Integrand size = 41, antiderivative size = 248 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(21 A-8 B+2 C) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {8 (216 A-83 B+20 C) \tan (c+d x)}{105 a^4 d}+\frac {(21 A-8 B+2 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(129 A-52 B+10 C) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {4 (216 A-83 B+20 C) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(2 A-B) \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]
1/2*(21*A-8*B+2*C)*arctanh(sin(d*x+c))/a^4/d-8/105*(216*A-83*B+20*C)*tan(d *x+c)/a^4/d+1/2*(21*A-8*B+2*C)*sec(d*x+c)*tan(d*x+c)/a^4/d-1/105*(129*A-52 *B+10*C)*sec(d*x+c)*tan(d*x+c)/a^4/d/(1+cos(d*x+c))^2-4/105*(216*A-83*B+20 *C)*sec(d*x+c)*tan(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*sec(d*x+c)*tan( d*x+c)/d/(a+a*cos(d*x+c))^4-1/5*(2*A-B)*sec(d*x+c)*tan(d*x+c)/a/d/(a+a*cos (d*x+c))^3
Time = 5.74 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.09 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {13440 (21 A-8 B+2 C) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 \cos \left (\frac {1}{2} (c+d x)\right ) (58161 A-22888 B+5290 C+8 (12813 A-4994 B+1130 C) \cos (c+d x)+60 (1177 A-456 B+106 C) \cos (2 (c+d x))+35928 A \cos (3 (c+d x))-13864 B \cos (3 (c+d x))+3280 C \cos (3 (c+d x))+11619 A \cos (4 (c+d x))-4472 B \cos (4 (c+d x))+1070 C \cos (4 (c+d x))+1728 A \cos (5 (c+d x))-664 B \cos (5 (c+d x))+160 C \cos (5 (c+d x))) \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right )}{1680 a^4 d (1+\cos (c+d x))^4} \]
-1/1680*(13440*(21*A - 8*B + 2*C)*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*Cos[( c + d*x)/2]*(58161*A - 22888*B + 5290*C + 8*(12813*A - 4994*B + 1130*C)*Co s[c + d*x] + 60*(1177*A - 456*B + 106*C)*Cos[2*(c + d*x)] + 35928*A*Cos[3* (c + d*x)] - 13864*B*Cos[3*(c + d*x)] + 3280*C*Cos[3*(c + d*x)] + 11619*A* Cos[4*(c + d*x)] - 4472*B*Cos[4*(c + d*x)] + 1070*C*Cos[4*(c + d*x)] + 172 8*A*Cos[5*(c + d*x)] - 664*B*Cos[5*(c + d*x)] + 160*C*Cos[5*(c + d*x)])*Se c[c + d*x]^2*Sin[(c + d*x)/2])/(a^4*d*(1 + Cos[c + d*x])^4)
Time = 1.80 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 3520, 3042, 3457, 3042, 3457, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3520 |
| \(\displaystyle \frac {\int \frac {(a (9 A-2 B+2 C)-a (5 A-5 B-2 C) \cos (c+d x)) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 A-2 B+2 C)-a (5 A-5 B-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a^2 (73 A-24 B+10 C)-28 a^2 (2 A-B) \cos (c+d x)\right ) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (73 A-24 B+10 C)-28 a^2 (2 A-B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a^3 (477 A-176 B+50 C)-3 a^3 (129 A-52 B+10 C) \cos (c+d x)\right ) \sec ^3(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 (477 A-176 B+50 C)-3 a^3 (129 A-52 B+10 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \left (105 a^4 (21 A-8 B+2 C)-8 a^4 (216 A-83 B+20 C) \cos (c+d x)\right ) \sec ^3(c+d x)dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {105 a^4 (21 A-8 B+2 C)-8 a^4 (216 A-83 B+20 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \int \sec ^3(c+d x)dx-8 a^4 (216 A-83 B+20 C) \int \sec ^2(c+d x)dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-8 a^4 (216 A-83 B+20 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {8 a^4 (216 A-83 B+20 C) \int 1d(-\tan (c+d x))}{d}+105 a^4 (21 A-8 B+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^4 (216 A-83 B+20 C) \tan (c+d x)}{d}}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (216 A-83 B+20 C) \tan (c+d x)}{d}}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (216 A-83 B+20 C) \tan (c+d x)}{d}}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A-8 B+2 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {8 a^4 (216 A-83 B+20 C) \tan (c+d x)}{d}}{a^2}-\frac {4 a^3 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {7 a (2 A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + ((-7*a*(2*A - B)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (-1/3*((129*A - 52*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(d*(1 + Cos[c + d *x])^2) + ((-4*a^3*(216*A - 83*B + 20*C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-8*a^4*(216*A - 83*B + 20*C)*Tan[c + d*x])/d + 105* a^4*(21*A - 8*B + 2*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/a^2)/(3*a^2))/(5*a^2))/(7*a^2)
3.4.71.3.1 Defintions of rubi rules used
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[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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] && (IntegerQ[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)]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 3.19 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.87
| method | result | size |
| parallelrisch | \(\frac {-70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {8 B}{21}+\frac {2 C}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A -\frac {8 B}{21}+\frac {2 C}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-35928 \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (\frac {5885 A}{2994}-\frac {380 B}{499}+\frac {265 C}{1497}\right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {1733 B}{4491}+\frac {410 C}{4491}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {1291 A}{3992}-\frac {559 B}{4491}+\frac {535 C}{17964}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {24 A}{499}-\frac {83 B}{4491}+\frac {20 C}{4491}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {4271 A}{1497}-\frac {4994 B}{4491}+\frac {1130 C}{4491}\right ) \cos \left (d x +c \right )+\frac {19387 A}{11976}-\frac {2861 B}{4491}+\frac {2645 C}{17964}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6720 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(216\) |
| derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-84 A +32 B -8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 A -32 B +8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(294\) |
| default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A +\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-84 A +32 B -8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-36 A +8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (84 A -32 B +8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(294\) |
| norman | \(\frac {-\frac {\left (A -B +C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {\left (9 A -7 B +5 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}-\frac {\left (159 A -71 B +11 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}-\frac {\left (167 A -65 B +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (267 A -155 B +71 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}-\frac {\left (537 A -231 B +65 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}+\frac {\left (1541 A -563 B +145 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}+\frac {\left (2055 A -655 B +151 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} a^{3}}-\frac {\left (21 A -8 B +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4} d}+\frac {\left (21 A -8 B +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4} d}\) | \(323\) |
| risch | \(-\frac {i \left (49980 A \,{\mathrm e}^{8 i \left (d x +c \right )}+320 C +3456 A -1328 B +166668 A \,{\mathrm e}^{4 i \left (d x +c \right )}-64384 B \,{\mathrm e}^{4 i \left (d x +c \right )}-24664 B \,{\mathrm e}^{2 i \left (d x +c \right )}+15160 C \,{\mathrm e}^{4 i \left (d x +c \right )}+5890 C \,{\mathrm e}^{2 i \left (d x +c \right )}-39200 B \,{\mathrm e}^{7 i \left (d x +c \right )}+119364 A \,{\mathrm e}^{3 i \left (d x +c \right )}+183162 A \,{\mathrm e}^{5 i \left (d x +c \right )}-70896 B \,{\mathrm e}^{5 i \left (d x +c \right )}+64053 A \,{\mathrm e}^{2 i \left (d x +c \right )}+21987 A \,{\mathrm e}^{i \left (d x +c \right )}-8456 B \,{\mathrm e}^{i \left (d x +c \right )}-59248 B \,{\mathrm e}^{6 i \left (d x +c \right )}-46032 B \,{\mathrm e}^{3 i \left (d x +c \right )}+155526 A \,{\mathrm e}^{6 i \left (d x +c \right )}+14140 C \,{\mathrm e}^{6 i \left (d x +c \right )}-840 B \,{\mathrm e}^{10 i \left (d x +c \right )}+2205 A \,{\mathrm e}^{10 i \left (d x +c \right )}+102900 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2030 C \,{\mathrm e}^{i \left (d x +c \right )}-19040 B \,{\mathrm e}^{8 i \left (d x +c \right )}+9800 C \,{\mathrm e}^{7 i \left (d x +c \right )}+17220 C \,{\mathrm e}^{5 i \left (d x +c \right )}+10920 C \,{\mathrm e}^{3 i \left (d x +c \right )}+210 C \,{\mathrm e}^{10 i \left (d x +c \right )}+4760 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1470 C \,{\mathrm e}^{9 i \left (d x +c \right )}+15435 A \,{\mathrm e}^{9 i \left (d x +c \right )}-5880 B \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{105 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}-\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{4} d}+\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{4} d}\) | \(538\) |
1/6720*(-70560*(1+cos(2*d*x+2*c))*(A-8/21*B+2/21*C)*ln(tan(1/2*d*x+1/2*c)- 1)+70560*(1+cos(2*d*x+2*c))*(A-8/21*B+2/21*C)*ln(tan(1/2*d*x+1/2*c)+1)-359 28*sec(1/2*d*x+1/2*c)^6*((5885/2994*A-380/499*B+265/1497*C)*cos(2*d*x+2*c) +(A-1733/4491*B+410/4491*C)*cos(3*d*x+3*c)+(1291/3992*A-559/4491*B+535/179 64*C)*cos(4*d*x+4*c)+(24/499*A-83/4491*B+20/4491*C)*cos(5*d*x+5*c)+(4271/1 497*A-4994/4491*B+1130/4491*C)*cos(d*x+c)+19387/11976*A-2861/4491*B+2645/1 7964*C)*tan(1/2*d*x+1/2*c))/d/a^4/(1+cos(2*d*x+2*c))
Time = 0.28 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.62 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A - 8 \, B + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (216 \, A - 83 \, B + 20 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (11619 \, A - 4472 \, B + 1070 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3411 \, A - 1318 \, B + 310 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1509 \, A - 592 \, B + 130 \, C\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right ) - 105 \, A\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^4, x, algorithm="fricas")
1/420*(105*((21*A - 8*B + 2*C)*cos(d*x + c)^6 + 4*(21*A - 8*B + 2*C)*cos(d *x + c)^5 + 6*(21*A - 8*B + 2*C)*cos(d*x + c)^4 + 4*(21*A - 8*B + 2*C)*cos (d*x + c)^3 + (21*A - 8*B + 2*C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 1 05*((21*A - 8*B + 2*C)*cos(d*x + c)^6 + 4*(21*A - 8*B + 2*C)*cos(d*x + c)^ 5 + 6*(21*A - 8*B + 2*C)*cos(d*x + c)^4 + 4*(21*A - 8*B + 2*C)*cos(d*x + c )^3 + (21*A - 8*B + 2*C)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(16*(2 16*A - 83*B + 20*C)*cos(d*x + c)^5 + (11619*A - 4472*B + 1070*C)*cos(d*x + c)^4 + 4*(3411*A - 1318*B + 310*C)*cos(d*x + c)^3 + 4*(1509*A - 592*B + 1 30*C)*cos(d*x + c)^2 + 210*(2*A - B)*cos(d*x + c) - 105*A)*sin(d*x + c))/( a^4*d*cos(d*x + c)^6 + 4*a^4*d*cos(d*x + c)^5 + 6*a^4*d*cos(d*x + c)^4 + 4 *a^4*d*cos(d*x + c)^3 + a^4*d*cos(d*x + c)^2)
Timed out. \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (234) = 468\).
Time = 0.23 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.24 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \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 {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + 5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^4, x, algorithm="maxima")
-1/840*(3*A*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^3/(co s(d*x + c) + 1)^3)/(a^4 - 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4* sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) + 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 2940*log(sin(d* x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 2940*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4) - B*(1680*sin(d*x + c)/((a^4 - a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/(cos(d*x + c) + 1) + 805*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4) + 5*C*((315*sin(d*x + c)/(cos(d*x + c) + 1) + 77*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 168*log(sin(d*x + c)/(cos(d*x + c) + 1 ) + 1)/a^4 + 168*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d
Time = 0.40 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.37 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (21 \, A - 8 \, B + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+a*cos(d*x+c))^4, x, algorithm="giac")
1/840*(420*(21*A - 8*B + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 420 *(21*A - 8*B + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + 840*(9*A*tan( 1/2*d*x + 1/2*c)^3 - 2*B*tan(1/2*d*x + 1/2*c)^3 - 7*A*tan(1/2*d*x + 1/2*c) + 2*B*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^4) - (15*A* a^24*tan(1/2*d*x + 1/2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24 *tan(1/2*d*x + 1/2*c)^7 + 189*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 147*B*a^24*t an(1/2*d*x + 1/2*c)^5 + 105*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 1365*A*a^24*ta n(1/2*d*x + 1/2*c)^3 - 805*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 385*C*a^24*tan( 1/2*d*x + 1/2*c)^3 + 11655*A*a^24*tan(1/2*d*x + 1/2*c) - 5145*B*a^24*tan(1 /2*d*x + 1/2*c) + 1575*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 1.62 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.28 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,A-2\,B\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A-2\,B\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (6\,A-4\,B+2\,C\right )}{4\,a^4}-\frac {3\,\left (5\,B-15\,A+C\right )}{8\,a^4}+\frac {5\,\left (A-B+C\right )}{4\,a^4}+\frac {20\,A-4\,C}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {6\,A-4\,B+2\,C}{40\,a^4}+\frac {3\,\left (A-B+C\right )}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,A-4\,B+2\,C}{8\,a^4}-\frac {5\,B-15\,A+C}{24\,a^4}+\frac {A-B+C}{4\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d}+\frac {2\,\mathrm {atanh}\left (\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,A}{2}-4\,B+C\right )}{21\,A-8\,B+2\,C}\right )\,\left (\frac {21\,A}{2}-4\,B+C\right )}{a^4\,d} \]
(tan(c/2 + (d*x)/2)^3*(9*A - 2*B) - tan(c/2 + (d*x)/2)*(7*A - 2*B))/(d*(a^ 4*tan(c/2 + (d*x)/2)^4 - 2*a^4*tan(c/2 + (d*x)/2)^2 + a^4)) - (tan(c/2 + ( d*x)/2)*((3*(6*A - 4*B + 2*C))/(4*a^4) - (3*(5*B - 15*A + C))/(8*a^4) + (5 *(A - B + C))/(4*a^4) + (20*A - 4*C)/(8*a^4)))/d - (tan(c/2 + (d*x)/2)^5*( (6*A - 4*B + 2*C)/(40*a^4) + (3*(A - B + C))/(40*a^4)))/d - (tan(c/2 + (d* x)/2)^3*((6*A - 4*B + 2*C)/(8*a^4) - (5*B - 15*A + C)/(24*a^4) + (A - B + C)/(4*a^4)))/d - (tan(c/2 + (d*x)/2)^7*(A - B + C))/(56*a^4*d) + (2*atanh( (2*tan(c/2 + (d*x)/2)*((21*A)/2 - 4*B + C))/(21*A - 8*B + 2*C))*((21*A)/2 - 4*B + C))/(a^4*d)