Integrand size = 21, antiderivative size = 163 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {5 a^2 \cot (c+d x)}{d}-\frac {3 a^2 \cot ^3(c+d x)}{d}-\frac {7 a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}+\frac {a^2 \tan (c+d x)}{d} \]
2*a^2*arctanh(sin(d*x+c))/d-5*a^2*cot(d*x+c)/d-3*a^2*cot(d*x+c)^3/d-7/5*a^ 2*cot(d*x+c)^5/d-2/7*a^2*cot(d*x+c)^7/d-2*a^2*csc(d*x+c)/d-2/3*a^2*csc(d*x +c)^3/d-2/5*a^2*csc(d*x+c)^5/d-2/7*a^2*csc(d*x+c)^7/d+a^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(428\) vs. \(2(163)=326\).
Time = 2.96 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.63 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2 \cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (1+\sec (c+d x))^2 \left (-6881280 \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6881280 \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-32 \csc (2 c) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x) (-9856 \sin (2 c)+17288 \sin (d x)-29056 \sin (2 d x)-7264 \sin (c-d x)+14208 \sin (c+d x)-19536 \sin (2 (c+d x))+7104 \sin (3 (c+d x))+7104 \sin (4 (c+d x))-7104 \sin (5 (c+d x))+1776 \sin (6 (c+d x))+17288 \sin (2 c+d x)+20384 \sin (3 c+d x)-23771 \sin (c+2 d x)+7104 \sin (2 (c+2 d x))-23771 \sin (3 c+2 d x)-8960 \sin (4 c+2 d x)+19984 \sin (c+3 d x)+8644 \sin (2 c+3 d x)+8644 \sin (4 c+3 d x)-6160 \sin (5 c+3 d x)+8644 \sin (3 c+4 d x)+8644 \sin (5 c+4 d x)+6720 \sin (6 c+4 d x)-12144 \sin (3 c+5 d x)-8644 \sin (4 c+5 d x)-8644 \sin (6 c+5 d x)-1680 \sin (7 c+5 d x)+3456 \sin (4 c+6 d x)+2161 \sin (5 c+6 d x)+2161 \sin (7 c+6 d x))\right )}{13762560 d} \]
(a^2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(-6881280*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6881280*Cos[c + d*x]*Log [Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 32*Csc[2*c]*Csc[(c + d*x)/2]^4*Csc [c + d*x]^3*(-9856*Sin[2*c] + 17288*Sin[d*x] - 29056*Sin[2*d*x] - 7264*Sin [c - d*x] + 14208*Sin[c + d*x] - 19536*Sin[2*(c + d*x)] + 7104*Sin[3*(c + d*x)] + 7104*Sin[4*(c + d*x)] - 7104*Sin[5*(c + d*x)] + 1776*Sin[6*(c + d* x)] + 17288*Sin[2*c + d*x] + 20384*Sin[3*c + d*x] - 23771*Sin[c + 2*d*x] + 7104*Sin[2*(c + 2*d*x)] - 23771*Sin[3*c + 2*d*x] - 8960*Sin[4*c + 2*d*x] + 19984*Sin[c + 3*d*x] + 8644*Sin[2*c + 3*d*x] + 8644*Sin[4*c + 3*d*x] - 6 160*Sin[5*c + 3*d*x] + 8644*Sin[3*c + 4*d*x] + 8644*Sin[5*c + 4*d*x] + 672 0*Sin[6*c + 4*d*x] - 12144*Sin[3*c + 5*d*x] - 8644*Sin[4*c + 5*d*x] - 8644 *Sin[6*c + 5*d*x] - 1680*Sin[7*c + 5*d*x] + 3456*Sin[4*c + 6*d*x] + 2161*S in[5*c + 6*d*x] + 2161*Sin[7*c + 6*d*x])))/(13762560*d)
Time = 0.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, 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, 3352, 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.
| \(\displaystyle \int \csc ^8(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^8(c+d x) \sec ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}{\sin \left (c+d x-\frac {\pi }{2}\right )^2 \cos \left (c+d x-\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \csc ^8(c+d x)+a^2 \csc ^8(c+d x) \sec ^2(c+d x)+2 a^2 \csc ^8(c+d x) \sec (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^2 \tan (c+d x)}{d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {7 a^2 \cot ^5(c+d x)}{5 d}-\frac {3 a^2 \cot ^3(c+d x)}{d}-\frac {5 a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d}\) |
(2*a^2*ArcTanh[Sin[c + d*x]])/d - (5*a^2*Cot[c + d*x])/d - (3*a^2*Cot[c + d*x]^3)/d - (7*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x]^3)/(3*d) - (2*a^2*Csc[c + d*x ]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) + (a^2*Tan[c + d*x])/d
3.1.36.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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 = 1.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.17
| method | result | size |
| parallelrisch | \(\frac {a^{2} \left (15 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+174 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+910 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1141 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-18375 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6720 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+9380 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3360 d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3360 d}\) | \(191\) |
| norman | \(\frac {\frac {a^{2}}{224 d}+\frac {29 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{560 d}+\frac {163 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{480 d}+\frac {67 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{24 d}+\frac {13 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{48 d}+\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{96 d}-\frac {175 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(192\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {1}{7 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {8}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {16}{35 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {64}{35 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {128 \cot \left (d x +c \right )}{35}\right )+2 a^{2} \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\) | \(193\) |
| default | \(\frac {a^{2} \left (-\frac {1}{7 \sin \left (d x +c \right )^{7} \cos \left (d x +c \right )}-\frac {8}{35 \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )}-\frac {16}{35 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {64}{35 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {128 \cot \left (d x +c \right )}{35}\right )+2 a^{2} \left (-\frac {1}{7 \sin \left (d x +c \right )^{7}}-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {16}{35}-\frac {\csc \left (d x +c \right )^{6}}{7}-\frac {6 \csc \left (d x +c \right )^{4}}{35}-\frac {8 \csc \left (d x +c \right )^{2}}{35}\right ) \cot \left (d x +c \right )}{d}\) | \(193\) |
| risch | \(-\frac {4 i a^{2} \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-420 \,{\mathrm e}^{10 i \left (d x +c \right )}+385 \,{\mathrm e}^{9 i \left (d x +c \right )}+560 \,{\mathrm e}^{8 i \left (d x +c \right )}-1274 \,{\mathrm e}^{7 i \left (d x +c \right )}+616 \,{\mathrm e}^{6 i \left (d x +c \right )}+454 \,{\mathrm e}^{5 i \left (d x +c \right )}-1816 \,{\mathrm e}^{4 i \left (d x +c \right )}+1249 \,{\mathrm e}^{3 i \left (d x +c \right )}+444 \,{\mathrm e}^{2 i \left (d x +c \right )}-759 \,{\mathrm e}^{i \left (d x +c \right )}+216\right )}{105 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(215\) |
a^2*(15*cot(1/2*d*x+1/2*c)^7+35*tan(1/2*d*x+1/2*c)^5+174*cot(1/2*d*x+1/2*c )^5+910*tan(1/2*d*x+1/2*c)^3-6720*ln(tan(1/2*d*x+1/2*c)-1)*tan(1/2*d*x+1/2 *c)^2+6720*ln(tan(1/2*d*x+1/2*c)+1)*tan(1/2*d*x+1/2*c)^2+1141*cot(1/2*d*x+ 1/2*c)^3-18375*tan(1/2*d*x+1/2*c)+6720*ln(tan(1/2*d*x+1/2*c)-1)-6720*ln(ta n(1/2*d*x+1/2*c)+1)+9380*cot(1/2*d*x+1/2*c))/(3360*d*tan(1/2*d*x+1/2*c)^2- 3360*d)
Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.67 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {432 \, a^{2} \cos \left (d x + c\right )^{6} - 654 \, a^{2} \cos \left (d x + c\right )^{5} - 636 \, a^{2} \cos \left (d x + c\right )^{4} + 1226 \, a^{2} \cos \left (d x + c\right )^{3} + 74 \, a^{2} \cos \left (d x + c\right )^{2} - 562 \, a^{2} \cos \left (d x + c\right ) - 105 \, {\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, {\left (a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 105 \, a^{2}}{105 \, {\left (d \cos \left (d x + c\right )^{5} - 2 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
-1/105*(432*a^2*cos(d*x + c)^6 - 654*a^2*cos(d*x + c)^5 - 636*a^2*cos(d*x + c)^4 + 1226*a^2*cos(d*x + c)^3 + 74*a^2*cos(d*x + c)^2 - 562*a^2*cos(d*x + c) - 105*(a^2*cos(d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c )^2 - a^2*cos(d*x + c))*log(sin(d*x + c) + 1)*sin(d*x + c) + 105*(a^2*cos( d*x + c)^5 - 2*a^2*cos(d*x + c)^4 + 2*a^2*cos(d*x + c)^2 - a^2*cos(d*x + c ))*log(-sin(d*x + c) + 1)*sin(d*x + c) + 105*a^2)/((d*cos(d*x + c)^5 - 2*d *cos(d*x + c)^4 + 2*d*cos(d*x + c)^2 - d*cos(d*x + c))*sin(d*x + c))
Timed out. \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.07 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^{2} {\left (\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} + 15\right )}}{\sin \left (d x + c\right )^{7}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3 \, a^{2} {\left (\frac {140 \, \tan \left (d x + c\right )^{6} + 70 \, \tan \left (d x + c\right )^{4} + 28 \, \tan \left (d x + c\right )^{2} + 5}{\tan \left (d x + c\right )^{7}} - 35 \, \tan \left (d x + c\right )\right )} + \frac {3 \, {\left (35 \, \tan \left (d x + c\right )^{6} + 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \]
-1/105*(a^2*(2*(105*sin(d*x + c)^6 + 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 + 15)/sin(d*x + c)^7 - 105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1)) + 3*a^2*((140*tan(d*x + c)^6 + 70*tan(d*x + c)^4 + 28*tan(d*x + c)^2 + 5)/tan(d*x + c)^7 - 35*tan(d*x + c)) + 3*(35*tan(d*x + c)^6 + 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 + 5)*a^2/tan(d*x + c)^7)/d
Time = 0.35 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6720 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 6720 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {6720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {10710 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{3360 \, d} \]
1/3360*(35*a^2*tan(1/2*d*x + 1/2*c)^3 + 6720*a^2*log(abs(tan(1/2*d*x + 1/2 *c) + 1)) - 6720*a^2*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 945*a^2*tan(1/2* d*x + 1/2*c) - 6720*a^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - (10710*a^2*tan(1/2*d*x + 1/2*c)^6 + 1330*a^2*tan(1/2*d*x + 1/2*c)^4 + 18 9*a^2*tan(1/2*d*x + 1/2*c)^2 + 15*a^2)/tan(1/2*d*x + 1/2*c)^7)/d
Time = 13.47 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98 \[ \int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32\,d}-\frac {-166\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {268\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {163\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {58\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {a^2}{7}}{d\,\left (32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\right )} \]
(a^2*tan(c/2 + (d*x)/2)^3)/(96*d) + (4*a^2*atanh(tan(c/2 + (d*x)/2)))/d + (9*a^2*tan(c/2 + (d*x)/2))/(32*d) - ((58*a^2*tan(c/2 + (d*x)/2)^2)/35 + (1 63*a^2*tan(c/2 + (d*x)/2)^4)/15 + (268*a^2*tan(c/2 + (d*x)/2)^6)/3 - 166*a ^2*tan(c/2 + (d*x)/2)^8 + a^2/7)/(d*(32*tan(c/2 + (d*x)/2)^7 - 32*tan(c/2 + (d*x)/2)^9))