Integrand size = 21, antiderivative size = 199 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {245 a^2 x}{128}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d} \]
-245/128*a^2*x+2*a^2*arctanh(sin(d*x+c))/d-2*a^2*sin(d*x+c)/d+139/128*a^2* cos(d*x+c)*sin(d*x+c)/d+11/192*a^2*cos(d*x+c)^3*sin(d*x+c)/d-17/48*a^2*cos (d*x+c)^5*sin(d*x+c)/d+1/8*a^2*cos(d*x+c)^7*sin(d*x+c)/d-2/3*a^2*sin(d*x+c )^3/d-2/5*a^2*sin(d*x+c)^5/d-2/7*a^2*sin(d*x+c)^7/d+a^2*tan(d*x+c)/d
Time = 1.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.72 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (168000 c+168000 d x+37800 \arctan (\tan (c+d x))-215040 \text {arctanh}(\sin (c+d x))+215040 \sin (c+d x)+71680 \sin ^3(c+d x)+43008 \sin ^5(c+d x)+30720 \sin ^7(c+d x)-55440 \sin (2 (c+d x))+2520 \sin (4 (c+d x))+560 \sin (6 (c+d x))-105 \sin (8 (c+d x))-107520 \tan (c+d x)\right )}{430080 d} \]
-1/430080*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(168000*c + 168000* d*x + 37800*ArcTan[Tan[c + d*x]] - 215040*ArcTanh[Sin[c + d*x]] + 215040*S in[c + d*x] + 71680*Sin[c + d*x]^3 + 43008*Sin[c + d*x]^5 + 30720*Sin[c + d*x]^7 - 55440*Sin[2*(c + d*x)] + 2520*Sin[4*(c + d*x)] + 560*Sin[6*(c + d *x)] - 105*Sin[8*(c + d*x)] - 107520*Tan[c + d*x]))/d
Time = 0.65 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 4360, 3042, 3351, 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.
| \(\displaystyle \int \sin ^8(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^8 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^6(c+d x) \tan ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^8 \left (a \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\cos ^8(c+d x) a^{10}+2 \cos ^7(c+d x) a^{10}-3 \cos ^6(c+d x) a^{10}-8 \cos ^5(c+d x) a^{10}+2 \cos ^4(c+d x) a^{10}+12 \cos ^3(c+d x) a^{10}+2 \cos ^2(c+d x) a^{10}+\sec ^2(c+d x) a^{10}-8 \cos (c+d x) a^{10}+2 \sec (c+d x) a^{10}-3 a^{10}\right )dx}{a^8}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^{10} \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^{10} \sin ^7(c+d x)}{7 d}-\frac {2 a^{10} \sin ^5(c+d x)}{5 d}-\frac {2 a^{10} \sin ^3(c+d x)}{3 d}-\frac {2 a^{10} \sin (c+d x)}{d}+\frac {a^{10} \tan (c+d x)}{d}+\frac {a^{10} \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac {17 a^{10} \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {11 a^{10} \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {139 a^{10} \sin (c+d x) \cos (c+d x)}{128 d}-\frac {245 a^{10} x}{128}}{a^8}\) |
((-245*a^10*x)/128 + (2*a^10*ArcTanh[Sin[c + d*x]])/d - (2*a^10*Sin[c + d* x])/d + (139*a^10*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (11*a^10*Cos[c + d* x]^3*Sin[c + d*x])/(192*d) - (17*a^10*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^10*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (2*a^10*Sin[c + d*x]^3)/(3*d) - (2*a^10*Sin[c + d*x]^5)/(5*d) - (2*a^10*Sin[c + d*x]^7)/(7*d) + (a^10*T an[c + d*x])/d)/a^8
3.1.29.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 2.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(\frac {a^{2} \left (-411600 d x \cos \left (d x +c \right )-430080 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+430080 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+270480 \sin \left (d x +c \right )+35392 \sin \left (4 d x +4 c \right )+105 \sin \left (9 d x +9 c \right )-271040 \sin \left (2 d x +2 c \right )+480 \sin \left (8 d x +8 c \right )-5568 \sin \left (6 d x +6 c \right )-455 \sin \left (7 d x +7 c \right )-3080 \sin \left (5 d x +5 c \right )+52920 \sin \left (3 d x +3 c \right )\right )}{215040 d \cos \left (d x +c \right )}\) | \(164\) |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(194\) |
| default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(194\) |
| parts | \(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )-\frac {35 d x}{16}-\frac {35 c}{16}\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(199\) |
| risch | \(-\frac {245 a^{2} x}{128}+\frac {33 i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {93 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{64 d}+\frac {93 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{64 d}-\frac {33 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {a^{2} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a^{2} \sin \left (7 d x +7 c \right )}{224 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}-\frac {9 a^{2} \sin \left (5 d x +5 c \right )}{160 d}-\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{128 d}+\frac {37 a^{2} \sin \left (3 d x +3 c \right )}{96 d}\) | \(246\) |
1/215040*a^2*(-411600*d*x*cos(d*x+c)-430080*ln(tan(1/2*d*x+1/2*c)-1)*cos(d *x+c)+430080*ln(tan(1/2*d*x+1/2*c)+1)*cos(d*x+c)+270480*sin(d*x+c)+35392*s in(4*d*x+4*c)+105*sin(9*d*x+9*c)-271040*sin(2*d*x+2*c)+480*sin(8*d*x+8*c)- 5568*sin(6*d*x+6*c)-455*sin(7*d*x+7*c)-3080*sin(5*d*x+5*c)+52920*sin(3*d*x +3*c))/d/cos(d*x+c)
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {25725 \, a^{2} d x \cos \left (d x + c\right ) - 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (1680 \, a^{2} \cos \left (d x + c\right )^{8} + 3840 \, a^{2} \cos \left (d x + c\right )^{7} - 4760 \, a^{2} \cos \left (d x + c\right )^{6} - 16896 \, a^{2} \cos \left (d x + c\right )^{5} + 770 \, a^{2} \cos \left (d x + c\right )^{4} + 31232 \, a^{2} \cos \left (d x + c\right )^{3} + 14595 \, a^{2} \cos \left (d x + c\right )^{2} - 45056 \, a^{2} \cos \left (d x + c\right ) + 13440 \, a^{2}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )} \]
-1/13440*(25725*a^2*d*x*cos(d*x + c) - 13440*a^2*cos(d*x + c)*log(sin(d*x + c) + 1) + 13440*a^2*cos(d*x + c)*log(-sin(d*x + c) + 1) - (1680*a^2*cos( d*x + c)^8 + 3840*a^2*cos(d*x + c)^7 - 4760*a^2*cos(d*x + c)^6 - 16896*a^2 *cos(d*x + c)^5 + 770*a^2*cos(d*x + c)^4 + 31232*a^2*cos(d*x + c)^3 + 1459 5*a^2*cos(d*x + c)^2 - 45056*a^2*cos(d*x + c) + 13440*a^2)*sin(d*x + c))/( d*cos(d*x + c))
Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {1024 \, {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{2} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 2240 \, {\left (105 \, d x + 105 \, c - \frac {87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{2}}{107520 \, d} \]
-1/107520*(1024*(30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 210*sin(d*x + c ))*a^2 - 35*(128*sin(2*d*x + 2*c)^3 + 840*d*x + 840*c + 3*sin(8*d*x + 8*c) + 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^2 + 2240*(105*d*x + 105* c - (87*tan(d*x + c)^5 + 136*tan(d*x + c)^3 + 57*tan(d*x + c))/(tan(d*x + c)^6 + 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1) - 48*tan(d*x + c))*a^2)/d
Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.13 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=-\frac {25725 \, {\left (d x + c\right )} a^{2} - 26880 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 26880 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {26880 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (39165 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 300265 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 989261 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1791073 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1814943 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 670131 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147735 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14595 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \]
-1/13440*(25725*(d*x + c)*a^2 - 26880*a^2*log(abs(tan(1/2*d*x + 1/2*c) + 1 )) + 26880*a^2*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 26880*a^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) + 2*(39165*a^2*tan(1/2*d*x + 1/2*c)^ 15 + 300265*a^2*tan(1/2*d*x + 1/2*c)^13 + 989261*a^2*tan(1/2*d*x + 1/2*c)^ 11 + 1791073*a^2*tan(1/2*d*x + 1/2*c)^9 + 1814943*a^2*tan(1/2*d*x + 1/2*c) ^7 + 670131*a^2*tan(1/2*d*x + 1/2*c)^5 + 147735*a^2*tan(1/2*d*x + 1/2*c)^3 + 14595*a^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d
Time = 15.70 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.47 \[ \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx=\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {245\,a^2\,x}{128}+\frac {\frac {501\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}+\frac {2633\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{48}+\frac {38047\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{240}+\frac {388613\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{1680}+\frac {13781\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{96}-\frac {32681\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{560}-\frac {1739\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{80}-\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}-\frac {11\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(4*a^2*atanh(tan(c/2 + (d*x)/2)))/d - (245*a^2*x)/128 + ((13781*a^2*tan(c/ 2 + (d*x)/2)^9)/96 - (1739*a^2*tan(c/2 + (d*x)/2)^5)/80 - (32681*a^2*tan(c /2 + (d*x)/2)^7)/560 - (61*a^2*tan(c/2 + (d*x)/2)^3)/16 + (388613*a^2*tan( c/2 + (d*x)/2)^11)/1680 + (38047*a^2*tan(c/2 + (d*x)/2)^13)/240 + (2633*a^ 2*tan(c/2 + (d*x)/2)^15)/48 + (501*a^2*tan(c/2 + (d*x)/2)^17)/64 - (11*a^2 *tan(c/2 + (d*x)/2))/64)/(d*(7*tan(c/2 + (d*x)/2)^2 + 20*tan(c/2 + (d*x)/2 )^4 + 28*tan(c/2 + (d*x)/2)^6 + 14*tan(c/2 + (d*x)/2)^8 - 14*tan(c/2 + (d* x)/2)^10 - 28*tan(c/2 + (d*x)/2)^12 - 20*tan(c/2 + (d*x)/2)^14 - 7*tan(c/2 + (d*x)/2)^16 - tan(c/2 + (d*x)/2)^18 + 1))