Integrand size = 33, antiderivative size = 224 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(21 A+2 C) \text {arctanh}(\sin (c+d x))}{2 a^4 d}-\frac {32 (54 A+5 C) \tan (c+d x)}{105 a^4 d}+\frac {(21 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a^4 d}-\frac {(129 A+10 C) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {16 (54 A+5 C) \sec (c+d x) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))}-\frac {(A+C) \sec (c+d x) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {2 A \sec (c+d x) \tan (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]
1/2*(21*A+2*C)*arctanh(sin(d*x+c))/a^4/d-32/105*(54*A+5*C)*tan(d*x+c)/a^4/ d+1/2*(21*A+2*C)*sec(d*x+c)*tan(d*x+c)/a^4/d-1/105*(129*A+10*C)*sec(d*x+c) *tan(d*x+c)/a^4/d/(1+cos(d*x+c))^2-16/105*(54*A+5*C)*sec(d*x+c)*tan(d*x+c) /a^4/d/(1+cos(d*x+c))-1/7*(A+C)*sec(d*x+c)*tan(d*x+c)/d/(a+a*cos(d*x+c))^4 -2/5*A*sec(d*x+c)*tan(d*x+c)/a/d/(a+a*cos(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(784\) vs. \(2(224)=448\).
Time = 8.74 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.50 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {8 (21 A+2 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^4}+\frac {8 (21 A+2 C) \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^4}+\frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (73206 A \sin \left (\frac {d x}{2}\right )+14140 C \sin \left (\frac {d x}{2}\right )-166668 A \sin \left (\frac {3 d x}{2}\right )-15160 C \sin \left (\frac {3 d x}{2}\right )+183162 A \sin \left (c-\frac {d x}{2}\right )+17220 C \sin \left (c-\frac {d x}{2}\right )-100842 A \sin \left (c+\frac {d x}{2}\right )-17220 C \sin \left (c+\frac {d x}{2}\right )+155526 A \sin \left (2 c+\frac {d x}{2}\right )+14140 C \sin \left (2 c+\frac {d x}{2}\right )+37380 A \sin \left (c+\frac {3 d x}{2}\right )+9800 C \sin \left (c+\frac {3 d x}{2}\right )-101148 A \sin \left (2 c+\frac {3 d x}{2}\right )-15160 C \sin \left (2 c+\frac {3 d x}{2}\right )+102900 A \sin \left (3 c+\frac {3 d x}{2}\right )+9800 C \sin \left (3 c+\frac {3 d x}{2}\right )-119364 A \sin \left (c+\frac {5 d x}{2}\right )-10920 C \sin \left (c+\frac {5 d x}{2}\right )+8820 A \sin \left (2 c+\frac {5 d x}{2}\right )+4760 C \sin \left (2 c+\frac {5 d x}{2}\right )-78204 A \sin \left (3 c+\frac {5 d x}{2}\right )-10920 C \sin \left (3 c+\frac {5 d x}{2}\right )+49980 A \sin \left (4 c+\frac {5 d x}{2}\right )+4760 C \sin \left (4 c+\frac {5 d x}{2}\right )-64053 A \sin \left (2 c+\frac {7 d x}{2}\right )-5890 C \sin \left (2 c+\frac {7 d x}{2}\right )-3885 A \sin \left (3 c+\frac {7 d x}{2}\right )+1470 C \sin \left (3 c+\frac {7 d x}{2}\right )-44733 A \sin \left (4 c+\frac {7 d x}{2}\right )-5890 C \sin \left (4 c+\frac {7 d x}{2}\right )+15435 A \sin \left (5 c+\frac {7 d x}{2}\right )+1470 C \sin \left (5 c+\frac {7 d x}{2}\right )-21987 A \sin \left (3 c+\frac {9 d x}{2}\right )-2030 C \sin \left (3 c+\frac {9 d x}{2}\right )-3675 A \sin \left (4 c+\frac {9 d x}{2}\right )+210 C \sin \left (4 c+\frac {9 d x}{2}\right )-16107 A \sin \left (5 c+\frac {9 d x}{2}\right )-2030 C \sin \left (5 c+\frac {9 d x}{2}\right )+2205 A \sin \left (6 c+\frac {9 d x}{2}\right )+210 C \sin \left (6 c+\frac {9 d x}{2}\right )-3456 A \sin \left (4 c+\frac {11 d x}{2}\right )-320 C \sin \left (4 c+\frac {11 d x}{2}\right )-840 A \sin \left (5 c+\frac {11 d x}{2}\right )-2616 A \sin \left (6 c+\frac {11 d x}{2}\right )-320 C \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{6720 d (a+a \cos (c+d x))^4} \]
(-8*(21*A + 2*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + ( d*x)/2]])/(d*(a + a*Cos[c + d*x])^4) + (8*(21*A + 2*C)*Cos[c/2 + (d*x)/2]^ 8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^4) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(73206*A*Sin[(d*x)/2 ] + 14140*C*Sin[(d*x)/2] - 166668*A*Sin[(3*d*x)/2] - 15160*C*Sin[(3*d*x)/2 ] + 183162*A*Sin[c - (d*x)/2] + 17220*C*Sin[c - (d*x)/2] - 100842*A*Sin[c + (d*x)/2] - 17220*C*Sin[c + (d*x)/2] + 155526*A*Sin[2*c + (d*x)/2] + 1414 0*C*Sin[2*c + (d*x)/2] + 37380*A*Sin[c + (3*d*x)/2] + 9800*C*Sin[c + (3*d* x)/2] - 101148*A*Sin[2*c + (3*d*x)/2] - 15160*C*Sin[2*c + (3*d*x)/2] + 102 900*A*Sin[3*c + (3*d*x)/2] + 9800*C*Sin[3*c + (3*d*x)/2] - 119364*A*Sin[c + (5*d*x)/2] - 10920*C*Sin[c + (5*d*x)/2] + 8820*A*Sin[2*c + (5*d*x)/2] + 4760*C*Sin[2*c + (5*d*x)/2] - 78204*A*Sin[3*c + (5*d*x)/2] - 10920*C*Sin[3 *c + (5*d*x)/2] + 49980*A*Sin[4*c + (5*d*x)/2] + 4760*C*Sin[4*c + (5*d*x)/ 2] - 64053*A*Sin[2*c + (7*d*x)/2] - 5890*C*Sin[2*c + (7*d*x)/2] - 3885*A*S in[3*c + (7*d*x)/2] + 1470*C*Sin[3*c + (7*d*x)/2] - 44733*A*Sin[4*c + (7*d *x)/2] - 5890*C*Sin[4*c + (7*d*x)/2] + 15435*A*Sin[5*c + (7*d*x)/2] + 1470 *C*Sin[5*c + (7*d*x)/2] - 21987*A*Sin[3*c + (9*d*x)/2] - 2030*C*Sin[3*c + (9*d*x)/2] - 3675*A*Sin[4*c + (9*d*x)/2] + 210*C*Sin[4*c + (9*d*x)/2] - 16 107*A*Sin[5*c + (9*d*x)/2] - 2030*C*Sin[5*c + (9*d*x)/2] + 2205*A*Sin[6*c + (9*d*x)/2] + 210*C*Sin[6*c + (9*d*x)/2] - 3456*A*Sin[4*c + (11*d*x)/2...
Time = 1.67 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 3521, 3042, 3457, 3042, 3457, 3042, 3457, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a \cos (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 3521 |
| \(\displaystyle \frac {\int \frac {(a (9 A+2 C)-a (5 A-2 C) \cos (c+d x)) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 A+2 C)-a (5 A-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a^2 (73 A+10 C)-56 a^2 A \cos (c+d x)\right ) \sec ^3(c+d x)}{(\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (73 A+10 C)-56 a^2 A \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a^3 (477 A+50 C)-3 a^3 (129 A+10 C) \cos (c+d x)\right ) \sec ^3(c+d x)}{\cos (c+d x) a+a}dx}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a^3 (477 A+50 C)-3 a^3 (129 A+10 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3457 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \left (105 a^4 (21 A+2 C)-32 a^4 (54 A+5 C) \cos (c+d x)\right ) \sec ^3(c+d x)dx}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {105 a^4 (21 A+2 C)-32 a^4 (54 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \int \sec ^3(c+d x)dx-32 a^4 (54 A+5 C) \int \sec ^2(c+d x)dx}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-32 a^4 (54 A+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {32 a^4 (54 A+5 C) \int 1d(-\tan (c+d x))}{d}+105 a^4 (21 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {32 a^4 (54 A+5 C) \tan (c+d x)}{d}}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a^4 (54 A+5 C) \tan (c+d x)}{d}}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a^4 (54 A+5 C) \tan (c+d x)}{d}}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (21 A+2 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {32 a^4 (54 A+5 C) \tan (c+d x)}{d}}{a^2}-\frac {16 a^3 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}-\frac {(129 A+10 C) \tan (c+d x) \sec (c+d x)}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {14 a A \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}-\frac {(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A + C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])^4) + ((-1 4*a*A*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (-1/3*((12 9*A + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(d*(1 + Cos[c + d*x])^2) + ((-16*a^ 3*(54*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((-32 *a^4*(54*A + 5*C)*Tan[c + d*x])/d + 105*a^4*(21*A + 2*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/a^2)/(3*a^2))/(5*a^2))/( 7*a^2)
3.1.72.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ [b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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 = 2.62 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.86
| method | result | size |
| parallelrisch | \(\frac {-70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {2 C}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+70560 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (A +\frac {2 C}{21}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-11619 \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\left (\frac {23540 A}{3873}+\frac {2120 C}{3873}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {3992 A}{1291}+\frac {3280 C}{11619}\right ) \cos \left (3 d x +3 c \right )+\left (A +\frac {1070 C}{11619}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {192 A}{1291}+\frac {160 C}{11619}\right ) \cos \left (5 d x +5 c \right )+\left (\frac {34168 A}{3873}+\frac {9040 C}{11619}\right ) \cos \left (d x +c \right )+\frac {19387 A}{3873}+\frac {5290 C}{11619}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6720 d \,a^{4} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(192\) |
| derivativedivides | \(\frac {\left (-84 A -8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {36 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A -\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (84 A +8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {36 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(222\) |
| default | \(\frac {\left (-84 A -8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {36 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{7}-\frac {9 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C -13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A -\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3}-111 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\left (84 A +8 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {4 A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {36 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(222\) |
| norman | \(\frac {-\frac {\left (A +C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {\left (9 A +5 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}-\frac {\left (159 A +11 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}-\frac {\left (167 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (267 A +71 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}-\frac {\left (537 A +65 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}+\frac {\left (1541 A +145 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 a d}+\frac {\left (2055 A +151 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{168 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} a^{3}}-\frac {\left (21 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{4} d}+\frac {\left (21 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{4} d}\) | \(293\) |
| risch | \(-\frac {i \left (2205 A \,{\mathrm e}^{10 i \left (d x +c \right )}+210 C \,{\mathrm e}^{10 i \left (d x +c \right )}+15435 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1470 C \,{\mathrm e}^{9 i \left (d x +c \right )}+49980 A \,{\mathrm e}^{8 i \left (d x +c \right )}+4760 C \,{\mathrm e}^{8 i \left (d x +c \right )}+102900 A \,{\mathrm e}^{7 i \left (d x +c \right )}+9800 C \,{\mathrm e}^{7 i \left (d x +c \right )}+155526 A \,{\mathrm e}^{6 i \left (d x +c \right )}+14140 C \,{\mathrm e}^{6 i \left (d x +c \right )}+183162 A \,{\mathrm e}^{5 i \left (d x +c \right )}+17220 C \,{\mathrm e}^{5 i \left (d x +c \right )}+166668 A \,{\mathrm e}^{4 i \left (d x +c \right )}+15160 C \,{\mathrm e}^{4 i \left (d x +c \right )}+119364 A \,{\mathrm e}^{3 i \left (d x +c \right )}+10920 C \,{\mathrm e}^{3 i \left (d x +c \right )}+64053 A \,{\mathrm e}^{2 i \left (d x +c \right )}+5890 C \,{\mathrm e}^{2 i \left (d x +c \right )}+21987 A \,{\mathrm e}^{i \left (d x +c \right )}+2030 C \,{\mathrm e}^{i \left (d x +c \right )}+3456 A +320 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}+\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{4} d}-\frac {21 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{4} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{4} d}\) | \(371\) |
1/6720*(-70560*(1+cos(2*d*x+2*c))*(A+2/21*C)*ln(tan(1/2*d*x+1/2*c)-1)+7056 0*(1+cos(2*d*x+2*c))*(A+2/21*C)*ln(tan(1/2*d*x+1/2*c)+1)-11619*sec(1/2*d*x +1/2*c)^6*((23540/3873*A+2120/3873*C)*cos(2*d*x+2*c)+(3992/1291*A+3280/116 19*C)*cos(3*d*x+3*c)+(A+1070/11619*C)*cos(4*d*x+4*c)+(192/1291*A+160/11619 *C)*cos(5*d*x+5*c)+(34168/3873*A+9040/11619*C)*cos(d*x+c)+19387/3873*A+529 0/11619*C)*tan(1/2*d*x+1/2*c))/d/a^4/(1+cos(2*d*x+2*c))
Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (21 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (64 \, {\left (54 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{5} + {\left (11619 \, A + 1070 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3411 \, A + 310 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1509 \, A + 130 \, C\right )} \cos \left (d x + c\right )^{2} + 420 \, A \cos \left (d x + c\right ) - 105 \, A\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} + 4 \, a^{4} d \cos \left (d x + c\right )^{5} + 6 \, a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + a^{4} d \cos \left (d x + c\right )^{2}\right )}} \]
1/420*(105*((21*A + 2*C)*cos(d*x + c)^6 + 4*(21*A + 2*C)*cos(d*x + c)^5 + 6*(21*A + 2*C)*cos(d*x + c)^4 + 4*(21*A + 2*C)*cos(d*x + c)^3 + (21*A + 2* C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 105*((21*A + 2*C)*cos(d*x + c)^ 6 + 4*(21*A + 2*C)*cos(d*x + c)^5 + 6*(21*A + 2*C)*cos(d*x + c)^4 + 4*(21* A + 2*C)*cos(d*x + c)^3 + (21*A + 2*C)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(64*(54*A + 5*C)*cos(d*x + c)^5 + (11619*A + 1070*C)*cos(d*x + c)^ 4 + 4*(3411*A + 310*C)*cos(d*x + c)^3 + 4*(1509*A + 130*C)*cos(d*x + c)^2 + 420*A*cos(d*x + c) - 105*A)*sin(d*x + c))/(a^4*d*cos(d*x + c)^6 + 4*a^4* d*cos(d*x + c)^5 + 6*a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + a^4*d *cos(d*x + c)^2)
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.66 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} - \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {2940 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} + 5 \, C {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \]
-1/840*(3*A*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) - 9*sin(d*x + c)^3/(co s(d*x + c) + 1)^3)/(a^4 - 2*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4* sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) + 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 2940*log(sin(d* x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 2940*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4) + 5*C*((315*sin(d*x + c)/(cos(d*x + c) + 1) + 77*sin(d*x + c )^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*sin( d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 168*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^4 + 168*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d
Time = 0.32 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (21 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {420 \, {\left (21 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
1/840*(420*(21*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 420*(21*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 + 840*(9*A*tan(1/2*d*x + 1/ 2*c)^3 - 7*A*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^4) - (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 + 189 *A*a^24*tan(1/2*d*x + 1/2*c)^5 + 105*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 1365* A*a^24*tan(1/2*d*x + 1/2*c)^3 + 385*C*a^24*tan(1/2*d*x + 1/2*c)^3 + 11655* A*a^24*tan(1/2*d*x + 1/2*c) + 1575*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 1.07 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {21\,A}{2}+C\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A+C\right )}{40\,a^4}+\frac {6\,A+2\,C}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A+C\right )}{4\,a^4}+\frac {3\,\left (6\,A+2\,C\right )}{4\,a^4}+\frac {3\,\left (15\,A-C\right )}{8\,a^4}+\frac {20\,A-4\,C}{8\,a^4}\right )}{d}-\frac {7\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9\,A\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A+C}{4\,a^4}+\frac {6\,A+2\,C}{8\,a^4}+\frac {15\,A-C}{24\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4\,d} \]
(2*atanh(tan(c/2 + (d*x)/2))*((21*A)/2 + C))/(a^4*d) - (tan(c/2 + (d*x)/2) ^5*((3*(A + C))/(40*a^4) + (6*A + 2*C)/(40*a^4)))/d - (tan(c/2 + (d*x)/2)* ((5*(A + C))/(4*a^4) + (3*(6*A + 2*C))/(4*a^4) + (3*(15*A - C))/(8*a^4) + (20*A - 4*C)/(8*a^4)))/d - (7*A*tan(c/2 + (d*x)/2) - 9*A*tan(c/2 + (d*x)/2 )^3)/(d*(a^4*tan(c/2 + (d*x)/2)^4 - 2*a^4*tan(c/2 + (d*x)/2)^2 + a^4)) - ( tan(c/2 + (d*x)/2)^3*((A + C)/(4*a^4) + (6*A + 2*C)/(8*a^4) + (15*A - C)/( 24*a^4)))/d - (tan(c/2 + (d*x)/2)^7*(A + C))/(56*a^4*d)