Integrand size = 40, antiderivative size = 398 \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (6 a b B-12 a^2 C-b^2 C\right ) x}{2 b^5}+\frac {a^2 \left (6 a^4 b B-15 a^2 b^3 B+12 b^5 B-12 a^5 C+29 a^3 b^2 C-20 a b^4 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-12 a^5 C+21 a^3 b^2 C-6 a b^4 C\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-6 a^4 C+10 a^2 b^2 C-b^4 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (b B-a C) \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
-1/2*(6*B*a*b-12*C*a^2-C*b^2)*x/b^5+a^2*(6*B*a^4*b-15*B*a^2*b^3+12*B*b^5-1 2*C*a^5+29*C*a^3*b^2-20*C*a*b^4)*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^ 5-12*C*a^5+21*C*a^3*b^2-6*C*a*b^4)*sin(d*x+c)/b^4/(a^2-b^2)^2/d-1/2*(3*B*a ^3*b-6*B*a*b^3-6*C*a^4+10*C*a^2*b^2-C*b^4)*cos(d*x+c)*sin(d*x+c)/b^3/(a^2- b^2)^2/d+1/2*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*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 = 5.31 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {16 a^2 \left (-6 a^4 b B+15 a^2 b^3 B-12 b^5 B+12 a^5 C-29 a^3 b^2 C+20 a b^4 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {-48 a^7 b B c+72 a^5 b^3 B c-24 a b^7 B c+96 a^8 c C-136 a^6 b^2 c C-12 a^4 b^4 c C+48 a^2 b^6 c C+4 b^8 c C-48 a^7 b B d x+72 a^5 b^3 B d x-24 a b^7 B d x+96 a^8 C d x-136 a^6 b^2 C d x-12 a^4 b^4 C d x+48 a^2 b^6 C d x+4 b^8 C d x+16 a b \left (a^2-b^2\right )^2 \left (-6 a b B+12 a^2 C+b^2 C\right ) (c+d x) \cos (c+d x)+4 \left (-a^2 b+b^3\right )^2 \left (-6 a b B+12 a^2 C+b^2 C\right ) (c+d x) \cos (2 (c+d x))+48 a^6 b^2 B \sin (c+d x)-84 a^4 b^4 B \sin (c+d x)+8 a^2 b^6 B \sin (c+d x)+4 b^8 B \sin (c+d x)-96 a^7 b C \sin (c+d x)+160 a^5 b^3 C \sin (c+d x)-32 a^3 b^5 C \sin (c+d x)-8 a b^7 C \sin (c+d x)+36 a^5 b^3 B \sin (2 (c+d x))-64 a^3 b^5 B \sin (2 (c+d x))+16 a b^7 B \sin (2 (c+d x))-72 a^6 b^2 C \sin (2 (c+d x))+130 a^4 b^4 C \sin (2 (c+d x))-48 a^2 b^6 C \sin (2 (c+d x))+2 b^8 C \sin (2 (c+d x))+4 a^4 b^4 B \sin (3 (c+d x))-8 a^2 b^6 B \sin (3 (c+d x))+4 b^8 B \sin (3 (c+d x))-8 a^5 b^3 C \sin (3 (c+d x))+16 a^3 b^5 C \sin (3 (c+d x))-8 a b^7 C \sin (3 (c+d x))+a^4 b^4 C \sin (4 (c+d x))-2 a^2 b^6 C \sin (4 (c+d x))+b^8 C \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{16 b^5 d} \]
((16*a^2*(-6*a^4*b*B + 15*a^2*b^3*B - 12*b^5*B + 12*a^5*C - 29*a^3*b^2*C + 20*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (-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 - 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*(-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*(-6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[2*(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] + 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)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*b^5*d)
Time = 2.25 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 3508, 3042, 3468, 25, 3042, 3526, 25, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {\cos ^4(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int -\frac {\cos ^2(c+d x) \left (-2 \left (-2 C a^2+b B a+b^2 C\right ) \cos ^2(c+d x)-2 b (b B-a C) \cos (c+d x)+3 a (b B-a C)\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (-2 C a^2+b B a+b^2 C\right ) \cos ^2(c+d x)-2 b (b B-a C) \cos (c+d x)+3 a (b B-a C)\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-2 \left (-2 C a^2+b B a+b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 b (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a (b B-a C)\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int -\frac {\cos (c+d x) \left (-2 \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right ) \cos ^2(c+d x)+b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \cos (c+d x)+2 a \left (-4 C a^3+2 b B a^2+7 b^2 C a-5 b^3 B\right )\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\cos (c+d x) \left (-2 \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right ) \cos ^2(c+d x)+b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \cos (c+d x)+2 a \left (-4 C a^3+2 b B a^2+7 b^2 C a-5 b^3 B\right )\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-2 \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-4 C a^3+2 b B a^2+7 b^2 C a-5 b^3 B\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \left (-\left (\left (-12 C a^5+6 b B a^4+21 b^2 C a^3-11 b^3 B a^2-6 b^4 C a+2 b^5 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^4+b B a^3+4 b^2 C a^2-4 b^3 B a+b^4 C\right ) \cos (c+d x)+a \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right )\right )}{a+b \cos (c+d x)}dx}{2 b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-\left (\left (-12 C a^5+6 b B a^4+21 b^2 C a^3-11 b^3 B a^2-6 b^4 C a+2 b^5 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^4+b B a^3+4 b^2 C a^2-4 b^3 B a+b^4 C\right ) \cos (c+d x)+a \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right )}{a+b \cos (c+d x)}dx}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\left (12 C a^5-6 b B a^4-21 b^2 C a^3+11 b^3 B a^2+6 b^4 C a-2 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-2 C a^4+b B a^3+4 b^2 C a^2-4 b^3 B a+b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\left (-12 C a^2+6 b B a-b^2 C\right ) \cos (c+d x) \left (a^2-b^2\right )^2+a b \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right )}{a+b \cos (c+d x)}dx}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\left (-12 C a^2+6 b B a-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2+a b \left (-6 C a^4+3 b B a^3+10 b^2 C a^2-6 b^3 B a-b^4 C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (-12 a^2 C+6 a b B-b^2 C\right )}{b}-\frac {a^2 \left (-12 a^5 C+6 a^4 b B+29 a^3 b^2 C-15 a^2 b^3 B-20 a b^4 C+12 b^5 B\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{b}}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (-12 a^2 C+6 a b B-b^2 C\right )}{b}-\frac {a^2 \left (-12 a^5 C+6 a^4 b B+29 a^3 b^2 C-15 a^2 b^3 B-20 a b^4 C+12 b^5 B\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (-12 a^2 C+6 a b B-b^2 C\right )}{b}-\frac {2 a^2 \left (-12 a^5 C+6 a^4 b B+29 a^3 b^2 C-15 a^2 b^3 B-20 a b^4 C+12 b^5 B\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}}{b \left (a^2-b^2\right )}+\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right ) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {-\frac {\left (-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right ) \sin (c+d x) \cos (c+d x)}{b d}-\frac {\frac {\frac {x \left (a^2-b^2\right )^2 \left (-12 a^2 C+6 a b B-b^2 C\right )}{b}-\frac {2 a^2 \left (-12 a^5 C+6 a^4 b B+29 a^3 b^2 C-15 a^2 b^3 B-20 a b^4 C+12 b^5 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}}}{b}-\frac {\left (-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right ) \sin (c+d x)}{b d}}{b}}{b \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}\) |
(a*(b*B - a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[ c + d*x])^2) + ((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 - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*Cos[c + d*x]*Sin[c + d*x])/( b*d)) - ((((a^2 - b^2)^2*(6*a*b*B - 12*a^2*C - b^2*C)*x)/b - (2*a^2*(6*a^4 *b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*C - 20*a*b^4*C)*Arc Tan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b ]*d))/b - ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*Sin[c + d*x])/(b*d))/b)/(b*(a^2 - b^2)))/(2*b*(a^2 - b^2))
3.9.7.3.1 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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((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 - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(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 = 2.11 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a^{2} \left (\frac {\frac {\left (4 B \,a^{2} b -B a \,b^{2}-8 B \,b^{3}-6 C \,a^{3}+C \,a^{2} b +10 C a \,b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}+2 a b +b^{2}\right ) \left (a -b \right )}+\frac {b a \left (4 B \,a^{2} b +B a \,b^{2}-8 B \,b^{3}-6 C \,a^{3}-C \,a^{2} b +10 C a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (6 B \,a^{4} b -15 B \,a^{2} b^{3}+12 B \,b^{5}-12 C \,a^{5}+29 C \,a^{3} b^{2}-20 C a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}-\frac {2 \left (\frac {\left (-B \,b^{2}+3 C a b +\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-B \,b^{2}+3 C a b -\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 B a b -12 a^{2} C -b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}}{d}\) | \(402\) |
| default | \(\frac {\frac {2 a^{2} \left (\frac {\frac {\left (4 B \,a^{2} b -B a \,b^{2}-8 B \,b^{3}-6 C \,a^{3}+C \,a^{2} b +10 C a \,b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}+2 a b +b^{2}\right ) \left (a -b \right )}+\frac {b a \left (4 B \,a^{2} b +B a \,b^{2}-8 B \,b^{3}-6 C \,a^{3}-C \,a^{2} b +10 C a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{2}}+\frac {\left (6 B \,a^{4} b -15 B \,a^{2} b^{3}+12 B \,b^{5}-12 C \,a^{5}+29 C \,a^{3} b^{2}-20 C a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}-\frac {2 \left (\frac {\left (-B \,b^{2}+3 C a b +\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-B \,b^{2}+3 C a b -\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 B a b -12 a^{2} C -b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}}{d}\) | \(402\) |
| risch | \(\text {Expression too large to display}\) | \(1509\) |
1/d*(2*a^2/b^5*((1/2*(4*B*a^2*b-B*a*b^2-8*B*b^3-6*C*a^3+C*a^2*b+10*C*a*b^2 )*a*b/(a^2+2*a*b+b^2)/(a-b)*tan(1/2*d*x+1/2*c)^3+1/2*b*a*(4*B*a^2*b+B*a*b^ 2-8*B*b^3-6*C*a^3-C*a^2*b+10*C*a*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(t an(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c)^2+a+b)^2+1/2*(6*B*a^4*b-15*B*a^ 2*b^3+12*B*b^5-12*C*a^5+29*C*a^3*b^2-20*C*a*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*C*a*b+1/2*b^2*C)*tan(1/2*d*x+1/2*c)^3+(-B*b^2+3*C*a*b-1/2*b^2 *C)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(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. 871 vs. \(2 (378) = 756\).
Time = 0.48 (sec) , antiderivative size = 1812, normalized size of antiderivative = 4.55 \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^3*(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 - 35*C*a^6*b^4 + 18*B*a^5*b^5 + 33*C*a ^4*b^6 - 18*B*a^3*b^7 - 9*C*a^2*b^8 + 6*B*a*b^9 - C*b^10)*d*x*cos(d*x + c) ^2 + 4*(12*C*a^9*b - 6*B*a^8*b^2 - 35*C*a^7*b^3 + 18*B*a^6*b^4 + 33*C*a^5* b^5 - 18*B*a^4*b^6 - 9*C*a^3*b^7 + 6*B*a^2*b^8 - C*a*b^9)*d*x*cos(d*x + c) + 2*(12*C*a^10 - 6*B*a^9*b - 35*C*a^8*b^2 + 18*B*a^7*b^3 + 33*C*a^6*b^4 - 18*B*a^5*b^5 - 9*C*a^4*b^6 + 6*B*a^3*b^7 - C*a^2*b^8)*d*x + (12*C*a^9 - 6 *B*a^8*b - 29*C*a^7*b^2 + 15*B*a^6*b^3 + 20*C*a^5*b^4 - 12*B*a^4*b^5 + (12 *C*a^7*b^2 - 6*B*a^6*b^3 - 29*C*a^5*b^4 + 15*B*a^4*b^5 + 20*C*a^3*b^6 - 12 *B*a^2*b^7)*cos(d*x + c)^2 + 2*(12*C*a^8*b - 6*B*a^7*b^2 - 29*C*a^6*b^3 + 15*B*a^5*b^4 + 20*C*a^4*b^5 - 12*B*a^3*b^6)*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 - 33*C*a^7*b^3 + 17*B*a^6*b^4 + 27*C*a^5*b^5 - 13*B*a^4*b^6 - 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 - 50*C* a^6*b^4 + 25*B*a^5*b^5 + 43*C*a^4*b^6 - 20*B*a^3*b^7 - 11*C*a^2*b^8 + 4*B* a*b^9)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^ 13)*d*cos(d*x + c)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^3*(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. 2712 vs. \(2 (378) = 756\).
Time = 0.63 (sec) , antiderivative size = 2712, normalized size of antiderivative = 6.81 \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
1/2*((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) + 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 - 12*C*a^10*b^5 + 6*B*a^9*b^6 - 100*C*a^9*b^6 + 51*B* a^8*b^7 + 47*C*a^8*b^7 - 24*B*a^7*b^8 + 158*C*a^7*b^8 - 84*B*a^6*b^9 - 68* C*a^6*b^9 + 36*B*a^5*b^10 - 111*C*a^5*b^10 + 63*B*a^4*b^11 + 42*C*a^4*b^11 - 24*B*a^3*b^12 + 28*C*a^3*b^12 - 18*B*a^2*b^13 - 8*C*a^2*b^13 + 6*B*a*b^ 14 + C*a*b^14 - C*b^15 - 12*C*a^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 6*B*...
Time = 14.15 (sec) , antiderivative size = 10598, normalized size of antiderivative = 26.63 \[ \int \frac {\cos ^3(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)*(2*B*b^6 - 12*C*a^6 + C*b^6 - 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)) - (tan(c/2 + (d*x)/2)^3*(2*B*b^7 + 36*C*a^7 + 3*C*b^7 - 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)^5*(3*C*b^7 - 36*C*a^7 - 2*B*b^7 + 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 + 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))/ (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(((((8*tan(c/2 + (d*x)/2)*(288* C^2*a^14 + C^2*b^14 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 36*B^2*a^2*b^12 - 72 *B^2*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432* B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^ 2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C...