Integrand size = 21, antiderivative size = 156 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {93 a^5 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {16 a^5 \tan (c+d x)}{d}+\frac {93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {28 a^5 \tan ^3(c+d x)}{3 d}+\frac {13 a^5 \tan ^5(c+d x)}{5 d}+\frac {a^5 \tan ^7(c+d x)}{7 d} \]
93/16*a^5*arctanh(sin(d*x+c))/d+16*a^5*tan(d*x+c)/d+93/16*a^5*sec(d*x+c)*t an(d*x+c)/d+85/24*a^5*sec(d*x+c)^3*tan(d*x+c)/d+5/6*a^5*sec(d*x+c)^5*tan(d *x+c)/d+28/3*a^5*tan(d*x+c)^3/d+13/5*a^5*tan(d*x+c)^5/d+1/7*a^5*tan(d*x+c) ^7/d
Time = 5.07 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {93 a^5 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {479 a^5 \tan (c+d x)}{35 d}+\frac {93 a^5 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {85 a^5 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {76 a^5 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {5 a^5 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {a^5 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac {479 a^5 \tan ^3(c+d x)}{105 d} \]
(93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (479*a^5*Tan[c + d*x])/(35*d) + (9 3*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (85*a^5*Sec[c + d*x]^3*Tan[c + d *x])/(24*d) + (76*a^5*Sec[c + d*x]^4*Tan[c + d*x])/(35*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (a^5*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + (4 79*a^5*Tan[c + d*x]^3)/(105*d)
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4278, 2009}
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.
| \(\displaystyle \int \sec ^3(c+d x) (a \sec (c+d x)+a)^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^5dx\) |
| \(\Big \downarrow \) 4278 |
| \(\displaystyle \int \left (a^5 \sec ^8(c+d x)+5 a^5 \sec ^7(c+d x)+10 a^5 \sec ^6(c+d x)+10 a^5 \sec ^5(c+d x)+5 a^5 \sec ^4(c+d x)+a^5 \sec ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {93 a^5 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^5 \tan ^7(c+d x)}{7 d}+\frac {13 a^5 \tan ^5(c+d x)}{5 d}+\frac {28 a^5 \tan ^3(c+d x)}{3 d}+\frac {16 a^5 \tan (c+d x)}{d}+\frac {5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {93 a^5 \tan (c+d x) \sec (c+d x)}{16 d}\) |
(93*a^5*ArcTanh[Sin[c + d*x]])/(16*d) + (16*a^5*Tan[c + d*x])/d + (93*a^5* Sec[c + d*x]*Tan[c + d*x])/(16*d) + (85*a^5*Sec[c + d*x]^3*Tan[c + d*x])/( 24*d) + (5*a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (28*a^5*Tan[c + d*x]^3 )/(3*d) + (13*a^5*Tan[c + d*x]^5)/(5*d) + (a^5*Tan[c + d*x]^7)/(7*d)
3.1.41.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f *x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && I GtQ[m, 0] && RationalQ[n]
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(-\frac {i a^{5} \left (9765 \,{\mathrm e}^{13 i \left (d x +c \right )}+62860 \,{\mathrm e}^{11 i \left (d x +c \right )}-16800 \,{\mathrm e}^{10 i \left (d x +c \right )}+118825 \,{\mathrm e}^{9 i \left (d x +c \right )}-162400 \,{\mathrm e}^{8 i \left (d x +c \right )}-374080 \,{\mathrm e}^{6 i \left (d x +c \right )}-118825 \,{\mathrm e}^{5 i \left (d x +c \right )}-305088 \,{\mathrm e}^{4 i \left (d x +c \right )}-62860 \,{\mathrm e}^{3 i \left (d x +c \right )}-107296 \,{\mathrm e}^{2 i \left (d x +c \right )}-9765 \,{\mathrm e}^{i \left (d x +c \right )}-15328\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {93 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}-\frac {93 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}\) | \(189\) |
| norman | \(\frac {-\frac {419 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {943 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {37169 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}+\frac {11904 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {8773 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{40 d}+\frac {155 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{2 d}-\frac {93 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}-\frac {93 a^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {93 a^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(190\) |
| parallelrisch | \(\frac {3395 a^{5} \left (\frac {279 \left (-\cos \left (d x +c \right )-\frac {3 \cos \left (3 d x +3 c \right )}{5}-\frac {\cos \left (5 d x +5 c \right )}{5}-\frac {\cos \left (7 d x +7 c \right )}{35}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{194}+\frac {279 \left (\frac {\cos \left (7 d x +7 c \right )}{35}+\frac {\cos \left (5 d x +5 c \right )}{5}+\frac {3 \cos \left (3 d x +3 c \right )}{5}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{194}+\sin \left (2 d x +2 c \right )+\frac {20592 \sin \left (3 d x +3 c \right )}{16975}+\frac {1796 \sin \left (4 d x +4 c \right )}{3395}+\frac {7664 \sin \left (5 d x +5 c \right )}{16975}+\frac {279 \sin \left (6 d x +6 c \right )}{3395}+\frac {7664 \sin \left (7 d x +7 c \right )}{118825}+\frac {432 \sin \left (d x +c \right )}{485}\right )}{24 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\) | \(234\) |
| derivativedivides | \(\frac {-a^{5} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+5 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-10 a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+10 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-5 a^{5} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(248\) |
| default | \(\frac {-a^{5} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )+5 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-10 a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+10 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-5 a^{5} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(248\) |
| parts | \(\frac {a^{5} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {a^{5} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}-\frac {5 a^{5} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {10 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {10 a^{5} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {5 a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(262\) |
-1/840*I*a^5*(9765*exp(13*I*(d*x+c))+62860*exp(11*I*(d*x+c))-16800*exp(10* I*(d*x+c))+118825*exp(9*I*(d*x+c))-162400*exp(8*I*(d*x+c))-374080*exp(6*I* (d*x+c))-118825*exp(5*I*(d*x+c))-305088*exp(4*I*(d*x+c))-62860*exp(3*I*(d* x+c))-107296*exp(2*I*(d*x+c))-9765*exp(I*(d*x+c))-15328)/d/(exp(2*I*(d*x+c ))+1)^7+93/16*a^5/d*ln(exp(I*(d*x+c))+I)-93/16*a^5/d*ln(exp(I*(d*x+c))-I)
Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, a^{5} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15328 \, a^{5} \cos \left (d x + c\right )^{6} + 9765 \, a^{5} \cos \left (d x + c\right )^{5} + 7664 \, a^{5} \cos \left (d x + c\right )^{4} + 5950 \, a^{5} \cos \left (d x + c\right )^{3} + 3648 \, a^{5} \cos \left (d x + c\right )^{2} + 1400 \, a^{5} \cos \left (d x + c\right ) + 240 \, a^{5}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
1/3360*(9765*a^5*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 9765*a^5*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*(15328*a^5*cos(d*x + c)^6 + 9765*a^5*cos( d*x + c)^5 + 7664*a^5*cos(d*x + c)^4 + 5950*a^5*cos(d*x + c)^3 + 3648*a^5* cos(d*x + c)^2 + 1400*a^5*cos(d*x + c) + 240*a^5)*sin(d*x + c))/(d*cos(d*x + c)^7)
\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=a^{5} \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int 5 \sec ^{4}{\left (c + d x \right )}\, dx + \int 10 \sec ^{5}{\left (c + d x \right )}\, dx + \int 10 \sec ^{6}{\left (c + d x \right )}\, dx + \int 5 \sec ^{7}{\left (c + d x \right )}\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
a**5*(Integral(sec(c + d*x)**3, x) + Integral(5*sec(c + d*x)**4, x) + Inte gral(10*sec(c + d*x)**5, x) + Integral(10*sec(c + d*x)**6, x) + Integral(5 *sec(c + d*x)**7, x) + Integral(sec(c + d*x)**8, x))
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (142) = 284\).
Time = 0.32 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.01 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{5} + 2240 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 5600 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{5} - 175 \, a^{5} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2100 \, a^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{5} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{3360 \, d} \]
1/3360*(96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35* tan(d*x + c))*a^5 + 2240*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d* x + c))*a^5 + 5600*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^5 - 175*a^5*(2*(15* sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3* sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log (sin(d*x + c) - 1)) - 2100*a^5*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin (d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin( d*x + c) - 1)) - 840*a^5*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d* x + c) + 1) + log(sin(d*x + c) - 1)))/d
Time = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {9765 \, a^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 9765 \, a^{5} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (9765 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 65100 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 184233 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 285696 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 260183 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 132020 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 43995 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{1680 \, d} \]
1/1680*(9765*a^5*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 9765*a^5*log(abs(tan (1/2*d*x + 1/2*c) - 1)) - 2*(9765*a^5*tan(1/2*d*x + 1/2*c)^13 - 65100*a^5* tan(1/2*d*x + 1/2*c)^11 + 184233*a^5*tan(1/2*d*x + 1/2*c)^9 - 285696*a^5*t an(1/2*d*x + 1/2*c)^7 + 260183*a^5*tan(1/2*d*x + 1/2*c)^5 - 132020*a^5*tan (1/2*d*x + 1/2*c)^3 + 43995*a^5*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c )^2 - 1)^7)/d
Time = 17.09 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.46 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx=\frac {93\,a^5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {\frac {93\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {155\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}+\frac {8773\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{40}-\frac {11904\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{35}+\frac {37169\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{120}-\frac {943\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {419\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(93*a^5*atanh(tan(c/2 + (d*x)/2)))/(8*d) - ((37169*a^5*tan(c/2 + (d*x)/2)^ 5)/120 - (943*a^5*tan(c/2 + (d*x)/2)^3)/6 - (11904*a^5*tan(c/2 + (d*x)/2)^ 7)/35 + (8773*a^5*tan(c/2 + (d*x)/2)^9)/40 - (155*a^5*tan(c/2 + (d*x)/2)^1 1)/2 + (93*a^5*tan(c/2 + (d*x)/2)^13)/8 + (419*a^5*tan(c/2 + (d*x)/2))/8)/ (d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/ 2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d *x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))