Integrand size = 41, antiderivative size = 174 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {3 (4 A-4 B+5 C) x}{8 a}-\frac {(3 A-4 B+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d} \]
3/8*(4*A-4*B+5*C)*x/a-(3*A-4*B+4*C)*sin(d*x+c)/a/d+3/8*(4*A-4*B+5*C)*cos(d *x+c)*sin(d*x+c)/a/d+1/4*(4*A-4*B+5*C)*cos(d*x+c)^3*sin(d*x+c)/a/d-(A-B+C) *cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))+1/3*(3*A-4*B+4*C)*sin(d*x+c)^3 /a/d
Leaf count is larger than twice the leaf count of optimal. \(393\) vs. \(2(174)=348\).
Time = 2.55 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.26 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (72 (4 A-4 B+5 C) d x \cos \left (\frac {d x}{2}\right )+72 (4 A-4 B+5 C) d x \cos \left (c+\frac {d x}{2}\right )-480 A \sin \left (\frac {d x}{2}\right )+552 B \sin \left (\frac {d x}{2}\right )-552 C \sin \left (\frac {d x}{2}\right )-96 A \sin \left (c+\frac {d x}{2}\right )+168 B \sin \left (c+\frac {d x}{2}\right )-168 C \sin \left (c+\frac {d x}{2}\right )-72 A \sin \left (c+\frac {3 d x}{2}\right )+144 B \sin \left (c+\frac {3 d x}{2}\right )-120 C \sin \left (c+\frac {3 d x}{2}\right )-72 A \sin \left (2 c+\frac {3 d x}{2}\right )+144 B \sin \left (2 c+\frac {3 d x}{2}\right )-120 C \sin \left (2 c+\frac {3 d x}{2}\right )+24 A \sin \left (2 c+\frac {5 d x}{2}\right )-16 B \sin \left (2 c+\frac {5 d x}{2}\right )+40 C \sin \left (2 c+\frac {5 d x}{2}\right )+24 A \sin \left (3 c+\frac {5 d x}{2}\right )-16 B \sin \left (3 c+\frac {5 d x}{2}\right )+40 C \sin \left (3 c+\frac {5 d x}{2}\right )+8 B \sin \left (3 c+\frac {7 d x}{2}\right )-5 C \sin \left (3 c+\frac {7 d x}{2}\right )+8 B \sin \left (4 c+\frac {7 d x}{2}\right )-5 C \sin \left (4 c+\frac {7 d x}{2}\right )+3 C \sin \left (4 c+\frac {9 d x}{2}\right )+3 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{192 a d (1+\cos (c+d x))} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(72*(4*A - 4*B + 5*C)*d*x*Cos[(d*x)/2] + 72*(4* A - 4*B + 5*C)*d*x*Cos[c + (d*x)/2] - 480*A*Sin[(d*x)/2] + 552*B*Sin[(d*x) /2] - 552*C*Sin[(d*x)/2] - 96*A*Sin[c + (d*x)/2] + 168*B*Sin[c + (d*x)/2] - 168*C*Sin[c + (d*x)/2] - 72*A*Sin[c + (3*d*x)/2] + 144*B*Sin[c + (3*d*x) /2] - 120*C*Sin[c + (3*d*x)/2] - 72*A*Sin[2*c + (3*d*x)/2] + 144*B*Sin[2*c + (3*d*x)/2] - 120*C*Sin[2*c + (3*d*x)/2] + 24*A*Sin[2*c + (5*d*x)/2] - 1 6*B*Sin[2*c + (5*d*x)/2] + 40*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d *x)/2] - 16*B*Sin[3*c + (5*d*x)/2] + 40*C*Sin[3*c + (5*d*x)/2] + 8*B*Sin[3 *c + (7*d*x)/2] - 5*C*Sin[3*c + (7*d*x)/2] + 8*B*Sin[4*c + (7*d*x)/2] - 5* C*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 3*C*Sin[5*c + (9*d*x)/ 2]))/(192*a*d*(1 + Cos[c + d*x]))
Time = 0.73 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.83, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.293, Rules used = {3042, 3520, 25, 3042, 3227, 3042, 3113, 2009, 3115, 3042, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a \cos (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int -\cos ^3(c+d x) (a (3 A-4 B+4 C)-a (4 A-4 B+5 C) \cos (c+d x))dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \cos ^3(c+d x) (a (3 A-4 B+4 C)-a (4 A-4 B+5 C) \cos (c+d x))dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a (3 A-4 B+4 C)-a (4 A-4 B+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {a (3 A-4 B+4 C) \int \cos ^3(c+d x)dx-a (4 A-4 B+5 C) \int \cos ^4(c+d x)dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a (3 A-4 B+4 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-a (4 A-4 B+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {-\frac {a (3 A-4 B+4 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-a (4 A-4 B+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-a (4 A-4 B+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {a (3 A-4 B+4 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {-a (4 A-4 B+5 C) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a (3 A-4 B+4 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-a (4 A-4 B+5 C) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a (3 A-4 B+4 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {-a (4 A-4 B+5 C) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {a (3 A-4 B+4 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {-\frac {a (3 A-4 B+4 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-a (4 A-4 B+5 C) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}\) |
-(((A - B + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))) - (- ((a*(3*A - 4*B + 4*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d) - a*(4*A - 4* B + 5*C)*((Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Si n[c + d*x])/(2*d)))/4))/a^2
3.4.39.3.1 Defintions of rubi rules used
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_)*((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 = 2.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.62
| method | result | size |
| parallelrisch | \(\frac {\left (\left (24 A -8 B +38 C \right ) \cos \left (2 d x +2 c \right )+\left (8 B -2 C \right ) \cos \left (3 d x +3 c \right )+3 C \cos \left (4 d x +4 c \right )+\left (-48 A +136 B -82 C \right ) \cos \left (d x +c \right )-168 A +248 B -221 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+144 x \left (A -B +\frac {5 C}{4}\right ) d}{96 a d}\) | \(108\) |
| derivativedivides | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 A}{2}-\frac {25 C}{8}+\frac {5 B}{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {7 A}{2}-\frac {115 C}{24}+\frac {31 B}{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {5 A}{2}-\frac {109 C}{24}+\frac {25 B}{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}-\frac {7 C}{8}+\frac {3 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (4 A -4 B +5 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(170\) |
| default | \(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 A}{2}-\frac {25 C}{8}+\frac {5 B}{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {7 A}{2}-\frac {115 C}{24}+\frac {31 B}{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {5 A}{2}-\frac {109 C}{24}+\frac {25 B}{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}-\frac {7 C}{8}+\frac {3 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (4 A -4 B +5 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\) | \(170\) |
| risch | \(\frac {3 x A}{2 a}-\frac {3 B x}{2 a}+\frac {15 C x}{8 a}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} C}{8 a d}+\frac {2 i B}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B}{8 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a d}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a d}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )} C}{8 a d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B}{8 a d}+\frac {C \sin \left (4 d x +4 c \right )}{32 a d}+\frac {\sin \left (3 d x +3 c \right ) B}{12 a d}-\frac {\sin \left (3 d x +3 c \right ) C}{12 a d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 a d}-\frac {\sin \left (2 d x +2 c \right ) B}{4 a d}+\frac {\sin \left (2 d x +2 c \right ) C}{2 a d}\) | \(314\) |
| norman | \(\frac {\frac {3 \left (4 A -4 B +5 C \right ) x}{8 a}-\frac {\left (A -B +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {15 \left (4 A -4 B +5 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 \left (4 A -4 B +5 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 \left (4 A -4 B +5 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 \left (4 A -4 B +5 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 \left (4 A -4 B +5 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (8 A -16 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (32 A -40 B +45 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {2 \left (33 A -43 B +43 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (66 A -98 B +95 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {\left (120 A -152 B +155 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(337\) |
1/96*(((24*A-8*B+38*C)*cos(2*d*x+2*c)+(8*B-2*C)*cos(3*d*x+3*c)+3*C*cos(4*d *x+4*c)+(-48*A+136*B-82*C)*cos(d*x+c)-168*A+248*B-221*C)*tan(1/2*d*x+1/2*c )+144*x*(A-B+5/4*C)*d)/a/d
Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} d x + {\left (6 \, C \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - C\right )} \cos \left (d x + c\right )^{3} + {\left (12 \, A - 4 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (12 \, A - 28 \, B + 19 \, C\right )} \cos \left (d x + c\right ) - 48 \, A + 64 \, B - 64 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="fricas")
1/24*(9*(4*A - 4*B + 5*C)*d*x*cos(d*x + c) + 9*(4*A - 4*B + 5*C)*d*x + (6* C*cos(d*x + c)^4 + 2*(4*B - C)*cos(d*x + c)^3 + (12*A - 4*B + 13*C)*cos(d* x + c)^2 - (12*A - 28*B + 19*C)*cos(d*x + c) - 48*A + 64*B - 64*C)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)
Leaf count of result is larger than twice the leaf count of optimal. 2688 vs. \(2 (153) = 306\).
Time = 2.62 (sec) , antiderivative size = 2688, normalized size of antiderivative = 15.45 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\text {Too large to display} \]
Piecewise((36*A*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a *d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d* x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/2) **8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*ta n(c/2 + d*x/2)**2 + 24*a*d) + 216*A*d*x*tan(c/2 + d*x/2)**4/(24*a*d*tan(c/ 2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**2/(24* a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d *x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 36*A*d*x/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 24*A*tan(c/2 + d*x/2)**9/(24*a*d*ta n(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)* *4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 168*A*tan(c/2 + d*x/2)**7/(24* a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d *x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 312*A*tan(c/2 + d*x/2)** 5/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c /2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 216*A*tan(c/2 + d* x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d *tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 48*A*tan(c/2 + d*x/2)/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 14...
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (166) = 332\).
Time = 0.30 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.02 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=-\frac {C {\left (\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 4 \, B {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, A {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \]
integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c)),x, algorithm="maxima")
-1/12*(C*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 109*sin(d*x + c)^3/(cos(d* x + c) + 1)^3 + 115*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 75*sin(d*x + c)^ 7/(cos(d*x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*a *sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) - 45*arctan(sin(d*x + c)/(co s(d*x + c) + 1))/a + 12*sin(d*x + c)/(a*(cos(d*x + c) + 1))) - 4*B*((9*sin (d*x + c)/(cos(d*x + c) + 1) + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15 *sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a + 3*a*sin(d*x + c)^2/(cos(d*x + c ) + 1)^2 + 3*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin(d*x + c)^6/(cos (d*x + c) + 1)^6) - 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 3*sin(d* x + c)/(a*(cos(d*x + c) + 1))) + 12*A*((sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a + 2*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c) /(cos(d*x + c) + 1))/a + sin(d*x + c)/(a*(cos(d*x + c) + 1))))/d
Time = 0.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.43 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\frac {9 \, {\left (d x + c\right )} {\left (4 \, A - 4 \, B + 5 \, C\right )}}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 60 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 124 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 115 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 100 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
1/24*(9*(d*x + c)*(4*A - 4*B + 5*C)/a - 24*(A*tan(1/2*d*x + 1/2*c) - B*tan (1/2*d*x + 1/2*c) + C*tan(1/2*d*x + 1/2*c))/a - 2*(36*A*tan(1/2*d*x + 1/2* c)^7 - 60*B*tan(1/2*d*x + 1/2*c)^7 + 75*C*tan(1/2*d*x + 1/2*c)^7 + 84*A*ta n(1/2*d*x + 1/2*c)^5 - 124*B*tan(1/2*d*x + 1/2*c)^5 + 115*C*tan(1/2*d*x + 1/2*c)^5 + 60*A*tan(1/2*d*x + 1/2*c)^3 - 100*B*tan(1/2*d*x + 1/2*c)^3 + 10 9*C*tan(1/2*d*x + 1/2*c)^3 + 12*A*tan(1/2*d*x + 1/2*c) - 36*B*tan(1/2*d*x + 1/2*c) + 21*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/ d
Time = 3.42 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {3\,x\,\left (4\,A-4\,B+5\,C\right )}{8\,a}-\frac {\left (3\,A-5\,B+\frac {25\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (7\,A-\frac {31\,B}{3}+\frac {115\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (5\,A-\frac {25\,B}{3}+\frac {109\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-3\,B+\frac {7\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d} \]
(3*x*(4*A - 4*B + 5*C))/(8*a) - (tan(c/2 + (d*x)/2)*(A - 3*B + (7*C)/4) + tan(c/2 + (d*x)/2)^7*(3*A - 5*B + (25*C)/4) + tan(c/2 + (d*x)/2)^3*(5*A - (25*B)/3 + (109*C)/12) + tan(c/2 + (d*x)/2)^5*(7*A - (31*B)/3 + (115*C)/12 ))/(d*(a + 4*a*tan(c/2 + (d*x)/2)^2 + 6*a*tan(c/2 + (d*x)/2)^4 + 4*a*tan(c /2 + (d*x)/2)^6 + a*tan(c/2 + (d*x)/2)^8)) - (tan(c/2 + (d*x)/2)*(A - B + C))/(a*d)