Integrand size = 33, antiderivative size = 369 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=-\frac {4 a C x}{b^5}-\frac {\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]
-4*a*C*x/b^5-(2*A*b^8-8*a^8*C+28*a^6*b^2*C-35*a^4*b^4*C+a^2*b^6*(3*A+20*C) )*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b) ^(7/2)/d-1/6*(5*A*b^4-(12*a^4-23*a^2*b^2+6*b^4)*C)*sin(d*x+c)/b^4/(a^2-b^2 )^2/d-1/3*(A*b^2+C*a^2)*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x +c))^3+1/6*(3*A*b^4-4*a^4*C+a^2*b^2*(2*A+9*C))*cos(d*x+c)^2*sin(d*x+c)/b^2 /(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/2*a*(2*A*b^6+4*a^6*C-11*a^4*b^2*C+3*a^ 2*b^4*(A+4*C))*sin(d*x+c)/b^4/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(849\) vs. \(2(369)=738\).
Time = 6.18 (sec) , antiderivative size = 849, normalized size of antiderivative = 2.30 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\frac {\frac {24 \left (-2 A b^8+8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-a^2 b^6 (3 A+20 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {-96 a^{10} c C+144 a^8 b^2 c C+144 a^6 b^4 c C-336 a^4 b^6 c C+144 a^2 b^8 c C-96 a^{10} C d x+144 a^8 b^2 C d x+144 a^6 b^4 C d x-336 a^4 b^6 C d x+144 a^2 b^8 C d x-72 a b \left (a^2-b^2\right )^3 \left (4 a^2+b^2\right ) C (c+d x) \cos (c+d x)-144 a^2 b^2 \left (a^2-b^2\right )^3 C (c+d x) \cos (2 (c+d x))-24 a^7 b^3 c C \cos (3 (c+d x))+72 a^5 b^5 c C \cos (3 (c+d x))-72 a^3 b^7 c C \cos (3 (c+d x))+24 a b^9 c C \cos (3 (c+d x))-24 a^7 b^3 C d x \cos (3 (c+d x))+72 a^5 b^5 C d x \cos (3 (c+d x))-72 a^3 b^7 C d x \cos (3 (c+d x))+24 a b^9 C d x \cos (3 (c+d x))+18 a^5 A b^5 \sin (c+d x)+39 a^3 A b^7 \sin (c+d x)+18 a A b^9 \sin (c+d x)+96 a^9 b C \sin (c+d x)-228 a^7 b^3 C \sin (c+d x)+135 a^5 b^5 C \sin (c+d x)+90 a^3 b^7 C \sin (c+d x)-18 a b^9 C \sin (c+d x)+6 a^4 A b^6 \sin (2 (c+d x))+54 a^2 A b^8 \sin (2 (c+d x))+120 a^8 b^2 C \sin (2 (c+d x))-336 a^6 b^4 C \sin (2 (c+d x))+300 a^4 b^6 C \sin (2 (c+d x))-18 a^2 b^8 C \sin (2 (c+d x))-6 b^{10} C \sin (2 (c+d x))+2 a^5 A b^5 \sin (3 (c+d x))-5 a^3 A b^7 \sin (3 (c+d x))+18 a A b^9 \sin (3 (c+d x))+44 a^7 b^3 C \sin (3 (c+d x))-125 a^5 b^5 C \sin (3 (c+d x))+114 a^3 b^7 C \sin (3 (c+d x))-18 a b^9 C \sin (3 (c+d x))+3 a^6 b^4 C \sin (4 (c+d x))-9 a^4 b^6 C \sin (4 (c+d x))+9 a^2 b^8 C \sin (4 (c+d x))-3 b^{10} C \sin (4 (c+d x))}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{24 b^5 d} \]
((24*(-2*A*b^8 + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - a^2*b^6*(3*A + 20 *C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7 /2) + (-96*a^10*c*C + 144*a^8*b^2*c*C + 144*a^6*b^4*c*C - 336*a^4*b^6*c*C + 144*a^2*b^8*c*C - 96*a^10*C*d*x + 144*a^8*b^2*C*d*x + 144*a^6*b^4*C*d*x - 336*a^4*b^6*C*d*x + 144*a^2*b^8*C*d*x - 72*a*b*(a^2 - b^2)^3*(4*a^2 + b^ 2)*C*(c + d*x)*Cos[c + d*x] - 144*a^2*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2* (c + d*x)] - 24*a^7*b^3*c*C*Cos[3*(c + d*x)] + 72*a^5*b^5*c*C*Cos[3*(c + d *x)] - 72*a^3*b^7*c*C*Cos[3*(c + d*x)] + 24*a*b^9*c*C*Cos[3*(c + d*x)] - 2 4*a^7*b^3*C*d*x*Cos[3*(c + d*x)] + 72*a^5*b^5*C*d*x*Cos[3*(c + d*x)] - 72* a^3*b^7*C*d*x*Cos[3*(c + d*x)] + 24*a*b^9*C*d*x*Cos[3*(c + d*x)] + 18*a^5* A*b^5*Sin[c + d*x] + 39*a^3*A*b^7*Sin[c + d*x] + 18*a*A*b^9*Sin[c + d*x] + 96*a^9*b*C*Sin[c + d*x] - 228*a^7*b^3*C*Sin[c + d*x] + 135*a^5*b^5*C*Sin[ c + d*x] + 90*a^3*b^7*C*Sin[c + d*x] - 18*a*b^9*C*Sin[c + d*x] + 6*a^4*A*b ^6*Sin[2*(c + d*x)] + 54*a^2*A*b^8*Sin[2*(c + d*x)] + 120*a^8*b^2*C*Sin[2* (c + d*x)] - 336*a^6*b^4*C*Sin[2*(c + d*x)] + 300*a^4*b^6*C*Sin[2*(c + d*x )] - 18*a^2*b^8*C*Sin[2*(c + d*x)] - 6*b^10*C*Sin[2*(c + d*x)] + 2*a^5*A*b ^5*Sin[3*(c + d*x)] - 5*a^3*A*b^7*Sin[3*(c + d*x)] + 18*a*A*b^9*Sin[3*(c + d*x)] + 44*a^7*b^3*C*Sin[3*(c + d*x)] - 125*a^5*b^5*C*Sin[3*(c + d*x)] + 114*a^3*b^7*C*Sin[3*(c + d*x)] - 18*a*b^9*C*Sin[3*(c + d*x)] + 3*a^6*b^4*C *Sin[4*(c + d*x)] - 9*a^4*b^6*C*Sin[4*(c + d*x)] + 9*a^2*b^8*C*Sin[4*(c...
Time = 2.19 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.15, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3527, 3042, 3526, 3042, 3510, 25, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-\left (\left (4 C a^2+A b^2-3 b^2 C\right ) \cos ^2(c+d x)\right )-3 a b (A+C) \cos (c+d x)+3 \left (C a^2+A b^2\right )\right )}{(a+b \cos (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-4 C a^2-A b^2+3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-3 a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (C a^2+A b^2\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (-\left (\left (5 A b^4-\left (12 a^4-23 b^2 a^2+6 b^4\right ) C\right ) \cos ^2(c+d x)\right )-2 a b \left (-C a^2+5 A b^2+6 b^2 C\right ) \cos (c+d x)+2 \left (-4 C a^4+b^2 (2 A+9 C) a^2+3 A b^4\right )\right )}{(a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (\left (12 a^4-23 b^2 a^2+6 b^4\right ) C-5 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (-C a^2+5 A b^2+6 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (-4 C a^4+b^2 (2 A+9 C) a^2+3 A b^4\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {b \left (a^2-b^2\right ) \left (5 A b^4-\left (12 a^4-23 b^2 a^2+6 b^4\right ) C\right ) \cos ^2(c+d x)+a \left (a^2-b^2\right ) \left (12 C a^4-25 b^2 C a^2+5 A b^4+18 b^4 C\right ) \cos (c+d x)+3 b \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}+\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (5 A b^4-\left (12 a^4-23 b^2 a^2+6 b^4\right ) C\right ) \cos ^2(c+d x)+a \left (a^2-b^2\right ) \left (12 C a^4-25 b^2 C a^2+5 A b^4+18 b^4 C\right ) \cos (c+d x)+3 b \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )}{a+b \cos (c+d x)}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {b \left (a^2-b^2\right ) \left (5 A b^4-\left (12 a^4-23 b^2 a^2+6 b^4\right ) C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (a^2-b^2\right ) \left (12 C a^4-25 b^2 C a^2+5 A b^4+18 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int \frac {3 \left (8 a b C \cos (c+d x) \left (a^2-b^2\right )^3+b^2 \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )\right )}{a+b \cos (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \int \frac {8 a b C \cos (c+d x) \left (a^2-b^2\right )^3+b^2 \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )}{a+b \cos (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \int \frac {8 a b C \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^3+b^2 \left (4 C a^6-11 b^2 C a^4+3 b^4 (A+4 C) a^2+2 A b^6\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx+8 a C x \left (a^2-b^2\right )^3\right )}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx+8 a C x \left (a^2-b^2\right )^3\right )}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {3 \left (\frac {2 \left (-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}+8 a C x \left (a^2-b^2\right )^3\right )}{b}+\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {-\frac {\left (-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 a \left (4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right ) \sin (c+d x)}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^2-b^2\right ) \left (5 A b^4-C \left (12 a^4-23 a^2 b^2+6 b^4\right )\right ) \sin (c+d x)}{d}+\frac {3 \left (8 a C x \left (a^2-b^2\right )^3+\frac {2 \left (-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}\right )}{b}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
-1/3*((A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b *Cos[c + d*x])^3) - (-1/2*((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*a*( 2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(b^2 *(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ((3*(8*a*(a^2 - b^2)^3*C*x + (2*(2* A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcT an[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d )))/b + ((a^2 - b^2)*(5*A*b^4 - (12*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Sin[c + d *x])/d)/(b^2*(a^2 - b^2)))/(2*b*(a^2 - b^2)))/(3*b*(a^2 - b^2))
3.6.86.3.1 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_)])*((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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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 = 3.17 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 C \left (-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{5}}-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{4}+3 A a \,b^{5}+6 A \,b^{6}+6 C \,a^{6}-2 C \,a^{5} b -18 C \,a^{4} b^{2}+5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (A \,a^{2} b^{4}+9 A \,b^{6}+9 C \,a^{6}-29 C \,a^{4} b^{2}+30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{4}-3 A a \,b^{5}+6 A \,b^{6}+6 C \,a^{6}+2 C \,a^{5} b -18 C \,a^{4} b^{2}-5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{3}}+\frac {\left (3 A \,a^{2} b^{6}+2 A \,b^{8}-8 C \,a^{8}+28 C \,a^{6} b^{2}-35 C \,a^{4} b^{4}+20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\) | \(499\) |
| default | \(\frac {-\frac {2 C \left (-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+4 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{b^{5}}-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{4}+3 A a \,b^{5}+6 A \,b^{6}+6 C \,a^{6}-2 C \,a^{5} b -18 C \,a^{4} b^{2}+5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (A \,a^{2} b^{4}+9 A \,b^{6}+9 C \,a^{6}-29 C \,a^{4} b^{2}+30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{4}-3 A a \,b^{5}+6 A \,b^{6}+6 C \,a^{6}+2 C \,a^{5} b -18 C \,a^{4} b^{2}-5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}^{3}}+\frac {\left (3 A \,a^{2} b^{6}+2 A \,b^{8}-8 C \,a^{8}+28 C \,a^{6} b^{2}-35 C \,a^{4} b^{4}+20 C \,a^{2} b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\) | \(499\) |
| risch | \(\text {Expression too large to display}\) | \(1818\) |
1/d*(-2*C/b^5*(-b*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+4*a*arctan(t an(1/2*d*x+1/2*c)))-2/b^5*((-1/2*(2*A*a^2*b^4+3*A*a*b^5+6*A*b^6+6*C*a^6-2* C*a^5*b-18*C*a^4*b^2+5*C*a^3*b^3+20*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a* b^2+b^3)*tan(1/2*d*x+1/2*c)^5-2/3*(A*a^2*b^4+9*A*b^6+9*C*a^6-29*C*a^4*b^2+ 30*C*a^2*b^4)*a*b/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2 *(2*A*a^2*b^4-3*A*a*b^5+6*A*b^6+6*C*a^6+2*C*a^5*b-18*C*a^4*b^2-5*C*a^3*b^3 +20*C*a^2*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(ta n(1/2*d*x+1/2*c)^2*a-b*tan(1/2*d*x+1/2*c)^2+a+b)^3+1/2*(3*A*a^2*b^6+2*A*b^ 8-8*C*a^8+28*C*a^6*b^2-35*C*a^4*b^4+20*C*a^2*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))))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (353) = 706\).
Time = 0.45 (sec) , antiderivative size = 1919, normalized size of antiderivative = 5.20 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
[-1/12*(48*(C*a^9*b^3 - 4*C*a^7*b^5 + 6*C*a^5*b^7 - 4*C*a^3*b^9 + C*a*b^11 )*d*x*cos(d*x + c)^3 + 144*(C*a^10*b^2 - 4*C*a^8*b^4 + 6*C*a^6*b^6 - 4*C*a ^4*b^8 + C*a^2*b^10)*d*x*cos(d*x + c)^2 + 144*(C*a^11*b - 4*C*a^9*b^3 + 6* C*a^7*b^5 - 4*C*a^5*b^7 + C*a^3*b^9)*d*x*cos(d*x + c) + 48*(C*a^12 - 4*C*a ^10*b^2 + 6*C*a^8*b^4 - 4*C*a^6*b^6 + C*a^4*b^8)*d*x + 3*(8*C*a^11 - 28*C* a^9*b^2 + 35*C*a^7*b^4 - (3*A + 20*C)*a^5*b^6 - 2*A*a^3*b^8 + (8*C*a^8*b^3 - 28*C*a^6*b^5 + 35*C*a^4*b^7 - (3*A + 20*C)*a^2*b^9 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 28*C*a^7*b^4 + 35*C*a^5*b^6 - (3*A + 20*C)*a^3*b ^8 - 2*A*a*b^10)*cos(d*x + c)^2 + 3*(8*C*a^10*b - 28*C*a^8*b^3 + 35*C*a^6* b^5 - (3*A + 20*C)*a^4*b^7 - 2*A*a^2*b^9)*cos(d*x + c))*sqrt(-a^2 + b^2)*l og((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*(24*C*a^11*b - 92*C*a^9*b^3 + (4*A + 133*C)*a ^7*b^5 + (7*A - 71*C)*a^5*b^7 - (11*A - 6*C)*a^3*b^9 + 6*(C*a^8*b^4 - 4*C* a^6*b^6 + 6*C*a^4*b^8 - 4*C*a^2*b^10 + C*b^12)*cos(d*x + c)^3 + (44*C*a^9* b^3 + (2*A - 169*C)*a^7*b^5 - (7*A - 239*C)*a^5*b^7 + (23*A - 132*C)*a^3*b ^9 - 18*(A - C)*a*b^11)*cos(d*x + c)^2 + 3*(20*C*a^10*b^2 - 77*C*a^8*b^4 + (A + 110*C)*a^6*b^6 + (8*A - 59*C)*a^4*b^8 - 3*(3*A - 2*C)*a^2*b^10)*cos( d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*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. 846 vs. \(2 (353) = 706\).
Time = 0.40 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.29 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
-1/3*(3*(8*C*a^8 - 28*C*a^6*b^2 + 35*C*a^4*b^4 - 3*A*a^2*b^6 - 20*C*a^2*b^ 6 - 2*A*b^8)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-( a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^6*b ^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*sqrt(a^2 - b^2)) + 12*(d*x + c)*C*a/b^5 - (18*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^ 5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c )^5 + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*C*a^3*b^6*tan(1/2*d*x + 1/2* c)^5 - 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2* c)^5 + 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 - 152*C*a^7*b^2*tan(1/2*d*x + 1/2*c )^3 + 4*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*C*a^5*b^4*tan(1/2*d*x + 1/2 *c)^3 + 32*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 - 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*d*x + 1/2* c) + 42*C*a^8*b*tan(1/2*d*x + 1/2*c) - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c) - 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c) - 2 4*C*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 105* C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c) + 60*C*a ^3*b^6*tan(1/2*d*x + 1/2*c) + 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 18*A*a*b ^8*tan(1/2*d*x + 1/2*c))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*(a*t...
Time = 13.80 (sec) , antiderivative size = 10081, normalized size of antiderivative = 27.32 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^3*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*b^6 - 7*A*a^3*b^5 + 10*A*a^4*b^4 - 72*C*a^2*b^6 - 60*C*a^3*b^5 + 273*C*a^4*b^4 + 47*C*a^5*b^3 - 236*C*a^6*b^2 - 18*A*a*b^7 - 12*C*a^7*b))/(3*b^4*(a + b)^2*(a - b)^3) + (tan(c/2 + (d*x)/2)^5*(72*C*a^8 + 18*C*b^8 + 45*A*a^2*b^6 + 7*A*a^3*b^5 + 10*A*a^4*b^4 - 72*C*a^2*b^6 + 60*C*a^3*b^5 + 273*C*a^4*b^4 - 47*C*a^5*b^3 - 236*C*a^6*b^2 + 18*A*a*b^7 + 12*C*a^7*b))/(3*b^4*(a + b)^3*(a - b)^2) + (tan(c/2 + (d*x)/2)*(8*C*a^7 - 2*C*b^7 - 3*A*a^2*b^5 + 2*A*a^3*b^4 + 6*C* a^2*b^5 + 26*C*a^3*b^4 - 11*C*a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^6 - 2*C*a*b ^6 + 4*C*a^6*b))/(b^4*(a + b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^7*(8*C*a^7 + 2*C*b^7 + 3*A*a^2*b^5 + 2*A*a^3*b^4 - 6*C*a^2*b^5 + 26*C*a^3*b^4 + 11*C* a^4*b^3 - 24*C*a^5*b^2 + 6*A*a*b^6 - 2*C*a*b^6 - 4*C*a^6*b))/(b^4*(a + b)^ 3*(a - b)))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^3) + tan(c/2 + (d*x)/2)^2*(6*a^2*b + 4*a^3 - 2*b^3) + tan(c/2 + (d*x)/2)^6*( 4*a^3 - 6*a^2*b + 2*b^3) + a^3 + b^3 + tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a ^2*b + a^3 - b^3))) - (8*C*a*atan(((4*C*a*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^ 16 + 128*C^2*a^16 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 64 *C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 82 4*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b ^3 - 768*C^2*a^14*b^2 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*...