Integrand size = 33, antiderivative size = 225 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {(23 A+6 C) \text {arctanh}(\sin (c+d x))}{2 a^3 d}+\frac {4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac {(23 A+6 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac {(13 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \cos (c+d x))^2}-\frac {(23 A+6 C) \sec ^2(c+d x) \tan (c+d x)}{3 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d} \]
-1/2*(23*A+6*C)*arctanh(sin(d*x+c))/a^3/d+4/5*(34*A+9*C)*tan(d*x+c)/a^3/d- 1/2*(23*A+6*C)*sec(d*x+c)*tan(d*x+c)/a^3/d-1/5*(A+C)*sec(d*x+c)^2*tan(d*x+ c)/d/(a+a*cos(d*x+c))^3-1/15*(13*A+3*C)*sec(d*x+c)^2*tan(d*x+c)/a/d/(a+a*c os(d*x+c))^2-1/3*(23*A+6*C)*sec(d*x+c)^2*tan(d*x+c)/d/(a^3+a^3*cos(d*x+c)) +4/15*(34*A+9*C)*tan(d*x+c)^3/a^3/d
Leaf count is larger than twice the leaf count of optimal. \(723\) vs. \(2(225)=450\).
Time = 5.78 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.21 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {3840 (23 A+6 C) \cos ^6\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 )+\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (-36 (69 A+49 C) \sin \left (\frac {d x}{2}\right )+2 (6311 A+1686 C) \sin \left (\frac {3 d x}{2}\right )-13340 A \sin \left (c-\frac {d x}{2}\right )-3480 C \sin \left (c-\frac {d x}{2}\right )+4140 A \sin \left (c+\frac {d x}{2}\right )+2100 C \sin \left (c+\frac {d x}{2}\right )-11684 A \sin \left (2 c+\frac {d x}{2}\right )-3144 C \sin \left (2 c+\frac {d x}{2}\right )-450 A \sin \left (c+\frac {3 d x}{2}\right )-960 C \sin \left (c+\frac {3 d x}{2}\right )+5022 A \sin \left (2 c+\frac {3 d x}{2}\right )+2232 C \sin \left (2 c+\frac {3 d x}{2}\right )-8050 A \sin \left (3 c+\frac {3 d x}{2}\right )-2100 C \sin \left (3 c+\frac {3 d x}{2}\right )+9230 A \sin \left (c+\frac {5 d x}{2}\right )+2460 C \sin \left (c+\frac {5 d x}{2}\right )+630 A \sin \left (2 c+\frac {5 d x}{2}\right )-390 C \sin \left (2 c+\frac {5 d x}{2}\right )+4230 A \sin \left (3 c+\frac {5 d x}{2}\right )+1710 C \sin \left (3 c+\frac {5 d x}{2}\right )-4370 A \sin \left (4 c+\frac {5 d x}{2}\right )-1140 C \sin \left (4 c+\frac {5 d x}{2}\right )+5347 A \sin \left (2 c+\frac {7 d x}{2}\right )+1422 C \sin \left (2 c+\frac {7 d x}{2}\right )+875 A \sin \left (3 c+\frac {7 d x}{2}\right )-60 C \sin \left (3 c+\frac {7 d x}{2}\right )+2747 A \sin \left (4 c+\frac {7 d x}{2}\right )+1032 C \sin \left (4 c+\frac {7 d x}{2}\right )-1725 A \sin \left (5 c+\frac {7 d x}{2}\right )-450 C \sin \left (5 c+\frac {7 d x}{2}\right )+2375 A \sin \left (3 c+\frac {9 d x}{2}\right )+630 C \sin \left (3 c+\frac {9 d x}{2}\right )+655 A \sin \left (4 c+\frac {9 d x}{2}\right )+60 C \sin \left (4 c+\frac {9 d x}{2}\right )+1375 A \sin \left (5 c+\frac {9 d x}{2}\right )+480 C \sin \left (5 c+\frac {9 d x}{2}\right )-345 A \sin \left (6 c+\frac {9 d x}{2}\right )-90 C \sin \left (6 c+\frac {9 d x}{2}\right )+544 A \sin \left (4 c+\frac {11 d x}{2}\right )+144 C \sin \left (4 c+\frac {11 d x}{2}\right )+200 A \sin \left (5 c+\frac {11 d x}{2}\right )+30 C \sin \left (5 c+\frac {11 d x}{2}\right )+344 A \sin \left (6 c+\frac {11 d x}{2}\right )+114 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{960 a^3 d (1+\cos (c+d x))^3} \]
(3840*(23*A + 6*C)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x )/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c /2]*Sec[c]*Sec[c + d*x]^3*(-36*(69*A + 49*C)*Sin[(d*x)/2] + 2*(6311*A + 16 86*C)*Sin[(3*d*x)/2] - 13340*A*Sin[c - (d*x)/2] - 3480*C*Sin[c - (d*x)/2] + 4140*A*Sin[c + (d*x)/2] + 2100*C*Sin[c + (d*x)/2] - 11684*A*Sin[2*c + (d *x)/2] - 3144*C*Sin[2*c + (d*x)/2] - 450*A*Sin[c + (3*d*x)/2] - 960*C*Sin[ c + (3*d*x)/2] + 5022*A*Sin[2*c + (3*d*x)/2] + 2232*C*Sin[2*c + (3*d*x)/2] - 8050*A*Sin[3*c + (3*d*x)/2] - 2100*C*Sin[3*c + (3*d*x)/2] + 9230*A*Sin[ c + (5*d*x)/2] + 2460*C*Sin[c + (5*d*x)/2] + 630*A*Sin[2*c + (5*d*x)/2] - 390*C*Sin[2*c + (5*d*x)/2] + 4230*A*Sin[3*c + (5*d*x)/2] + 1710*C*Sin[3*c + (5*d*x)/2] - 4370*A*Sin[4*c + (5*d*x)/2] - 1140*C*Sin[4*c + (5*d*x)/2] + 5347*A*Sin[2*c + (7*d*x)/2] + 1422*C*Sin[2*c + (7*d*x)/2] + 875*A*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] + 2747*A*Sin[4*c + (7*d*x)/2] + 1032*C*Sin[4*c + (7*d*x)/2] - 1725*A*Sin[5*c + (7*d*x)/2] - 450*C*Sin[5*c + (7*d*x)/2] + 2375*A*Sin[3*c + (9*d*x)/2] + 630*C*Sin[3*c + (9*d*x)/2] + 655*A*Sin[4*c + (9*d*x)/2] + 60*C*Sin[4*c + (9*d*x)/2] + 1375*A*Sin[5*c + (9*d*x)/2] + 480*C*Sin[5*c + (9*d*x)/2] - 345*A*Sin[6*c + (9*d*x)/2] - 90* C*Sin[6*c + (9*d*x)/2] + 544*A*Sin[4*c + (11*d*x)/2] + 144*C*Sin[4*c + (11 *d*x)/2] + 200*A*Sin[5*c + (11*d*x)/2] + 30*C*Sin[5*c + (11*d*x)/2] + 344* A*Sin[6*c + (11*d*x)/2] + 114*C*Sin[6*c + (11*d*x)/2]))/(960*a^3*d*(1 +...
Time = 1.38 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3521, 3042, 3457, 3042, 3457, 27, 3042, 3227, 3042, 4254, 2009, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {(a (8 A+3 C)-5 a A \cos (c+d x)) \sec ^4(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (8 A+3 C)-5 a A \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\left (9 a^2 (7 A+2 C)-4 a^2 (13 A+3 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {9 a^2 (7 A+2 C)-4 a^2 (13 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int 3 \left (4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \cos (c+d x)\right ) \sec ^4(c+d x)dx}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \cos (c+d x)\right ) \sec ^4(c+d x)dx}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {4 a^3 (34 A+9 C)-5 a^3 (23 A+6 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A+9 C) \int \sec ^4(c+d x)dx-5 a^3 (23 A+6 C) \int \sec ^3(c+d x)dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (4 a^3 (34 A+9 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx-5 a^3 (23 A+6 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {4 a^3 (34 A+9 C) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}-5 a^3 (23 A+6 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 a^3 (34 A+9 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^3 (34 A+9 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^3 (34 A+9 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 (23 A+6 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 a^3 (34 A+9 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {5 a^2 (23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {a (13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{5 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}\) |
-1/5*((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^3) + (- 1/3*(a*(13*A + 3*C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^2 ) + ((-5*a^2*(23*A + 6*C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d *x])) + (3*(-5*a^3*(23*A + 6*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d* x]*Tan[c + d*x])/(2*d)) - (4*a^3*(34*A + 9*C)*(-Tan[c + d*x] - Tan[c + d*x ]^3/3))/d))/a^2)/(3*a^2))/(5*a^2)
3.1.64.3.1 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[((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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[a*(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)) I nt[(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)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(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, 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 = 2.70 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.95
| method | result | size |
| parallelrisch | \(\frac {8280 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {6 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-8280 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {6 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+1287 \left (\sec ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {3098 A}{1287}+\frac {92 C}{143}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {5528 A}{1287}+\frac {496 C}{429}\right ) \cos \left (2 d x +2 c \right )+\left (A +\frac {38 C}{143}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {272 A}{1287}+\frac {8 C}{143}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {7814 A}{1287}+\frac {236 C}{143}\right ) \cos \left (d x +c \right )+\frac {4321 A}{1287}+\frac {382 C}{429}\right )}{240 d \,a^{3} \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(213\) |
| derivativedivides | \(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (-46 A -12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {34 A +4 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {4 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {8 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {34 A +4 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (46 A +12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {4 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{4 d \,a^{3}}\) | \(238\) |
| default | \(\frac {\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (-46 A -12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {34 A +4 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {4 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {8 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {34 A +4 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (46 A +12 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {4 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {8 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{4 d \,a^{3}}\) | \(238\) |
| norman | \(\frac {\frac {\left (A +C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {\left (47 A +27 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {\left (59 A +15 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {\left (93 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {7 \left (137 A +37 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}-\frac {\left (629 A +149 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 a d}+\frac {\left (679 A +219 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}-\frac {\left (1849 A +429 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} a^{2}}+\frac {\left (23 A +6 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{3} d}-\frac {\left (23 A +6 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{3} d}\) | \(293\) |
| risch | \(\frac {i \left (345 A \,{\mathrm e}^{10 i \left (d x +c \right )}+90 C \,{\mathrm e}^{10 i \left (d x +c \right )}+1725 A \,{\mathrm e}^{9 i \left (d x +c \right )}+450 C \,{\mathrm e}^{9 i \left (d x +c \right )}+4370 A \,{\mathrm e}^{8 i \left (d x +c \right )}+1140 C \,{\mathrm e}^{8 i \left (d x +c \right )}+8050 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2100 C \,{\mathrm e}^{7 i \left (d x +c \right )}+11684 A \,{\mathrm e}^{6 i \left (d x +c \right )}+3144 C \,{\mathrm e}^{6 i \left (d x +c \right )}+13340 A \,{\mathrm e}^{5 i \left (d x +c \right )}+3480 C \,{\mathrm e}^{5 i \left (d x +c \right )}+12622 A \,{\mathrm e}^{4 i \left (d x +c \right )}+3372 C \,{\mathrm e}^{4 i \left (d x +c \right )}+9230 A \,{\mathrm e}^{3 i \left (d x +c \right )}+2460 C \,{\mathrm e}^{3 i \left (d x +c \right )}+5347 A \,{\mathrm e}^{2 i \left (d x +c \right )}+1422 C \,{\mathrm e}^{2 i \left (d x +c \right )}+2375 A \,{\mathrm e}^{i \left (d x +c \right )}+630 C \,{\mathrm e}^{i \left (d x +c \right )}+544 A +144 C \right )}{15 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}-\frac {23 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{3} d}+\frac {23 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{3} d}\) | \(372\) |
1/240*(8280*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*(A+6/23*C)*ln(tan(1/2*d*x+1/2* c)-1)-8280*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*(A+6/23*C)*ln(tan(1/2*d*x+1/2*c )+1)+1287*sec(1/2*d*x+1/2*c)^4*tan(1/2*d*x+1/2*c)*((3098/1287*A+92/143*C)* cos(3*d*x+3*c)+(5528/1287*A+496/429*C)*cos(2*d*x+2*c)+(A+38/143*C)*cos(4*d *x+4*c)+(272/1287*A+8/143*C)*cos(5*d*x+5*c)+(7814/1287*A+236/143*C)*cos(d* x+c)+4321/1287*A+382/429*C))/d/a^3/(cos(3*d*x+3*c)+3*cos(d*x+c))
Time = 0.30 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.36 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {15 \, {\left ({\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \, {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (23 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (16 \, {\left (34 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{5} + 9 \, {\left (143 \, A + 38 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (869 \, A + 234 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (19 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} - 15 \, A \cos \left (d x + c\right ) + 10 \, A\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} + 3 \, a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + a^{3} d \cos \left (d x + c\right )^{3}\right )}} \]
-1/60*(15*((23*A + 6*C)*cos(d*x + c)^6 + 3*(23*A + 6*C)*cos(d*x + c)^5 + 3 *(23*A + 6*C)*cos(d*x + c)^4 + (23*A + 6*C)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 15*((23*A + 6*C)*cos(d*x + c)^6 + 3*(23*A + 6*C)*cos(d*x + c)^5 + 3*(23*A + 6*C)*cos(d*x + c)^4 + (23*A + 6*C)*cos(d*x + c)^3)*log(-sin(d* x + c) + 1) - 2*(16*(34*A + 9*C)*cos(d*x + c)^5 + 9*(143*A + 38*C)*cos(d*x + c)^4 + (869*A + 234*C)*cos(d*x + c)^3 + 5*(19*A + 6*C)*cos(d*x + c)^2 - 15*A*cos(d*x + c) + 10*A)*sin(d*x + c))/(a^3*d*cos(d*x + c)^6 + 3*a^3*d*c os(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + a^3*d*cos(d*x + c)^3)
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.87 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (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 {690 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {690 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 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 {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {60 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \]
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 - 690*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 + 690*log(sin(d*x + c)/(cos(d*x + c) + 1) - 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 - 60*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 + 60*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^3))/d
Time = 0.34 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=-\frac {\frac {30 \, {\left (23 \, A + 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (23 \, A + 6 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac {20 \, {\left (51 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 76 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 50 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 255 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]
-1/60*(30*(23*A + 6*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - 30*(23*A + 6*C)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^3 + 20*(51*A*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 - 12*C*tan (1/2*d*x + 1/2*c)^3 + 33*A*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*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 + 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) + 255*C*a^ 12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 1.05 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{3\,a^3}+\frac {6\,A+2\,C}{12\,a^3}\right )}{d}-\frac {\left (17\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {76\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^3}+\frac {6\,A+2\,C}{a^3}+\frac {15\,A-C}{4\,a^3}\right )}{d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (23\,A+6\,C\right )}{a^3\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A+C\right )}{20\,a^3\,d} \]
(tan(c/2 + (d*x)/2)^3*((A + C)/(3*a^3) + (6*A + 2*C)/(12*a^3)))/d - (tan(c /2 + (d*x)/2)^5*(17*A + 2*C) - tan(c/2 + (d*x)/2)^3*((76*A)/3 + 4*C) + tan (c/2 + (d*x)/2)*(11*A + 2*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)*( (5*(A + C))/(2*a^3) + (6*A + 2*C)/a^3 + (15*A - C)/(4*a^3)))/d - (atanh(ta n(c/2 + (d*x)/2))*(23*A + 6*C))/(a^3*d) + (tan(c/2 + (d*x)/2)^5*(A + C))/( 20*a^3*d)