Integrand size = 27, antiderivative size = 130 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5 \text {arctanh}(\sin (c+d x))}{128 a d}+\frac {\sec ^8(c+d x)}{8 a d}+\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{192 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{48 a d}-\frac {\sec ^7(c+d x) \tan (c+d x)}{8 a d} \]
5/128*arctanh(sin(d*x+c))/a/d+1/8*sec(d*x+c)^8/a/d+5/128*sec(d*x+c)*tan(d* x+c)/a/d+5/192*sec(d*x+c)^3*tan(d*x+c)/a/d+1/48*sec(d*x+c)^5*tan(d*x+c)/a/ d-1/8*sec(d*x+c)^7*tan(d*x+c)/a/d
Time = 0.74 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \text {arctanh}(\sin (c+d x))-\frac {4}{(-1+\sin (c+d x))^3}+\frac {9}{(-1+\sin (c+d x))^2}-\frac {15}{-1+\sin (c+d x)}+\frac {6}{(1+\sin (c+d x))^4}+\frac {8}{(1+\sin (c+d x))^3}+\frac {6}{(1+\sin (c+d x))^2}}{384 a d} \]
(15*ArcTanh[Sin[c + d*x]] - 4/(-1 + Sin[c + d*x])^3 + 9/(-1 + Sin[c + d*x] )^2 - 15/(-1 + Sin[c + d*x]) + 6/(1 + Sin[c + d*x])^4 + 8/(1 + Sin[c + d*x ])^3 + 6/(1 + Sin[c + d*x])^2)/(384*a*d)
Time = 0.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3314, 3042, 3086, 15, 3091, 3042, 4255, 3042, 4255, 3042, 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.
| \(\displaystyle \int \frac {\tan (c+d x) \sec ^6(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)}{\cos (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3314 |
| \(\displaystyle \frac {\int \sec ^8(c+d x) \tan (c+d x)dx}{a}-\frac {\int \sec ^7(c+d x) \tan ^2(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec (c+d x)^8 \tan (c+d x)dx}{a}-\frac {\int \sec (c+d x)^7 \tan (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {\int \sec ^7(c+d x)d\sec (c+d x)}{a d}-\frac {\int \sec (c+d x)^7 \tan (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\int \sec (c+d x)^7 \tan (c+d x)^2dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac {1}{8} \int \sec ^7(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac {1}{8} \int \csc \left (c+d x+\frac {\pi }{2}\right )^7dx}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \int \sec ^5(c+d x)dx-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \int \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\sec ^8(c+d x)}{8 a d}-\frac {\frac {1}{8} \left (-\frac {5}{6} \left (\frac {3}{4} \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )-\frac {\tan (c+d x) \sec ^5(c+d x)}{6 d}\right )+\frac {\tan (c+d x) \sec ^7(c+d x)}{8 d}}{a}\) |
Sec[c + d*x]^8/(8*a*d) - ((Sec[c + d*x]^7*Tan[c + d*x])/(8*d) + (-1/6*(Sec [c + d*x]^5*Tan[c + d*x])/d - (5*((Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3 *(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4))/6) /8)/a
3.9.88.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/(( a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[1/(b*d) Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] & & IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
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 = 0.97 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
| default | \(\frac {-\frac {1}{96 \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {3}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {1}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d a}\) | \(103\) |
| risch | \(-\frac {i \left (113 \,{\mathrm e}^{5 i \left (d x +c \right )}-30 i {\mathrm e}^{2 i \left (d x +c \right )}+170 i {\mathrm e}^{10 i \left (d x +c \right )}+70 \,{\mathrm e}^{3 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}+15 \,{\mathrm e}^{13 i \left (d x +c \right )}+113 \,{\mathrm e}^{9 i \left (d x +c \right )}+70 \,{\mathrm e}^{11 i \left (d x +c \right )}+396 i {\mathrm e}^{8 i \left (d x +c \right )}-396 i {\mathrm e}^{6 i \left (d x +c \right )}-170 i {\mathrm e}^{4 i \left (d x +c \right )}-3468 \,{\mathrm e}^{7 i \left (d x +c \right )}+30 i {\mathrm e}^{12 i \left (d x +c \right )}\right )}{192 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{6} d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d a}\) | \(231\) |
| norman | \(\frac {\frac {95 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {95 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}+\frac {95 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {5 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {59 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {59 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {149 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {149 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {163 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}+\frac {163 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d a}-\frac {625 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}-\frac {625 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a d}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a d}\) | \(315\) |
| parallelrisch | \(\frac {\left (-75 \sin \left (5 d x +5 c \right )-15 \sin \left (7 d x +7 c \right )-450 \cos \left (2 d x +2 c \right )-180 \cos \left (4 d x +4 c \right )-30 \cos \left (6 d x +6 c \right )-75 \sin \left (d x +c \right )-135 \sin \left (3 d x +3 c \right )-300\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (75 \sin \left (5 d x +5 c \right )+15 \sin \left (7 d x +7 c \right )+450 \cos \left (2 d x +2 c \right )+180 \cos \left (4 d x +4 c \right )+30 \cos \left (6 d x +6 c \right )+75 \sin \left (d x +c \right )+135 \sin \left (3 d x +3 c \right )+300\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-105 \sin \left (5 d x +5 c \right )-33 \sin \left (7 d x +7 c \right )-1216 \cos \left (2 d x +2 c \right )-536 \cos \left (4 d x +4 c \right )-96 \cos \left (6 d x +6 c \right )+627 \sin \left (d x +c \right )+43 \sin \left (3 d x +3 c \right )+2808}{384 a d \left (20+\sin \left (7 d x +7 c \right )+5 \sin \left (5 d x +5 c \right )+9 \sin \left (3 d x +3 c \right )+5 \sin \left (d x +c \right )+2 \cos \left (6 d x +6 c \right )+12 \cos \left (4 d x +4 c \right )+30 \cos \left (2 d x +2 c \right )\right )}\) | \(339\) |
1/d/a*(-1/96/(sin(d*x+c)-1)^3+3/128/(sin(d*x+c)-1)^2-5/128/(sin(d*x+c)-1)- 5/256*ln(sin(d*x+c)-1)+1/64/(1+sin(d*x+c))^4+1/48/(1+sin(d*x+c))^3+1/64/(1 +sin(d*x+c))^2+5/256*ln(1+sin(d*x+c)))
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.28 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
-1/768*(30*cos(d*x + c)^6 - 10*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 15*(cos (d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 15*(cos (d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) - 2*(15* cos(d*x + c)^4 + 10*cos(d*x + c)^2 + 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos(d*x + c)^6)
\[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\sin {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )^{2} + 33 \, \sin \left (d x + c\right ) + 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
-1/768*(2*(15*sin(d*x + c)^6 + 15*sin(d*x + c)^5 - 40*sin(d*x + c)^4 - 40* sin(d*x + c)^3 + 33*sin(d*x + c)^2 + 33*sin(d*x + c) + 48)/(a*sin(d*x + c) ^7 + a*sin(d*x + c)^6 - 3*a*sin(d*x + c)^5 - 3*a*sin(d*x + c)^4 + 3*a*sin( d*x + c)^3 + 3*a*sin(d*x + c)^2 - a*sin(d*x + c) - a) - 15*log(sin(d*x + c ) + 1)/a + 15*log(sin(d*x + c) - 1)/a)/d
Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} - 225 \, \sin \left (d x + c\right )^{2} + 321 \, \sin \left (d x + c\right ) - 167\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 500 \, \sin \left (d x + c\right )^{3} + 702 \, \sin \left (d x + c\right )^{2} + 340 \, \sin \left (d x + c\right ) - 35}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
1/3072*(60*log(abs(sin(d*x + c) + 1))/a - 60*log(abs(sin(d*x + c) - 1))/a + 2*(55*sin(d*x + c)^3 - 225*sin(d*x + c)^2 + 321*sin(d*x + c) - 167)/(a*( sin(d*x + c) - 1)^3) - (125*sin(d*x + c)^4 + 500*sin(d*x + c)^3 + 702*sin( d*x + c)^2 + 340*sin(d*x + c) - 35)/(a*(sin(d*x + c) + 1)^4))/d
Time = 19.51 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.98 \[ \int \frac {\sec ^6(c+d x) \tan (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d}+\frac {-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}+\frac {149\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}-\frac {625\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {95\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{16}-\frac {625\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}+\frac {149\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}+\frac {163\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]
(5*atanh(tan(c/2 + (d*x)/2)))/(64*a*d) + ((59*tan(c/2 + (d*x)/2)^2)/32 - ( 5*tan(c/2 + (d*x)/2))/64 + (163*tan(c/2 + (d*x)/2)^3)/96 + (149*tan(c/2 + (d*x)/2)^4)/96 - (625*tan(c/2 + (d*x)/2)^5)/192 + (95*tan(c/2 + (d*x)/2)^6 )/16 + (95*tan(c/2 + (d*x)/2)^7)/16 + (95*tan(c/2 + (d*x)/2)^8)/16 - (625* tan(c/2 + (d*x)/2)^9)/192 + (149*tan(c/2 + (d*x)/2)^10)/96 + (163*tan(c/2 + (d*x)/2)^11)/96 + (59*tan(c/2 + (d*x)/2)^12)/32 - (5*tan(c/2 + (d*x)/2)^ 13)/64)/(d*(a + 2*a*tan(c/2 + (d*x)/2) - 5*a*tan(c/2 + (d*x)/2)^2 - 12*a*t an(c/2 + (d*x)/2)^3 + 9*a*tan(c/2 + (d*x)/2)^4 + 30*a*tan(c/2 + (d*x)/2)^5 - 5*a*tan(c/2 + (d*x)/2)^6 - 40*a*tan(c/2 + (d*x)/2)^7 - 5*a*tan(c/2 + (d *x)/2)^8 + 30*a*tan(c/2 + (d*x)/2)^9 + 9*a*tan(c/2 + (d*x)/2)^10 - 12*a*ta n(c/2 + (d*x)/2)^11 - 5*a*tan(c/2 + (d*x)/2)^12 + 2*a*tan(c/2 + (d*x)/2)^1 3 + a*tan(c/2 + (d*x)/2)^14))