Integrand size = 41, antiderivative size = 237 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(23 A-13 B+6 C) x}{2 a^3}+\frac {4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac {(23 A-13 B+6 C) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(13 A-8 B+3 C) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(23 A-13 B+6 C) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d} \]
-1/2*(23*A-13*B+6*C)*x/a^3+4/5*(34*A-19*B+9*C)*sin(d*x+c)/a^3/d-1/2*(23*A- 13*B+6*C)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*(A-B+C)*cos(d*x+c)^2*sin(d*x+c)/ d/(a+a*sec(d*x+c))^3-1/15*(13*A-8*B+3*C)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a* sec(d*x+c))^2-1/3*(23*A-13*B+6*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a^3+a^3*sec(d *x+c))-4/15*(34*A-19*B+9*C)*sin(d*x+c)^3/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(655\) vs. \(2(237)=474\).
Time = 5.27 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.76 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (-600 (23 A-13 B+6 C) d x \cos \left (\frac {d x}{2}\right )-600 (23 A-13 B+6 C) d x \cos \left (c+\frac {d x}{2}\right )-6900 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 )-1800 C d x \cos \left (c+\frac {3 d x}{2}\right )-6900 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 )-1800 C d x \cos \left (2 c+\frac {3 d x}{2}\right )-1380 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 )-360 C d x \cos \left (2 c+\frac {5 d x}{2}\right )-1380 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 )-360 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+20410 A \sin \left (\frac {d x}{2}\right )-12760 B \sin \left (\frac {d x}{2}\right )+7020 C \sin \left (\frac {d x}{2}\right )-11110 A \sin \left (c+\frac {d x}{2}\right )+7560 B \sin \left (c+\frac {d x}{2}\right )-4500 C \sin \left (c+\frac {d x}{2}\right )+15380 A \sin \left (c+\frac {3 d x}{2}\right )-9230 B \sin \left (c+\frac {3 d x}{2}\right )+4860 C \sin \left (c+\frac {3 d x}{2}\right )-380 A \sin \left (2 c+\frac {3 d x}{2}\right )+930 B \sin \left (2 c+\frac {3 d x}{2}\right )-900 C \sin \left (2 c+\frac {3 d x}{2}\right )+4777 A \sin \left (2 c+\frac {5 d x}{2}\right )-2782 B \sin \left (2 c+\frac {5 d x}{2}\right )+1452 C \sin \left (2 c+\frac {5 d x}{2}\right )+1625 A \sin \left (3 c+\frac {5 d x}{2}\right )-750 B \sin \left (3 c+\frac {5 d x}{2}\right )+300 C \sin \left (3 c+\frac {5 d x}{2}\right )+230 A \sin \left (3 c+\frac {7 d x}{2}\right )-105 B \sin \left (3 c+\frac {7 d x}{2}\right )+60 C \sin \left (3 c+\frac {7 d x}{2}\right )+230 A \sin \left (4 c+\frac {7 d x}{2}\right )-105 B \sin \left (4 c+\frac {7 d x}{2}\right )+60 C \sin \left (4 c+\frac {7 d x}{2}\right )-20 A \sin \left (4 c+\frac {9 d x}{2}\right )+15 B \sin \left (4 c+\frac {9 d x}{2}\right )-20 A \sin \left (5 c+\frac {9 d x}{2}\right )+15 B \sin \left (5 c+\frac {9 d x}{2}\right )+5 A \sin \left (5 c+\frac {11 d x}{2}\right )+5 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{3840 a^3 d} \]
(Sec[c/2]*Sec[(c + d*x)/2]^5*(-600*(23*A - 13*B + 6*C)*d*x*Cos[(d*x)/2] - 600*(23*A - 13*B + 6*C)*d*x*Cos[c + (d*x)/2] - 6900*A*d*x*Cos[c + (3*d*x)/ 2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 1800*C*d*x*Cos[c + (3*d*x)/2] - 6900* A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 1800*C*d*x* Cos[2*c + (3*d*x)/2] - 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 360*C*d*x*Cos[2*c + (5*d*x)/2] - 1380*A*d*x*Cos[3*c + (5*d *x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 360*C*d*x*Cos[3*c + (5*d*x)/2] + 20410*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] + 7020*C*Sin[(d*x)/2] - 11110 *A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 4500*C*Sin[c + (d*x)/2] + 15380*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 4860*C*Sin[c + (3 *d*x)/2] - 380*A*Sin[2*c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 900*C *Sin[2*c + (3*d*x)/2] + 4777*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c + (5* d*x)/2] + 1452*C*Sin[2*c + (5*d*x)/2] + 1625*A*Sin[3*c + (5*d*x)/2] - 750* B*Sin[3*c + (5*d*x)/2] + 300*C*Sin[3*c + (5*d*x)/2] + 230*A*Sin[3*c + (7*d *x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 60*C*Sin[3*c + (7*d*x)/2] + 230*A*Si n[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 60*C*Sin[4*c + (7*d*x)/2 ] - 20*A*Sin[4*c + (9*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] - 20*A*Sin[5*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2] + 5*A*Sin[5*c + (11*d*x)/2] + 5*A* Sin[6*c + (11*d*x)/2]))/(3840*a^3*d)
Time = 1.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 4572, 3042, 4508, 3042, 4508, 27, 3042, 4274, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4572 |
| \(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (8 A-3 B+3 C)-5 a (A-B) \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (8 A-3 B+3 C)-5 a (A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^3(c+d x) \left (3 a^2 (21 A-11 B+6 C)-4 a^2 (13 A-8 B+3 C) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 a^2 (21 A-11 B+6 C)-4 a^2 (13 A-8 B+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int 3 \cos ^3(c+d x) \left (4 a^3 (34 A-19 B+9 C)-5 a^3 (23 A-13 B+6 C) \sec (c+d x)\right )dx}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \cos ^3(c+d x) \left (4 a^3 (34 A-19 B+9 C)-5 a^3 (23 A-13 B+6 C) \sec (c+d x)\right )dx}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {4 a^3 (34 A-19 B+9 C)-5 a^3 (23 A-13 B+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A-19 B+9 C) \int \cos ^3(c+d x)dx-5 a^3 (23 A-13 B+6 C) \int \cos ^2(c+d x)dx\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A-19 B+9 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-5 a^3 (23 A-13 B+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {4 a^3 (34 A-19 B+9 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-5 a^3 (23 A-13 B+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A-13 B+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 a^3 (34 A-19 B+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A-13 B+6 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {4 a^3 (34 A-19 B+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {4 a^3 (34 A-19 B+9 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-5 a^3 (23 A-13 B+6 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {5 a^2 (23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
-1/5*((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^3) + (-1/3*(a*(13*A - 8*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^2) + ((-5*a^2*(23*A - 13*B + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(d *(a + a*Sec[c + d*x])) + (3*(-5*a^3*(23*A - 13*B + 6*C)*(x/2 + (Cos[c + d* x]*Sin[c + d*x])/(2*d)) - (4*a^3*(34*A - 19*B + 9*C)*(-Sin[c + d*x] + Sin[ c + d*x]^3/3))/d))/a^2)/(3*a^2))/(5*a^2)
3.5.73.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[e + f*x])^n*Simp[b*B*n - a*A*(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B , 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(-(a*A - b*B + a*C))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m - n)))*Csc[ e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {43 \left (\frac {8 \left (427 A -232 B +117 C \right ) \cos \left (2 d x +2 c \right )}{215}+\left (A -\frac {18 B}{43}+\frac {12 C}{43}\right ) \cos \left (3 d x +3 c \right )+\frac {3 \left (-A +B \right ) \cos \left (4 d x +4 c \right )}{43}+\frac {A \cos \left (5 d x +5 c \right )}{43}+\frac {2 \left (5458 A -3003 B +1458 C \right ) \cos \left (d x +c \right )}{215}+\frac {181 A}{5}-\frac {4303 B}{215}+\frac {2088 C}{215}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-2208 \left (A -\frac {13 B}{23}+\frac {6 C}{23}\right ) x d}{192 a^{3} d}\) | \(139\) |
| derivativedivides | \(\frac {-\frac {16 \left (\left (-\frac {17 A}{4}+\frac {7 B}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}+3 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}+\frac {5 B}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A -13 B +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{4 d \,a^{3}}\) | \(234\) |
| default | \(\frac {-\frac {16 \left (\left (-\frac {17 A}{4}+\frac {7 B}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {19 A}{3}+3 B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {11 A}{4}+\frac {5 B}{4}-\frac {C}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-4 \left (23 A -13 B +6 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{4 d \,a^{3}}\) | \(234\) |
| norman | \(\frac {\frac {\left (23 A -13 B +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (23 A -13 B +6 C \right ) x}{2 a}+\frac {\left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{20 a d}-\frac {\left (22 A -17 B +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{30 a d}-\frac {\left (23 A -13 B +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a}-\frac {\left (23 A -13 B +6 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2 a}-\frac {\left (93 A -51 B +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (127 A -77 B +39 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 a d}+\frac {\left (207 A -112 B +52 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 a d}-\frac {\left (228 A -131 B +60 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}+\frac {\left (597 A -377 B +177 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\) | \(329\) |
| risch | \(-\frac {23 A x}{2 a^{3}}+\frac {13 B x}{2 a^{3}}-\frac {3 x C}{a^{3}}+\frac {3 i B \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}-\frac {27 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 a^{3} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{3} d}+\frac {2 i \left (225 A \,{\mathrm e}^{4 i \left (d x +c \right )}-150 B \,{\mathrm e}^{4 i \left (d x +c \right )}+90 C \,{\mathrm e}^{4 i \left (d x +c \right )}+810 A \,{\mathrm e}^{3 i \left (d x +c \right )}-525 B \,{\mathrm e}^{3 i \left (d x +c \right )}+300 C \,{\mathrm e}^{3 i \left (d x +c \right )}+1160 A \,{\mathrm e}^{2 i \left (d x +c \right )}-745 B \,{\mathrm e}^{2 i \left (d x +c \right )}+420 C \,{\mathrm e}^{2 i \left (d x +c \right )}+760 A \,{\mathrm e}^{i \left (d x +c \right )}-485 B \,{\mathrm e}^{i \left (d x +c \right )}+270 C \,{\mathrm e}^{i \left (d x +c \right )}+197 A -127 B +72 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{3} d}+\frac {3 i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{3} d}-\frac {3 i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {27 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{3} d}\) | \(427\) |
1/192*(43*(8/215*(427*A-232*B+117*C)*cos(2*d*x+2*c)+(A-18/43*B+12/43*C)*co s(3*d*x+3*c)+3/43*(-A+B)*cos(4*d*x+4*c)+1/43*A*cos(5*d*x+5*c)+2/215*(5458* A-3003*B+1458*C)*cos(d*x+c)+181/5*A-4303/215*B+2088/215*C)*tan(1/2*d*x+1/2 *c)*sec(1/2*d*x+1/2*c)^4-2208*(A-13/23*B+6/23*C)*x*d)/a^3/d
Time = 0.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (23 \, A - 13 \, B + 6 \, C\right )} d x - {\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \, {\left (A - B\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (19 \, A - 9 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (869 \, A - 479 \, B + 234 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (429 \, A - 239 \, B + 114 \, C\right )} \cos \left (d x + c\right ) + 544 \, A - 304 \, B + 144 \, C\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3, x, algorithm="fricas")
-1/30*(15*(23*A - 13*B + 6*C)*d*x*cos(d*x + c)^3 + 45*(23*A - 13*B + 6*C)* d*x*cos(d*x + c)^2 + 45*(23*A - 13*B + 6*C)*d*x*cos(d*x + c) + 15*(23*A - 13*B + 6*C)*d*x - (10*A*cos(d*x + c)^5 - 15*(A - B)*cos(d*x + c)^4 + 5*(19 *A - 9*B + 6*C)*cos(d*x + c)^3 + (869*A - 479*B + 234*C)*cos(d*x + c)^2 + 3*(429*A - 239*B + 114*C)*cos(d*x + c) + 544*A - 304*B + 144*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)
\[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
(Integral(A*cos(c + d*x)**3/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(B*cos(c + d*x)**3*sec(c + d*x)/(sec(c + d*x)** 3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(C*cos(c + d*x)* *3*sec(c + d*x)**2/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (223) = 446\).
Time = 0.34 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.31 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {A {\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 \, C {\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)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3, x, algorithm="maxima")
1/60*(A*(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*C*(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 = 0.35 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (d x + c\right )} {\left (23 \, A - 13 \, B + 6 \, C\right )}}{a^{3}} - \frac {20 \, {\left (51 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 76 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 465 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
integrate(cos(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3, x, algorithm="giac")
-1/60*(30*(d*x + c)*(23*A - 13*B + 6*C)/a^3 - 20*(51*A*tan(1/2*d*x + 1/2*c )^5 - 21*B*tan(1/2*d*x + 1/2*c)^5 + 6*C*tan(1/2*d*x + 1/2*c)^5 + 76*A*tan( 1/2*d*x + 1/2*c)^3 - 36*B*tan(1/2*d*x + 1/2*c)^3 + 12*C*tan(1/2*d*x + 1/2* c)^3 + 33*A*tan(1/2*d*x + 1/2*c) - 15*B*tan(1/2*d*x + 1/2*c) + 6*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 - 50*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 - 30*C*a^12*tan(1/2*d*x + 1/2*c)^3 + 735*A*a^12*tan(1/2*d*x + 1/2*c) - 465*B*a^12*tan(1/2*d*x + 1/2*c) + 255*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 15.90 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\left (17\,A-7\,B+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {76\,A}{3}-12\,B+4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A-5\,B+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {6\,A-4\,B+2\,C}{a^3}-\frac {5\,B-15\,A+C}{4\,a^3}+\frac {5\,\left (A-B+C\right )}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,A-4\,B+2\,C}{12\,a^3}+\frac {A-B+C}{3\,a^3}\right )}{d}-\frac {x\,\left (23\,A-13\,B+6\,C\right )}{2\,a^3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]
(tan(c/2 + (d*x)/2)*(11*A - 5*B + 2*C) + tan(c/2 + (d*x)/2)^5*(17*A - 7*B + 2*C) + tan(c/2 + (d*x)/2)^3*((76*A)/3 - 12*B + 4*C))/(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)*((6*A - 4*B + 2*C)/a^3 - (5*B - 15*A + C)/(4*a^3) + (5*(A - B + C))/(2*a^3)))/d - (tan(c/2 + (d*x)/2)^3*((6*A - 4*B + 2*C)/( 12*a^3) + (A - B + C)/(3*a^3)))/d - (x*(23*A - 13*B + 6*C))/(2*a^3) + (tan (c/2 + (d*x)/2)^5*(A - B + C))/(20*a^3*d)