b^4 - 3*(2*A - 11*C)*a^5*b^5 - 18*B*a^4*b^6 + 3*(2*A - 3*C)*a^3*b^7 + 6*B*a^2*b^8 - (2*A + C)*a*b^9)*d*x*cos(d*x + c) + 2*(12*C*a^10 - 6*B*a^9*b + (2*A - 35*C)*a^8*b^2 + 18*B*a^7*b^3 - 3*(2*A - 11*C)*a^6*b^4 - 18*B*a^5*b^5 + 3*(2*A - 3*C)*a^4*b^6 + 6*B*a^3*b^7 - (2*A + C)*a^2*b^8)*d*x - (12*C*a^9 - 6*B*a^8*b + (2*A - 29*C)*a^7*b^2 + 15*B*a ^6*b^3 - 5*(A - 4*C)*a^5*b^4 - 12*B*a^4*b^5 + 6*A*a^3*b^6 + (12*C*a^7*b^2 - 6*B*a^6*b^3 + (2*A - 29*C)*a^5*b^4 + 15*B*a^4*b^5 - 5*(A - 4*C)*a^3*b^6 - 12*B*a^2*b^7 + 6*A*a*b^8)*cos(d*x + c)^2 + 2*(12*C*a^8*b - 6*B*a^7*b^2 + (2*A - 29*C)*a^6*b^3 + 15*B*a^5*b^4 - 5*(A - 4*C)*a^4*b^5 - 12*B*a^3*b^6 + 6*A*a^2*b^7)*cos(d*x + c))*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)) - 2*(12*C*a^9*b - 6*B*a^8*b^2 + (2*A - 33*C)*a^7*b^3 + 17*B*a^6*b^4 - (7*A - 27*C)*a^5*b^5 - 13*B*a^4*b^6 + (5*A - 6*C)*a^3*b^7 + 2*B*a^2*b^8 - (C*a^6 *b^4 - 3*C*a^4*b^6 + 3*C*a^2*b^8 - C*b^10)*cos(d*x + c)^3 + 2*(2*C*a^7*b^3 - B*a^6*b^4 - 6*C*a^5*b^5 + 3*B*a^4*b^6 + 6*C*a^3*b^7 - 3*B*a^2*b^8 - 2*C *a*b^9 + B*b^10)*cos(d*x + c)^2 + (18*C*a^8*b^2 - 9*B*a^7*b^3 + (3*A - ...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(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*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. 3417 vs. \(2 (437) = 874\).
Time = 0.70 (sec) , antiderivative size = 3417, normalized size of antiderivative = 7.49 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3, x, algorithm="giac")
-1/2*(((2*a^4*b^2 - a^3*b^3 - 4*a^2*b^4 + 4*a*b^5 + 2*b^6)*sqrt(a^2 - b^2) *A*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) - 3*(2*a^5*b - a^4*b^2 - 4*a ^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*sqrt(a^2 - b^2)*B*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) + (12*a^6 - 6*a^5*b - 23*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^ 4 - a*b^5 + b^6)*sqrt(a^2 - b^2)*C*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) + (4*a^9*b^6 - 2*a^8*b^7 - 17*a^7*b^8 + 8*a^6*b^9 + 30*a^5*b^10 - 12*a ^4*b^11 - 25*a^3*b^12 + 8*a^2*b^13 + 8*a*b^14 - 2*b^15)*sqrt(a^2 - b^2)*A* abs(-a + b) - 3*(4*a^10*b^5 - 2*a^9*b^6 - 17*a^8*b^7 + 8*a^7*b^8 + 28*a^6* b^9 - 12*a^5*b^10 - 21*a^4*b^11 + 8*a^3*b^12 + 6*a^2*b^13 - 2*a*b^14)*sqrt (a^2 - b^2)*B*abs(-a + b) + (24*a^11*b^4 - 12*a^10*b^5 - 100*a^9*b^6 + 47* a^8*b^7 + 158*a^7*b^8 - 68*a^6*b^9 - 111*a^5*b^10 + 42*a^4*b^11 + 28*a^3*b ^12 - 8*a^2*b^13 + a*b^14 - b^15)*sqrt(a^2 - b^2)*C*abs(-a + b))*(pi*floor (1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 1/2*c)/sqrt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 + sqrt(-16*(a^5*b^4 + a^4*b^5 - 2*a^3*b^6 - 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2))/(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2* a^2*b^7 + a*b^8 - b^9))))/((a^4*b^5 - 2*a^2*b^7 + b^9)^2*(a^2 - 2*a*b + b^ 2) + (a^7*b^4 - 2*a^6*b^5 - a^5*b^6 + 4*a^4*b^7 - a^3*b^8 - 2*a^2*b^9 + a* b^10)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) - (24*C*a^11*b^4 - 12*B*a^10*b^5 - 1 2*C*a^10*b^5 + 4*A*a^9*b^6 + 6*B*a^9*b^6 - 100*C*a^9*b^6 - 2*A*a^8*b^7 ...
Time = 19.14 (sec) , antiderivative size = 16028, normalized size of antiderivative = 35.15 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^5*(3*C*b^7 - 36*C*a^7 - 2*B*b^7 - 6*A*a^2*b^5 + 15*A* a^3*b^4 + 3*A*a^4*b^3 - 6*A*a^5*b^2 + 10*B*a^2*b^5 + 16*B*a^3*b^4 - 35*B*a ^4*b^3 - 9*B*a^5*b^2 + 5*C*a^2*b^5 - 26*C*a^3*b^4 - 29*C*a^4*b^3 + 67*C*a^ 5*b^2 - 4*B*a*b^6 + 18*B*a^6*b + 4*C*a*b^6 + 18*C*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^3*(2*B*b^7 + 36*C*a^7 + 3*C*b^ 7 - 6*A*a^2*b^5 - 15*A*a^3*b^4 + 3*A*a^4*b^3 + 6*A*a^5*b^2 - 10*B*a^2*b^5 + 16*B*a^3*b^4 + 35*B*a^4*b^3 - 9*B*a^5*b^2 + 5*C*a^2*b^5 + 26*C*a^3*b^4 - 29*C*a^4*b^3 - 67*C*a^5*b^2 - 4*B*a*b^6 - 18*B*a^6*b - 4*C*a*b^6 + 18*C*a ^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) + (tan(c/2 + (d*x)/2)^7*(C*b^ 6 - 12*C*a^6 - 2*B*b^6 + 6*A*a^2*b^4 + A*a^3*b^3 - 2*A*a^4*b^2 + 4*B*a^2*b ^4 - 12*B*a^3*b^3 - 3*B*a^4*b^2 - 8*C*a^2*b^4 - 10*C*a^3*b^3 + 23*C*a^4*b^ 2 + 2*B*a*b^5 + 6*B*a^5*b + 5*C*a*b^5 + 6*C*a^5*b))/((a*b^4 - b^5)*(a + b) ^2) + (tan(c/2 + (d*x)/2)*(2*B*b^6 - 12*C*a^6 + C*b^6 + 6*A*a^2*b^4 - A*a^ 3*b^3 - 2*A*a^4*b^2 - 4*B*a^2*b^4 - 12*B*a^3*b^3 + 3*B*a^4*b^2 - 8*C*a^2*b ^4 + 10*C*a^3*b^3 + 23*C*a^4*b^2 + 2*B*a*b^5 + 6*B*a^5*b - 5*C*a*b^5 - 6*C *a^5*b))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)))/(d*(2*a*b + tan(c/2 + (d*x)/ 2)^4*(6*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^2*(4*a*b + 4*a^2) - tan(c/2 + (d *x)/2)^6*(4*a*b - 4*a^2) + tan(c/2 + (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^ 18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A...