Integrand size = 21, antiderivative size = 138 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=-\frac {33 a^3 x}{8}+\frac {3 a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \sin ^3(c+d x)}{d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
-33/8*a^3*x+3/2*a^3*arctanh(sin(d*x+c))/d-2*a^3*sin(d*x+c)/d+7/8*a^3*cos(d *x+c)*sin(d*x+c)/d+1/4*a^3*cos(d*x+c)^3*sin(d*x+c)/d-a^3*sin(d*x+c)^3/d+3* a^3*tan(d*x+c)/d+1/2*a^3*sec(d*x+c)*tan(d*x+c)/d
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=\frac {a^3 \sec ^2(c+d x) \left (-264 c-264 d x+192 \text {arctanh}(\sin (c+d x)) \cos ^2(c+d x)-264 (c+d x) \cos (2 (c+d x))-16 \sin (c+d x)+225 \sin (2 (c+d x))-72 \sin (3 (c+d x))+18 \sin (4 (c+d x))+8 \sin (5 (c+d x))+\sin (6 (c+d x))\right )}{128 d} \]
(a^3*Sec[c + d*x]^2*(-264*c - 264*d*x + 192*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 - 264*(c + d*x)*Cos[2*(c + d*x)] - 16*Sin[c + d*x] + 225*Sin[2*(c + d*x)] - 72*Sin[3*(c + d*x)] + 18*Sin[4*(c + d*x)] + 8*Sin[5*(c + d*x)] + Sin[6*(c + d*x)]))/(128*d)
Time = 0.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 25, 25, 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.
| \(\displaystyle \int \sin ^4(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^4 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin (c+d x) \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sin (c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^4 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\cos ^4(c+d x) a^7+3 \cos ^3(c+d x) a^7+\sec ^3(c+d x) a^7+\cos ^2(c+d x) a^7+3 \sec ^2(c+d x) a^7-5 \cos (c+d x) a^7+\sec (c+d x) a^7-5 a^7\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^7 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^7 \sin ^3(c+d x)}{d}-\frac {2 a^7 \sin (c+d x)}{d}+\frac {3 a^7 \tan (c+d x)}{d}+\frac {a^7 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {7 a^7 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^7 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {33 a^7 x}{8}}{a^4}\) |
((-33*a^7*x)/8 + (3*a^7*ArcTanh[Sin[c + d*x]])/(2*d) - (2*a^7*Sin[c + d*x] )/d + (7*a^7*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^7*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a^7*Sin[c + d*x]^3)/d + (3*a^7*Tan[c + d*x])/d + (a^7*Sec [c + d*x]*Tan[c + d*x])/(2*d))/a^4
3.1.50.3.1 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 = 2.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11
| method | result | size |
| parallelrisch | \(\frac {\left (\left (-12 \cos \left (2 d x +2 c \right )-12\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (12 \cos \left (2 d x +2 c \right )+12\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-33 d x \cos \left (2 d x +2 c \right )-33 d x +\sin \left (5 d x +5 c \right )+\frac {\sin \left (6 d x +6 c \right )}{8}-2 \sin \left (d x +c \right )+\frac {225 \sin \left (2 d x +2 c \right )}{8}-9 \sin \left (3 d x +3 c \right )+\frac {9 \sin \left (4 d x +4 c \right )}{4}\right ) a^{3}}{8 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(153\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(192\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(192\) |
| parts | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{3}}{2}+\frac {3 \sin \left (d x +c \right )}{2}-\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )}{d}\) | \(200\) |
| risch | \(-\frac {33 a^{3} x}{8}-\frac {i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {11 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {11 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}+\frac {i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {a^{3} \sin \left (4 d x +4 c \right )}{32 d}\) | \(230\) |
| norman | \(\frac {-\frac {33 a^{3} x}{8}-\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}+\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8}+\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}+\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}-\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {33 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8}+\frac {21 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {27 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}+\frac {79 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 d}+\frac {25 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}-\frac {83 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {45 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}-\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(294\) |
1/8*((-12*cos(2*d*x+2*c)-12)*ln(tan(1/2*d*x+1/2*c)-1)+(12*cos(2*d*x+2*c)+1 2)*ln(tan(1/2*d*x+1/2*c)+1)-33*d*x*cos(2*d*x+2*c)-33*d*x+sin(5*d*x+5*c)+1/ 8*sin(6*d*x+6*c)-2*sin(d*x+c)+225/8*sin(2*d*x+2*c)-9*sin(3*d*x+3*c)+9/4*si n(4*d*x+4*c))*a^3/d/(1+cos(2*d*x+2*c))
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=-\frac {33 \, a^{3} d x \cos \left (d x + c\right )^{2} - 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, a^{3} \cos \left (d x + c\right )^{5} + 8 \, a^{3} \cos \left (d x + c\right )^{4} + 7 \, a^{3} \cos \left (d x + c\right )^{3} - 24 \, a^{3} \cos \left (d x + c\right )^{2} + 24 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{2}} \]
-1/8*(33*a^3*d*x*cos(d*x + c)^2 - 6*a^3*cos(d*x + c)^2*log(sin(d*x + c) + 1) + 6*a^3*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - (2*a^3*cos(d*x + c)^5 + 8*a^3*cos(d*x + c)^4 + 7*a^3*cos(d*x + c)^3 - 24*a^3*cos(d*x + c)^2 + 24* a^3*cos(d*x + c) + 4*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=a^{3} \left (\int 3 \sin ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(3*sin(c + d*x)**4*sec(c + d*x), x) + Integral(3*sin(c + d*x )**4*sec(c + d*x)**2, x) + Integral(sin(c + d*x)**4*sec(c + d*x)**3, x) + Integral(sin(c + d*x)**4, x))
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.32 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=-\frac {16 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{3} - {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 48 \, {\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{3} + 8 \, a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, \sin \left (d x + c\right )\right )}}{32 \, d} \]
-1/32*(16*(2*sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1) + 6*sin(d*x + c))*a^3 - (12*d*x + 12*c + sin(4*d*x + 4*c) - 8*sin(2* d*x + 2*c))*a^3 + 48*(3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2* tan(d*x + c))*a^3 + 8*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) + 3*log(sin (d*x + c) + 1) - 3*log(sin(d*x + c) - 1) - 4*sin(d*x + c)))/d
Time = 0.40 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.30 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=-\frac {33 \, {\left (d x + c\right )} a^{3} - 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {8 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 81 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 79 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
-1/8*(33*(d*x + c)*a^3 - 12*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 12*a^ 3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 8*(5*a^3*tan(1/2*d*x + 1/2*c)^3 - 7 *a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(25*a^3*tan( 1/2*d*x + 1/2*c)^7 + 81*a^3*tan(1/2*d*x + 1/2*c)^5 + 79*a^3*tan(1/2*d*x + 1/2*c)^3 + 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 14.38 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48 \[ \int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx=\frac {3\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {33\,a^3\,x}{8}+\frac {-\frac {45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{4}-\frac {83\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {25\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {79\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {27\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\frac {21\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(3*a^3*atanh(tan(c/2 + (d*x)/2)))/d - (33*a^3*x)/8 + ((27*a^3*tan(c/2 + (d *x)/2)^3)/4 + (79*a^3*tan(c/2 + (d*x)/2)^5)/2 + (25*a^3*tan(c/2 + (d*x)/2) ^7)/2 - (83*a^3*tan(c/2 + (d*x)/2)^9)/4 - (45*a^3*tan(c/2 + (d*x)/2)^11)/4 + (21*a^3*tan(c/2 + (d*x)/2))/4)/(d*(2*tan(c/2 + (d*x)/2)^2 - tan(c/2 + ( d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^8 + 2*tan(c/2 + (d *x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))