Integrand size = 41, antiderivative size = 237 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {(6 A-13 B+23 C) x}{2 a^3}+\frac {4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac {(6 A-13 B+23 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(3 A-8 B+13 C) \cos ^4(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(6 A-13 B+23 C) \cos ^3(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac {4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d} \]
-1/2*(6*A-13*B+23*C)*x/a^3+4/5*(9*A-19*B+34*C)*sin(d*x+c)/a^3/d-1/2*(6*A-1 3*B+23*C)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*(A-B+C)*cos(d*x+c)^5*sin(d*x+c)/ d/(a+a*cos(d*x+c))^3-1/15*(3*A-8*B+13*C)*cos(d*x+c)^4*sin(d*x+c)/a/d/(a+a* cos(d*x+c))^2-1/3*(6*A-13*B+23*C)*cos(d*x+c)^3*sin(d*x+c)/d/(a^3+a^3*cos(d *x+c))-4/15*(9*A-19*B+34*C)*sin(d*x+c)^3/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(663\) vs. \(2(237)=474\).
Time = 4.91 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.80 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-600 (6 A-13 B+23 C) d x \cos \left (\frac {d x}{2}\right )-600 (6 A-13 B+23 C) d x \cos \left (c+\frac {d x}{2}\right )-1800 A d x \cos \left (c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (c+\frac {3 d x}{2}\right )-6900 C d x \cos \left (c+\frac {3 d x}{2}\right )-1800 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-6900 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-1380 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+7020 A \sin \left (\frac {d x}{2}\right )-12760 B \sin \left (\frac {d x}{2}\right )+20410 C \sin \left (\frac {d x}{2}\right )-4500 A \sin \left (c+\frac {d x}{2}\right )+7560 B \sin \left (c+\frac {d x}{2}\right )-11110 C \sin \left (c+\frac {d x}{2}\right )+4860 A \sin \left (c+\frac {3 d x}{2}\right )-9230 B \sin \left (c+\frac {3 d x}{2}\right )+15380 C \sin \left (c+\frac {3 d x}{2}\right )-900 A \sin \left (2 c+\frac {3 d x}{2}\right )+930 B \sin \left (2 c+\frac {3 d x}{2}\right )-380 C \sin \left (2 c+\frac {3 d x}{2}\right )+1452 A \sin \left (2 c+\frac {5 d x}{2}\right )-2782 B \sin \left (2 c+\frac {5 d x}{2}\right )+4777 C \sin \left (2 c+\frac {5 d x}{2}\right )+300 A \sin \left (3 c+\frac {5 d x}{2}\right )-750 B \sin \left (3 c+\frac {5 d x}{2}\right )+1625 C \sin \left (3 c+\frac {5 d x}{2}\right )+60 A \sin \left (3 c+\frac {7 d x}{2}\right )-105 B \sin \left (3 c+\frac {7 d x}{2}\right )+230 C \sin \left (3 c+\frac {7 d x}{2}\right )+60 A \sin \left (4 c+\frac {7 d x}{2}\right )-105 B \sin \left (4 c+\frac {7 d x}{2}\right )+230 C \sin \left (4 c+\frac {7 d x}{2}\right )+15 B \sin \left (4 c+\frac {9 d x}{2}\right )-20 C \sin \left (4 c+\frac {9 d x}{2}\right )+15 B \sin \left (5 c+\frac {9 d x}{2}\right )-20 C \sin \left (5 c+\frac {9 d x}{2}\right )+5 C \sin \left (5 c+\frac {11 d x}{2}\right )+5 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{480 a^3 d (1+\cos (c+d x))^3} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(-600*(6*A - 13*B + 23*C)*d*x*Cos[(d*x)/2] - 60 0*(6*A - 13*B + 23*C)*d*x*Cos[c + (d*x)/2] - 1800*A*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 6900*C*d*x*Cos[c + (3*d*x)/2] - 1800*A* d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 6900*C*d*x*Co s[2*c + (3*d*x)/2] - 360*A*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 1380*C*d*x*Cos[2*c + (5*d*x)/2] - 360*A*d*x*Cos[3*c + (5*d*x) /2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 1380*C*d*x*Cos[3*c + (5*d*x)/2] + 7 020*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] + 20410*C*Sin[(d*x)/2] - 4500*A* Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 11110*C*Sin[c + (d*x)/2] + 48 60*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 15380*C*Sin[c + (3*d *x)/2] - 900*A*Sin[2*c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 380*C*S in[2*c + (3*d*x)/2] + 1452*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c + (5*d* x)/2] + 4777*C*Sin[2*c + (5*d*x)/2] + 300*A*Sin[3*c + (5*d*x)/2] - 750*B*S in[3*c + (5*d*x)/2] + 1625*C*Sin[3*c + (5*d*x)/2] + 60*A*Sin[3*c + (7*d*x) /2] - 105*B*Sin[3*c + (7*d*x)/2] + 230*C*Sin[3*c + (7*d*x)/2] + 60*A*Sin[4 *c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 230*C*Sin[4*c + (7*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] - 20*C*Sin[4*c + (9*d*x)/2] + 15*B*Sin[5*c + ( 9*d*x)/2] - 20*C*Sin[5*c + (9*d*x)/2] + 5*C*Sin[5*c + (11*d*x)/2] + 5*C*Si n[6*c + (11*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)
Time = 1.30 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3520, 3042, 3456, 25, 3042, 3456, 27, 3042, 3227, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int \frac {\cos ^4(c+d x) (5 a (B-C)+a (3 A-3 B+8 C) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (5 a (B-C)+a (3 A-3 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {\frac {\int -\frac {\cos ^3(c+d x) \left (4 a^2 (3 A-8 B+13 C)-3 a^2 (6 A-11 B+21 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos ^3(c+d x) \left (4 a^2 (3 A-8 B+13 C)-3 a^2 (6 A-11 B+21 C) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (4 a^2 (3 A-8 B+13 C)-3 a^2 (6 A-11 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle \frac {-\frac {\frac {\int 3 \cos ^2(c+d x) \left (5 a^3 (6 A-13 B+23 C)-4 a^3 (9 A-19 B+34 C) \cos (c+d x)\right )dx}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {3 \int \cos ^2(c+d x) \left (5 a^3 (6 A-13 B+23 C)-4 a^3 (9 A-19 B+34 C) \cos (c+d x)\right )dx}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {3 \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (5 a^3 (6 A-13 B+23 C)-4 a^3 (9 A-19 B+34 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (5 a^3 (6 A-13 B+23 C) \int \cos ^2(c+d x)dx-4 a^3 (9 A-19 B+34 C) \int \cos ^3(c+d x)dx\right )}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (5 a^3 (6 A-13 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-4 a^3 (9 A-19 B+34 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (\frac {4 a^3 (9 A-19 B+34 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}+5 a^3 (6 A-13 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (5 a^3 (6 A-13 B+23 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {4 a^3 (9 A-19 B+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {-\frac {\frac {3 \left (5 a^3 (6 A-13 B+23 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {4 a^3 (9 A-19 B+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}+\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {\frac {5 a^2 (6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 \left (\frac {4 a^3 (9 A-19 B+34 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}+5 a^3 (6 A-13 B+23 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}}{3 a^2}-\frac {a (3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
-1/5*((A - B + C)*Cos[c + d*x]^5*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^3) + (-1/3*(a*(3*A - 8*B + 13*C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])^2) - ((5*a^2*(6*A - 13*B + 23*C)*Cos[c + d*x]^3*Sin[c + d*x])/(d* (a + a*Cos[c + d*x])) + (3*(5*a^3*(6*A - 13*B + 23*C)*(x/2 + (Cos[c + d*x] *Sin[c + d*x])/(2*d)) + (4*a^3*(9*A - 19*B + 34*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d))/a^2)/(3*a^2))/(5*a^2)
3.4.55.3.1 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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[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[(a*A - b*B + a*C)*Cos[e + f*x]*(a + b* Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x ] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c *(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c ^2 - d^2, 0] && LtQ[m, -2^(-1)]
Time = 2.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.58
| method | result | size |
| parallelrisch | \(\frac {\left (\frac {2 \left (39 A -\frac {232 B}{3}+\frac {427 C}{3}\right ) \cos \left (2 d x +2 c \right )}{5}+\left (A -\frac {3 B}{2}+\frac {43 C}{12}\right ) \cos \left (3 d x +3 c \right )+\frac {\left (B -C \right ) \cos \left (4 d x +4 c \right )}{4}+\frac {C \cos \left (5 d x +5 c \right )}{12}+\frac {\left (243 A -\frac {1001 B}{2}+\frac {2729 C}{3}\right ) \cos \left (d x +c \right )}{5}+\frac {174 A}{5}-\frac {4303 B}{60}+\frac {7783 C}{60}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-48 x d \left (A -\frac {13 B}{6}+\frac {23 C}{6}\right )}{16 a^{3} d}\) | \(138\) |
| derivativedivides | \(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-31 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {A}{2}+\frac {7 B}{4}-\frac {17 C}{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +3 B -\frac {19 C}{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {A}{2}+\frac {5 B}{4}-\frac {11 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-4 \left (6 A -13 B +23 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(234\) |
| default | \(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}-2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+17 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-31 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {16 \left (\left (-\frac {A}{2}+\frac {7 B}{4}-\frac {17 C}{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +3 B -\frac {19 C}{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {A}{2}+\frac {5 B}{4}-\frac {11 C}{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-4 \left (6 A -13 B +23 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(234\) |
| risch | \(-\frac {3 A x}{a^{3}}+\frac {13 B x}{2 a^{3}}-\frac {23 C x}{2 a^{3}}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{3} d}+\frac {2 i \left (90 A \,{\mathrm e}^{4 i \left (d x +c \right )}-150 B \,{\mathrm e}^{4 i \left (d x +c \right )}+225 C \,{\mathrm e}^{4 i \left (d x +c \right )}+300 A \,{\mathrm e}^{3 i \left (d x +c \right )}-525 B \,{\mathrm e}^{3 i \left (d x +c \right )}+810 C \,{\mathrm e}^{3 i \left (d x +c \right )}+420 A \,{\mathrm e}^{2 i \left (d x +c \right )}-745 B \,{\mathrm e}^{2 i \left (d x +c \right )}+1160 C \,{\mathrm e}^{2 i \left (d x +c \right )}+270 A \,{\mathrm e}^{i \left (d x +c \right )}-485 B \,{\mathrm e}^{i \left (d x +c \right )}+760 C \,{\mathrm e}^{i \left (d x +c \right )}+72 A -127 B +197 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} C}{8 a^{3} d}-\frac {i C \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{3} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{3} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{3} d}+\frac {i C \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{3} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} C}{8 a^{3} d}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} C}{8 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{3} d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} C}{8 a^{3} d}\) | \(427\) |
| norman | \(\frac {-\frac {\left (6 A -13 B +23 C \right ) x}{2 a}+\frac {\left (A -B +C \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {\left (4 A -9 B +16 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}-\frac {3 \left (6 A -13 B +23 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {15 \left (6 A -13 B +23 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {10 \left (6 A -13 B +23 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {15 \left (6 A -13 B +23 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {3 \left (6 A -13 B +23 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {\left (6 A -13 B +23 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (6 A -11 B +16 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 a d}+\frac {\left (25 A -51 B +93 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (42 A -89 B +158 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}+\frac {\left (210 A -437 B +786 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {3 \left (326 A -691 B +1236 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 a d}+\frac {\left (387 A -820 B +1465 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}+\frac {\left (808 A -1703 B +3048 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} a^{2}}\) | \(454\) |
1/16*((2/5*(39*A-232/3*B+427/3*C)*cos(2*d*x+2*c)+(A-3/2*B+43/12*C)*cos(3*d *x+3*c)+1/4*(B-C)*cos(4*d*x+4*c)+1/12*C*cos(5*d*x+5*c)+1/5*(243*A-1001/2*B +2729/3*C)*cos(d*x+c)+174/5*A-4303/60*B+7783/60*C)*tan(1/2*d*x+1/2*c)*sec( 1/2*d*x+1/2*c)^4-48*x*d*(A-13/6*B+23/6*C))/a^3/d
Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (6 \, A - 13 \, B + 23 \, C\right )} d x - {\left (10 \, C \cos \left (d x + c\right )^{5} + 15 \, {\left (B - C\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (6 \, A - 9 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (234 \, A - 479 \, B + 869 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (114 \, A - 239 \, B + 429 \, C\right )} \cos \left (d x + c\right ) + 144 \, A - 304 \, B + 544 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="fricas")
-1/30*(15*(6*A - 13*B + 23*C)*d*x*cos(d*x + c)^3 + 45*(6*A - 13*B + 23*C)* d*x*cos(d*x + c)^2 + 45*(6*A - 13*B + 23*C)*d*x*cos(d*x + c) + 15*(6*A - 1 3*B + 23*C)*d*x - (10*C*cos(d*x + c)^5 + 15*(B - C)*cos(d*x + c)^4 + 5*(6* A - 9*B + 19*C)*cos(d*x + c)^3 + (234*A - 479*B + 869*C)*cos(d*x + c)^2 + 3*(114*A - 239*B + 429*C)*cos(d*x + c) + 144*A - 304*B + 544*C)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 2373 vs. \(2 (233) = 466\).
Time = 7.62 (sec) , antiderivative size = 2373, normalized size of antiderivative = 10.01 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\text {Too large to display} \]
Piecewise((-180*A*d*x*tan(c/2 + d*x/2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3 *d) - 540*A*d*x*tan(c/2 + d*x/2)**4/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a **3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 540*A*d*x*tan(c/2 + d*x/2)**2/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d* tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) - 180*A* d*x/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180* a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 3*A*tan(c/2 + d*x/2)**11/(60*a** 3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan( c/2 + d*x/2)**2 + 60*a**3*d) - 21*A*tan(c/2 + d*x/2)**9/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2 )**2 + 60*a**3*d) + 174*A*tan(c/2 + d*x/2)**7/(60*a**3*d*tan(c/2 + d*x/2)* *6 + 180*a**3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60* a**3*d) + 798*A*tan(c/2 + d*x/2)**5/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a **3*d*tan(c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 975*A*tan(c/2 + d*x/2)**3/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan( c/2 + d*x/2)**4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 375*A*tan( c/2 + d*x/2)/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)* *4 + 180*a**3*d*tan(c/2 + d*x/2)**2 + 60*a**3*d) + 390*B*d*x*tan(c/2 + d*x /2)**6/(60*a**3*d*tan(c/2 + d*x/2)**6 + 180*a**3*d*tan(c/2 + d*x/2)**4 ...
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (223) = 446\).
Time = 0.30 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.31 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {C {\left (\frac {20 \, {\left (\frac {33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {50 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {1380 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + 3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="maxima")
1/60*(C*(20*(33*sin(d*x + c)/(cos(d*x + c) + 1) + 76*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 51*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^3 + 3*a^3*sin( d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^ 4 + a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (735*sin(d*x + c)/(cos(d*x + c) + 1) - 50*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos (d*x + c) + 1)^5)/a^3 - 1380*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) - B*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c ) + 1)^3)/(a^3 + 2*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1) - 40*si n(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5) /a^3 - 780*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) + 3*A*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/ (cos(d*x + c) + 1))/a^3))/d
Time = 0.32 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )} {\left (6 \, A - 13 \, B + 23 \, C\right )}}{a^{3}} - \frac {20 \, {\left (6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 51 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 76 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 33 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 50 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^3, x, algorithm="giac")
-1/60*(30*(d*x + c)*(6*A - 13*B + 23*C)/a^3 - 20*(6*A*tan(1/2*d*x + 1/2*c) ^5 - 21*B*tan(1/2*d*x + 1/2*c)^5 + 51*C*tan(1/2*d*x + 1/2*c)^5 + 12*A*tan( 1/2*d*x + 1/2*c)^3 - 36*B*tan(1/2*d*x + 1/2*c)^3 + 76*C*tan(1/2*d*x + 1/2* c)^3 + 6*A*tan(1/2*d*x + 1/2*c) - 15*B*tan(1/2*d*x + 1/2*c) + 33*C*tan(1/2 *d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^3) - (3*A*a^12*tan(1/2*d* x + 1/2*c)^5 - 3*B*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x + 1/ 2*c)^5 - 30*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 40*B*a^12*tan(1/2*d*x + 1/2*c) ^3 - 50*C*a^12*tan(1/2*d*x + 1/2*c)^3 + 255*A*a^12*tan(1/2*d*x + 1/2*c) - 465*B*a^12*tan(1/2*d*x + 1/2*c) + 735*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 1.47 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx=\frac {\left (2\,A-7\,B+17\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A-12\,B+\frac {76\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-5\,B+11\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,A-4\,B+6\,C}{a^3}-\frac {A+5\,B-15\,C}{4\,a^3}+\frac {5\,\left (A-B+C\right )}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,A-4\,B+6\,C}{12\,a^3}+\frac {A-B+C}{3\,a^3}\right )}{d}-\frac {x\,\left (6\,A-13\,B+23\,C\right )}{2\,a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]
(tan(c/2 + (d*x)/2)*(2*A - 5*B + 11*C) + tan(c/2 + (d*x)/2)^5*(2*A - 7*B + 17*C) + tan(c/2 + (d*x)/2)^3*(4*A - 12*B + (76*C)/3))/(d*(3*a^3*tan(c/2 + (d*x)/2)^2 + 3*a^3*tan(c/2 + (d*x)/2)^4 + a^3*tan(c/2 + (d*x)/2)^6 + a^3) ) + (tan(c/2 + (d*x)/2)*((2*A - 4*B + 6*C)/a^3 - (A + 5*B - 15*C)/(4*a^3) + (5*(A - B + C))/(2*a^3)))/d - (tan(c/2 + (d*x)/2)^3*((2*A - 4*B + 6*C)/( 12*a^3) + (A - B + C)/(3*a^3)))/d - (x*(6*A - 13*B + 23*C))/(2*a^3) + (tan (c/2 + (d*x)/2)^5*(A - B + C))/(20*a^3*d)