Integrand size = 33, antiderivative size = 257 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {2 (11 A+2 C) \text {arctanh}(\sin (c+d x))}{a^4 d}+\frac {4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac {2 (11 A+2 C) \sec (c+d x) \tan (c+d x)}{a^4 d}-\frac {(178 A+31 C) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {4 (11 A+2 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^4 d (1+\cos (c+d x))}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 (8 A+C) \sec ^2(c+d x) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d} \]
-2*(11*A+2*C)*arctanh(sin(d*x+c))/a^4/d+4/35*(454*A+83*C)*tan(d*x+c)/a^4/d -2*(11*A+2*C)*sec(d*x+c)*tan(d*x+c)/a^4/d-1/105*(178*A+31*C)*sec(d*x+c)^2* tan(d*x+c)/a^4/d/(1+cos(d*x+c))^2-4/3*(11*A+2*C)*sec(d*x+c)^2*tan(d*x+c)/a ^4/d/(1+cos(d*x+c))-1/7*(A+C)*sec(d*x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))^4 -2/35*(8*A+C)*sec(d*x+c)^2*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^3+4/105*(454*A+ 83*C)*tan(d*x+c)^3/a^4/d
Time = 4.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.40 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (15 (A+C) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+6 (31 A+17 C) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+4 (412 A+139 C) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+8 (2512 A+559 C) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+15 (A+C) \cos \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+6 (31 A+17 C) \cos ^3\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+4 (412 A+139 C) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {c}{2}\right )+280 \cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (6 (11 A+2 C) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec (c+d x) \left (32 A+3 C-6 A \sec (c+d x)+A \sec ^2(c+d x)\right ) \sin (d x)-6 A \sec (c+d x) \tan (c)+A \sec ^2(c+d x) \tan (c)\right )\right )}{105 a^4 d (1+\cos (c+d x))^4} \]
(2*Cos[(c + d*x)/2]*(15*(A + C)*Sec[c/2]*Sin[(d*x)/2] + 6*(31*A + 17*C)*Co s[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 4*(412*A + 139*C)*Cos[(c + d*x)/2 ]^4*Sec[c/2]*Sin[(d*x)/2] + 8*(2512*A + 559*C)*Cos[(c + d*x)/2]^6*Sec[c/2] *Sin[(d*x)/2] + 15*(A + C)*Cos[(c + d*x)/2]*Tan[c/2] + 6*(31*A + 17*C)*Cos [(c + d*x)/2]^3*Tan[c/2] + 4*(412*A + 139*C)*Cos[(c + d*x)/2]^5*Tan[c/2] + 280*Cos[(c + d*x)/2]^7*(6*(11*A + 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d *x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]* (32*A + 3*C - 6*A*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x] - 6*A*Sec[c + d*x]*Tan[c] + A*Sec[c + d*x]^2*Tan[c])))/(105*a^4*d*(1 + Cos[c + d*x])^4)
Time = 1.78 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3521, 3042, 3457, 3042, 3457, 27, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {(a (10 A+3 C)-a (6 A-C) \cos (c+d x)) \sec ^4(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (10 A+3 C)-a (6 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\left (7 a^2 (14 A+3 C)-10 a^2 (8 A+C) \cos (c+d x)\right ) \sec ^4(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {7 a^2 (14 A+3 C)-10 a^2 (8 A+C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 \left (3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {4 \int \frac {\left (3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \int \frac {3 a^3 (69 A+13 C)-a^3 (178 A+31 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {\int 3 \left (a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \cos (c+d x)\right ) \sec ^4(c+d x)dx}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \int \left (a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \cos (c+d x)\right ) \sec ^4(c+d x)dx}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \int \frac {a^4 (454 A+83 C)-35 a^4 (11 A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (a^4 (454 A+83 C) \int \sec ^4(c+d x)dx-35 a^4 (11 A+2 C) \int \sec ^3(c+d x)dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (a^4 (454 A+83 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx-35 a^4 (11 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-\frac {a^4 (454 A+83 C) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}-35 a^4 (11 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {a^4 (454 A+83 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a^4 (454 A+83 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a^4 (454 A+83 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {4 \left (\frac {3 \left (-35 a^4 (11 A+2 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {a^4 (454 A+83 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}\right )}{3 a^2}-\frac {(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + (( -2*a*(8*A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (-1/3*((178*A + 31*C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(1 + Cos[c + d*x])^ 2) + (4*((-35*a^3*(11*A + 2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[ c + d*x])) + (3*(-35*a^4*(11*A + 2*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[ c + d*x]*Tan[c + d*x])/(2*d)) - (a^4*(454*A + 83*C)*(-Tan[c + d*x] - Tan[c + d*x]^3/3))/d))/a^2))/(3*a^2))/(5*a^2))/(7*a^2)
3.1.73.3.1 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[((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 = 2.58 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.89
| method | result | size |
| parallelrisch | \(\frac {110880 \left (A +\frac {2 C}{11}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-110880 \left (A +\frac {2 C}{11}\right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+19816 \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (\frac {43329 A}{19816}+\frac {3979 C}{9908}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {18263 A}{4954}+\frac {6727 C}{9908}\right ) \cos \left (2 d x +2 c \right )+\left (A +\frac {454 C}{2477}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {6109 A}{19816}+\frac {559 C}{9908}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {227 A}{4954}+\frac {83 C}{9908}\right ) \cos \left (6 d x +6 c \right )+\left (\frac {49441 A}{9908}+\frac {4571 C}{4954}\right ) \cos \left (d x +c \right )+\frac {6803 A}{2477}+\frac {2497 C}{4954}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1680 d \,a^{4} \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(230\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {11 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {59 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {104 A +8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (176 A +32 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {20 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (-176 A -32 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {104 A +8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {8 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {20 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(266\) |
| default | \(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}+\frac {11 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{5}+\frac {59 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}+209 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {104 A +8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (176 A +32 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {20 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (-176 A -32 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {104 A +8 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {8 A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {20 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{8 d \,a^{4}}\) | \(266\) |
| risch | \(\frac {4 i \left (210 C \,{\mathrm e}^{12 i \left (d x +c \right )}+27335 A \,{\mathrm e}^{10 i \left (d x +c \right )}+1470 C \,{\mathrm e}^{11 i \left (d x +c \right )}+61985 A \,{\mathrm e}^{9 i \left (d x +c \right )}+11270 C \,{\mathrm e}^{9 i \left (d x +c \right )}+107646 A \,{\mathrm e}^{8 i \left (d x +c \right )}+19572 C \,{\mathrm e}^{8 i \left (d x +c \right )}+332 C +1816 A +121431 A \,{\mathrm e}^{4 i \left (d x +c \right )}+22262 C \,{\mathrm e}^{4 i \left (d x +c \right )}+6498 C \,{\mathrm e}^{2 i \left (d x +c \right )}+74361 A \,{\mathrm e}^{3 i \left (d x +c \right )}+159306 A \,{\mathrm e}^{5 i \left (d x +c \right )}+35499 A \,{\mathrm e}^{2 i \left (d x +c \right )}+11557 A \,{\mathrm e}^{i \left (d x +c \right )}+169774 A \,{\mathrm e}^{6 i \left (d x +c \right )}+30908 C \,{\mathrm e}^{6 i \left (d x +c \right )}+8085 A \,{\mathrm e}^{11 i \left (d x +c \right )}+4970 C \,{\mathrm e}^{10 i \left (d x +c \right )}+149842 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2114 C \,{\mathrm e}^{i \left (d x +c \right )}+1155 A \,{\mathrm e}^{12 i \left (d x +c \right )}+27524 C \,{\mathrm e}^{7 i \left (d x +c \right )}+29232 C \,{\mathrm e}^{5 i \left (d x +c \right )}+13622 C \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {22 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{4} d}+\frac {22 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{4} d}\) | \(420\) |
1/1680*(110880*(A+2/11*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+1 /2*c)-1)-110880*(A+2/11*C)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+ 1/2*c)+1)+19816*sec(1/2*d*x+1/2*c)^6*((43329/19816*A+3979/9908*C)*cos(3*d* x+3*c)+(18263/4954*A+6727/9908*C)*cos(2*d*x+2*c)+(A+454/2477*C)*cos(4*d*x+ 4*c)+(6109/19816*A+559/9908*C)*cos(5*d*x+5*c)+(227/4954*A+83/9908*C)*cos(6 *d*x+6*c)+(49441/9908*A+4571/4954*C)*cos(d*x+c)+6803/2477*A+2497/4954*C)*t an(1/2*d*x+1/2*c))/d/a^4/(cos(3*d*x+3*c)+3*cos(d*x+c))
Time = 0.27 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.45 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {105 \, {\left ({\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{7} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (11 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (8 \, {\left (454 \, A + 83 \, C\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (6109 \, A + 1118 \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (3592 \, A + 659 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (799 \, A + 148 \, C\right )} \cos \left (d x + c\right )^{3} + 35 \, {\left (14 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} - 70 \, A \cos \left (d x + c\right ) + 35 \, A\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} + 4 \, a^{4} d \cos \left (d x + c\right )^{6} + 6 \, a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + a^{4} d \cos \left (d x + c\right )^{3}\right )}} \]
-1/105*(105*((11*A + 2*C)*cos(d*x + c)^7 + 4*(11*A + 2*C)*cos(d*x + c)^6 + 6*(11*A + 2*C)*cos(d*x + c)^5 + 4*(11*A + 2*C)*cos(d*x + c)^4 + (11*A + 2 *C)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 105*((11*A + 2*C)*cos(d*x + c) ^7 + 4*(11*A + 2*C)*cos(d*x + c)^6 + 6*(11*A + 2*C)*cos(d*x + c)^5 + 4*(11 *A + 2*C)*cos(d*x + c)^4 + (11*A + 2*C)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) - (8*(454*A + 83*C)*cos(d*x + c)^6 + 2*(6109*A + 1118*C)*cos(d*x + c) ^5 + 4*(3592*A + 659*C)*cos(d*x + c)^4 + 8*(799*A + 148*C)*cos(d*x + c)^3 + 35*(14*A + 3*C)*cos(d*x + c)^2 - 70*A*cos(d*x + c) + 35*A)*sin(d*x + c)) /(a^4*d*cos(d*x + c)^7 + 4*a^4*d*cos(d*x + c)^6 + 6*a^4*d*cos(d*x + c)^5 + 4*a^4*d*cos(d*x + c)^4 + a^4*d*cos(d*x + c)^3)
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.79 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} - \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \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 {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {18480 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + C {\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 )}}{840 \, d} \]
1/840*(A*(560*(27*sin(d*x + c)/(cos(d*x + c) + 1) - 62*sin(d*x + c)^3/(cos (d*x + c) + 1)^3 + 39*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^4 - 3*a^4*si n(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1 )^4 - a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (21945*sin(d*x + c)/(cos( d*x + c) + 1) + 2065*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 231*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 - 1 8480*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 18480*log(sin(d*x + c) /(cos(d*x + c) + 1) - 1)/a^4) + C*(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 - 33 60*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 3360*log(sin(d*x + c)/(c os(d*x + c) + 1) - 1)/a^4))/d
Time = 0.32 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.15 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {1680 \, {\left (11 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {1680 \, {\left (11 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {560 \, {\left (39 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 62 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
-1/840*(1680*(11*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 1680*(1 1*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + 560*(39*A*tan(1/2*d*x + 1/2*c)^5 + 3*C*tan(1/2*d*x + 1/2*c)^5 - 62*A*tan(1/2*d*x + 1/2*c)^3 - 6* C*tan(1/2*d*x + 1/2*c)^3 + 27*A*tan(1/2*d*x + 1/2*c) + 3*C*tan(1/2*d*x + 1 /2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a^4) - (15*A*a^24*tan(1/2*d*x + 1/2 *c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 + 231*A*a^24*tan(1/2*d*x + 1/2*c) ^5 + 147*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 2065*A*a^24*tan(1/2*d*x + 1/2*c)^ 3 + 805*C*a^24*tan(1/2*d*x + 1/2*c)^3 + 21945*A*a^24*tan(1/2*d*x + 1/2*c) + 5145*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 1.10 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.18 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{2\,a^4}+\frac {21\,A+C}{2\,a^4}+\frac {5\,\left (7\,A+3\,C\right )}{4\,a^4}+\frac {35\,A-5\,C}{8\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A+C}{10\,a^4}+\frac {7\,A+3\,C}{40\,a^4}\right )}{d}-\frac {\left (26\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {124\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,\left (A+C\right )}{12\,a^4}+\frac {21\,A+C}{24\,a^4}+\frac {7\,A+3\,C}{6\,a^4}\right )}{d}-\frac {4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (11\,A+2\,C\right )}{a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
(tan(c/2 + (d*x)/2)*((5*(A + C))/(2*a^4) + (21*A + C)/(2*a^4) + (5*(7*A + 3*C))/(4*a^4) + (35*A - 5*C)/(8*a^4)))/d + (tan(c/2 + (d*x)/2)^5*((A + C)/ (10*a^4) + (7*A + 3*C)/(40*a^4)))/d - (tan(c/2 + (d*x)/2)^5*(26*A + 2*C) - tan(c/2 + (d*x)/2)^3*((124*A)/3 + 4*C) + tan(c/2 + (d*x)/2)*(18*A + 2*C)) /(d*(3*a^4*tan(c/2 + (d*x)/2)^2 - 3*a^4*tan(c/2 + (d*x)/2)^4 + a^4*tan(c/2 + (d*x)/2)^6 - a^4)) + (tan(c/2 + (d*x)/2)^3*((5*(A + C))/(12*a^4) + (21* A + C)/(24*a^4) + (7*A + 3*C)/(6*a^4)))/d - (4*atanh(tan(c/2 + (d*x)/2))*( 11*A + 2*C))/(a^4*d) + (tan(c/2 + (d*x)/2)^7*(A + C))/(56*a^4*d)