Integrand size = 41, antiderivative size = 349 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {C x}{b^4}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:
C*x/b^4-(3*a^2*b^5*B+2*b^7*B-a^3*b^4*(A-8*C)+2*a^7*C-7*a^5*b^2*C-4*a*b^6*( A+2*C))*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4 /(a+b)^(7/2)/d-1/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b^2) /d/(a+b*cos(d*x+c))^3-1/6*a*(2*A*b^4-5*B*a*b^3-3*a^4*C+a^2*b^2*(3*A+8*C))* sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2-1/6*(4*A*b^6+B*a^3*b^3-16* B*a*b^5+9*a^6*C+2*a^2*b^4*(7*A+17*C)-a^4*b^2*(3*A+28*C))*sin(d*x+c)/b^3/(a ^2-b^2)^3/d/(a+b*cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(863\) vs. \(2(349)=698\).
Time = 6.98 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.47 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {-\frac {24 \left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {24 a^9 c C-36 a^7 b^2 c C-36 a^5 b^4 c C+84 a^3 b^6 c C-36 a b^8 c C+24 a^9 C d x-36 a^7 b^2 C d x-36 a^5 b^4 C d x+84 a^3 b^6 C d x-36 a b^8 C d x+18 b \left (a^2-b^2\right )^3 \left (4 a^2+b^2\right ) C (c+d x) \cos (c+d x)+36 a b^2 \left (a^2-b^2\right )^3 C (c+d x) \cos (2 (c+d x))+6 a^6 b^3 c C \cos (3 (c+d x))-18 a^4 b^5 c C \cos (3 (c+d x))+18 a^2 b^7 c C \cos (3 (c+d x))-6 b^9 c C \cos (3 (c+d x))+6 a^6 b^3 C d x \cos (3 (c+d x))-18 a^4 b^5 C d x \cos (3 (c+d x))+18 a^2 b^7 C d x \cos (3 (c+d x))-6 b^9 C d x \cos (3 (c+d x))-51 a^4 A b^5 \sin (c+d x)-18 a^2 A b^7 \sin (c+d x)-6 A b^9 \sin (c+d x)+18 a^5 b^4 B \sin (c+d x)+39 a^3 b^6 B \sin (c+d x)+18 a b^8 B \sin (c+d x)-24 a^8 b C \sin (c+d x)+57 a^6 b^3 C \sin (c+d x)-72 a^4 b^5 C \sin (c+d x)-36 a^2 b^7 C \sin (c+d x)+6 a^5 A b^4 \sin (2 (c+d x))-54 a^3 A b^6 \sin (2 (c+d x))-12 a A b^8 \sin (2 (c+d x))+6 a^4 b^5 B \sin (2 (c+d x))+54 a^2 b^7 B \sin (2 (c+d x))-30 a^7 b^2 C \sin (2 (c+d x))+90 a^5 b^4 C \sin (2 (c+d x))-120 a^3 b^6 C \sin (2 (c+d x))+a^4 A b^5 \sin (3 (c+d x))-10 a^2 A b^7 \sin (3 (c+d x))-6 A b^9 \sin (3 (c+d x))+2 a^5 b^4 B \sin (3 (c+d x))-5 a^3 b^6 B \sin (3 (c+d x))+18 a b^8 B \sin (3 (c+d x))-11 a^6 b^3 C \sin (3 (c+d x))+32 a^4 b^5 C \sin (3 (c+d x))-36 a^2 b^7 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{24 b^4 d} \] Input:
Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b* Cos[c + d*x])^4,x]
Output:
((-24*(3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/ (-a^2 + b^2)^(7/2) + (24*a^9*c*C - 36*a^7*b^2*c*C - 36*a^5*b^4*c*C + 84*a^ 3*b^6*c*C - 36*a*b^8*c*C + 24*a^9*C*d*x - 36*a^7*b^2*C*d*x - 36*a^5*b^4*C* d*x + 84*a^3*b^6*C*d*x - 36*a*b^8*C*d*x + 18*b*(a^2 - b^2)^3*(4*a^2 + b^2) *C*(c + d*x)*Cos[c + d*x] + 36*a*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2*(c + d*x)] + 6*a^6*b^3*c*C*Cos[3*(c + d*x)] - 18*a^4*b^5*c*C*Cos[3*(c + d*x)] + 18*a^2*b^7*c*C*Cos[3*(c + d*x)] - 6*b^9*c*C*Cos[3*(c + d*x)] + 6*a^6*b^3* C*d*x*Cos[3*(c + d*x)] - 18*a^4*b^5*C*d*x*Cos[3*(c + d*x)] + 18*a^2*b^7*C* d*x*Cos[3*(c + d*x)] - 6*b^9*C*d*x*Cos[3*(c + d*x)] - 51*a^4*A*b^5*Sin[c + d*x] - 18*a^2*A*b^7*Sin[c + d*x] - 6*A*b^9*Sin[c + d*x] + 18*a^5*b^4*B*Si n[c + d*x] + 39*a^3*b^6*B*Sin[c + d*x] + 18*a*b^8*B*Sin[c + d*x] - 24*a^8* b*C*Sin[c + d*x] + 57*a^6*b^3*C*Sin[c + d*x] - 72*a^4*b^5*C*Sin[c + d*x] - 36*a^2*b^7*C*Sin[c + d*x] + 6*a^5*A*b^4*Sin[2*(c + d*x)] - 54*a^3*A*b^6*S in[2*(c + d*x)] - 12*a*A*b^8*Sin[2*(c + d*x)] + 6*a^4*b^5*B*Sin[2*(c + d*x )] + 54*a^2*b^7*B*Sin[2*(c + d*x)] - 30*a^7*b^2*C*Sin[2*(c + d*x)] + 90*a^ 5*b^4*C*Sin[2*(c + d*x)] - 120*a^3*b^6*C*Sin[2*(c + d*x)] + a^4*A*b^5*Sin[ 3*(c + d*x)] - 10*a^2*A*b^7*Sin[3*(c + d*x)] - 6*A*b^9*Sin[3*(c + d*x)] + 2*a^5*b^4*B*Sin[3*(c + d*x)] - 5*a^3*b^6*B*Sin[3*(c + d*x)] + 18*a*b^8*B*S in[3*(c + d*x)] - 11*a^6*b^3*C*Sin[3*(c + d*x)] + 32*a^4*b^5*C*Sin[3*(c...
Time = 1.82 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.16, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {3042, 3526, 3042, 3510, 25, 3042, 3500, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {\cos (c+d x) \left (-3 \left (a^2-b^2\right ) C \cos ^2(c+d x)+3 b (b B-a (A+C)) \cos (c+d x)+2 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (a^2-b^2\right ) C \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {\frac {\int -\frac {6 b \left (a^2-b^2\right )^2 C \cos ^2(c+d x)-\left (3 C a^5-b^2 (3 A+10 C) a^3+b^3 B a^2+4 b^4 (2 A+3 C) a-6 b^5 B\right ) \cos (c+d x)+2 b \left (-3 C a^4+b^2 (3 A+8 C) a^2-5 b^3 B a+2 A b^4\right )}{(a+b \cos (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}+\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {6 b \left (a^2-b^2\right )^2 C \cos ^2(c+d x)-\left (3 C a^5-b^2 (3 A+10 C) a^3+b^3 B a^2+4 b^4 (2 A+3 C) a-6 b^5 B\right ) \cos (c+d x)+2 b \left (-3 C a^4+b^2 (3 A+8 C) a^2-5 b^3 B a+2 A b^4\right )}{(a+b \cos (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {6 b \left (a^2-b^2\right )^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-3 C a^5+b^2 (3 A+10 C) a^3-b^3 B a^2-4 b^4 (2 A+3 C) a+6 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left (-3 C a^4+b^2 (3 A+8 C) a^2-5 b^3 B a+2 A b^4\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\int \frac {3 \left (2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \cos (c+d x) b\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \int \frac {2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \cos (c+d x) b}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \int \frac {2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \sin \left (c+d x+\frac {\pi }{2}\right ) b}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \left (\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{a+b \cos (c+d x)}dx-2 C x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \left (\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-2 C x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \left (\frac {2 \left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}-2 C x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {a \sin (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {3 \left (\frac {2 \left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-2 C x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]
Output:
-1/3*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Cos[c + d*x])^3) - ((a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*( 3*A + 8*C))*Sin[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ( (-3*(-2*(a^2 - b^2)^3*C*x + (2*(3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)))/(b*(a^2 - b^2)) - ((4 *A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b ^2*(3*A + 28*C))*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])))/(2*b^ 2*(a^2 - b^2)))/(3*b*(a^2 - b^2))
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 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]
Time = 1.35 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A a \,b^{5}+2 A \,b^{6}-2 B \,a^{3} b^{3}-3 B \,a^{2} b^{4}-6 B a \,b^{5}+2 a^{6} C -C \,a^{5} b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-B \,a^{3} b^{3}-9 B a \,b^{5}+3 a^{6} C -11 a^{4} b^{2} C +18 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A a \,b^{5}-2 A \,b^{6}+2 B \,a^{3} b^{3}-3 B \,a^{2} b^{4}+6 B a \,b^{5}-2 a^{6} C -C \,a^{5} b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}+\frac {2 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(561\) |
| default | \(\frac {\frac {\frac {2 \left (-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A a \,b^{5}+2 A \,b^{6}-2 B \,a^{3} b^{3}-3 B \,a^{2} b^{4}-6 B a \,b^{5}+2 a^{6} C -C \,a^{5} b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-B \,a^{3} b^{3}-9 B a \,b^{5}+3 a^{6} C -11 a^{4} b^{2} C +18 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A a \,b^{5}-2 A \,b^{6}+2 B \,a^{3} b^{3}-3 B \,a^{2} b^{4}+6 B a \,b^{5}-2 a^{6} C -C \,a^{5} b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 a^{2} C \,b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{b^{4}}+\frac {2 C \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(561\) |
| risch | \(\text {Expression too large to display}\) | \(2421\) |
Input:
int(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/d*(2/b^4*((-1/2*(A*a^3*b^3+6*A*a^2*b^4+2*A*a*b^5+2*A*b^6-2*B*a^3*b^3-3*B *a^2*b^4-6*B*a*b^5+2*C*a^6-C*a^5*b-6*C*a^4*b^2+4*C*a^3*b^3+12*C*a^2*b^4)*b /(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(7*A*a^2*b^4+3*A *b^6-B*a^3*b^3-9*B*a*b^5+3*C*a^6-11*C*a^4*b^2+18*C*a^2*b^4)*b/(a^2+2*a*b+b ^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*(A*a^3*b^3-6*A*a^2*b^4+2*A*a* b^5-2*A*b^6+2*B*a^3*b^3-3*B*a^2*b^4+6*B*a*b^5-2*C*a^6-C*a^5*b+6*C*a^4*b^2+ 4*C*a^3*b^3-12*C*a^2*b^4)*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*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)^3+1/2*(A*a^3*b^4 +4*A*a*b^6-3*B*a^2*b^5-2*B*b^7-2*C*a^7+7*C*a^5*b^2-8*C*a^3*b^4+8*C*a*b^6)/ (a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x +1/2*c)/((a-b)*(a+b))^(1/2)))+2*C/b^4*arctan(tan(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 988 vs. \(2 (334) = 668\).
Time = 0.27 (sec) , antiderivative size = 2045, normalized size of antiderivative = 5.86 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="fricas")
Output:
[1/12*(12*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11)*d *x*cos(d*x + c)^3 + 36*(C*a^9*b^2 - 4*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C*a^3*b^ 8 + C*a*b^10)*d*x*cos(d*x + c)^2 + 36*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^ 5 - 4*C*a^4*b^7 + C*a^2*b^9)*d*x*cos(d*x + c) + 12*(C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*d*x + 3*(2*C*a^10 - 7*C*a^8*b^2 - (A - 8*C)*a^6*b^4 + 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7 + (2*C *a^7*b^3 - 7*C*a^5*b^5 - (A - 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4*(A + 2*C)*a*b ^9 + 2*B*b^10)*cos(d*x + c)^3 + 3*(2*C*a^8*b^2 - 7*C*a^6*b^4 - (A - 8*C)*a ^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*a*b^9)*cos(d*x + c)^2 + 3 *(2*C*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)* a^3*b^7 + 2*B*a^2*b^8)*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*(6*C*a^10*b - 23*C*a^8*b^3 - 4*B*a^7*b^4 + (13*A + 43*C)*a^6*b^5 - 7*B*a^5*b^6 - (11*A + 26*C)*a^4*b^7 + 11*B*a^3*b^8 - 2*A*a^2*b^9 + (11* C*a^8*b^3 - 2*B*a^7*b^4 - (A + 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C) *a^4*b^7 - 23*B*a^3*b^8 - 4*(A + 9*C)*a^2*b^9 + 18*B*a*b^10 - 6*A*b^11)*co s(d*x + c)^2 + 3*(5*C*a^9*b^2 - (A + 20*C)*a^7*b^4 - B*a^6*b^5 + 5*(2*A + 7*C)*a^5*b^6 - 8*B*a^4*b^7 - (7*A + 20*C)*a^3*b^8 + 9*B*a^2*b^9 - 2*A*a*b^ 10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*...
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))* *4,x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="maxima")
Output:
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. 1104 vs. \(2 (334) = 668\).
Time = 0.20 (sec) , antiderivative size = 1104, normalized size of antiderivative = 3.16 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4, x, algorithm="giac")
Output:
1/3*(3*(2*C*a^7 - 7*C*a^5*b^2 - A*a^3*b^4 + 8*C*a^3*b^4 + 3*B*a^2*b^5 - 4* A*a*b^6 - 8*C*a*b^6 + 2*B*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^ 2 - b^2)))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) + 3* (d*x + c)*C/b^4 - (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 45*C*a^5*b^3*tan(1/2*d* x + 1/2*c)^5 + 12*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 + 3*B*a^4*b^4*tan(1/2*d *x + 1/2*c)^5 - 6*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^3*b^5*tan(1/2* d*x + 1/2*c)^5 - 6*B*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^3*b^5*tan(1/2 *d*x + 1/2*c)^5 + 12*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^2*b^6*tan(1 /2*d*x + 1/2*c)^5 + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*A*a*b^7*tan(1/ 2*d*x + 1/2*c)^5 - 18*B*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^8*tan(1/2*d*x + 1/2*c)^5 + 12*C*a^8*tan(1/2*d*x + 1/2*c)^3 - 56*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 + 28*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 32*B*a^3*b^5*tan(1/2*d *x + 1/2*c)^3 - 16*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 - 72*C*a^2*b^6*tan(1/2 *d*x + 1/2*c)^3 + 36*B*a*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^8*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^8*tan(1/2*d*x + 1/2*c) + 15*C*a^7*b*tan(1/2*d*x + 1/2* c) - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c) - 3*A*a^5*b^3*tan(1/2*d*x + 1/2*c...
Time = 8.56 (sec) , antiderivative size = 11947, normalized size of antiderivative = 34.23 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^4,x)
Output:
(2*C*atan(((C*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2*a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^ 6*b^8 + 12*B^2*a^2*b^12 + 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^ 11 - 92*C^2*a^4*b^10 - 120*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A* B*a^5*b^9 + 64*A*C*a^2*b^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8 *b^6 - 4*A*C*a^10*b^4 - 16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*a^9*b^5))/(a*b^16 + b^17 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10*a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b ^7 - a^11*b^6) + (C*((8*(4*B*b^21 + 4*C*b^21 + 8*A*a^2*b^19 + 22*A*a^3*b^1 8 - 22*A*a^4*b^17 - 18*A*a^5*b^16 + 18*A*a^6*b^15 + 2*A*a^7*b^14 - 2*A*a^8 *b^13 + 2*A*a^9*b^12 - 2*A*a^10*b^11 - 6*B*a^2*b^19 + 6*B*a^3*b^18 - 6*B*a ^4*b^17 + 6*B*a^5*b^16 + 14*B*a^6*b^15 - 14*B*a^7*b^14 - 6*B*a^8*b^13 + 6* B*a^9*b^12 - 12*C*a^2*b^19 + 64*C*a^3*b^18 + 20*C*a^4*b^17 - 110*C*a^5*b^1 6 - 30*C*a^6*b^15 + 110*C*a^7*b^14 + 30*C*a^8*b^13 - 70*C*a^9*b^12 - 14*C* a^10*b^11 + 26*C*a^11*b^10 + 2*C*a^12*b^9 - 4*C*a^13*b^8 - 8*A*a*b^20 - 4* B*a*b^20 - 16*C*a*b^20))/(a*b^19 + b^20 - 5*a^2*b^18 - 5*a^3*b^17 + 10*a^4 *b^16 + 10*a^5*b^15 - 10*a^6*b^14 - 10*a^7*b^13 + 5*a^8*b^12 + 5*a^9*b^11 - a^10*b^10 - a^11*b^9) - (C*tan(c/2 + (d*x)/2)*(8*a*b^21 - 8*a^2*b^20 ...
Time = 0.17 (sec) , antiderivative size = 3079, normalized size of antiderivative = 8.82 \[ \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x)
Output:
( - 12*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**7*b**3*c + 42*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*c os(c + d*x)*sin(c + d*x)**2*a**5*b**5*c + 6*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**7 - 48*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan(( c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**3*b**7*c + 6*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt (a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**9 + 48*sqrt(a**2 - b** 2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a*b**9*c - 12*sqrt(a**2 - b**2)*atan((tan((c + d*x )/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)* *2*b**11 + 36*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2 )*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**9*b*c - 114*sqrt(a**2 - b**2)*atan ((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x) *a**7*b**3*c - 18*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d* x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**6*b**5 + 102*sqrt(a**2 - b**2) *atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**5*b**5*c - 24*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**4*b**7 - 96*sqrt(a**2 ...