Integrand size = 27, antiderivative size = 106 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {11 \text {arctanh}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac {2 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac {17 \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))} \]
-11/2*arctanh(cos(d*x+c))/a^3/d+3*cot(d*x+c)/a^3/d-1/2*cot(d*x+c)*csc(d*x+ c)/a^3/d+2/3*cos(d*x+c)/a^3/d/(1+sin(d*x+c))^2+17/3*cos(d*x+c)/a^3/d/(1+si n(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(106)=212\).
Time = 4.98 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.91 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-32 \sin \left (\frac {1}{2} (c+d x)\right )-3 \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right )^3 \sin \left (\frac {1}{2} (c+d x)\right )+16 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-272 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2+36 \cot \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3-132 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+132 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3-36 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \tan \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^3\right )}{24 a^3 d (1+\sin (c+d x))^3} \]
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-32*Sin[(c + d*x)/2] - 3*(1 + Co t[(c + d*x)/2])^3*Sin[(c + d*x)/2] + 16*(Cos[(c + d*x)/2] + Sin[(c + d*x)/ 2]) - 272*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 36*Co t[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 132*Log[Cos[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + 132*Log[Sin[(c + d*x)/2 ]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 36*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*Tan[(c + d*x)/2] + 3*Cos[(c + d*x)/2]*(1 + Tan[(c + d*x)/2]) ^3))/(24*a^3*d*(1 + Sin[c + d*x])^3)
Time = 0.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3353, 3042, 3445, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^3 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3353 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) (a-a \sin (c+d x))}{(\sin (c+d x) a+a)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a-a \sin (c+d x)}{\sin (c+d x)^3 (\sin (c+d x) a+a)^2}dx}{a^2}\) |
| \(\Big \downarrow \) 3445 |
| \(\displaystyle \frac {\int \left (\frac {\csc ^3(c+d x)}{a}-\frac {3 \csc ^2(c+d x)}{a}+\frac {5 \csc (c+d x)}{a}-\frac {5}{a (\sin (c+d x)+1)}-\frac {2}{a (\sin (c+d x)+1)^2}\right )dx}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {11 \text {arctanh}(\cos (c+d x))}{2 a d}+\frac {3 \cot (c+d x)}{a d}+\frac {17 \cos (c+d x)}{3 a d (\sin (c+d x)+1)}+\frac {2 \cos (c+d x)}{3 a d (\sin (c+d x)+1)^2}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}}{a^2}\) |
((-11*ArcTanh[Cos[c + d*x]])/(2*a*d) + (3*Cot[c + d*x])/(a*d) - (Cot[c + d *x]*Csc[c + d*x])/(2*a*d) + (2*Cos[c + d*x])/(3*a*d*(1 + Sin[c + d*x])^2) + (17*Cos[c + d*x])/(3*a*d*(1 + Sin[c + d*x])))/a^2
3.4.20.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/b^2 Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /; Fre eQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] || !IGtQ[ n, 0])
Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[si n[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; FreeQ[{ a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ [m] && IntegerQ[n]
Time = 0.53 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+22 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {32}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {56}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(117\) |
| default | \(\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {6}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+22 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {32}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {56}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{4 d \,a^{3}}\) | \(117\) |
| parallelrisch | \(\frac {132 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+564 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+942 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+502}{24 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}\) | \(123\) |
| risch | \(\frac {99 i {\mathrm e}^{5 i \left (d x +c \right )}+33 \,{\mathrm e}^{6 i \left (d x +c \right )}-210 i {\mathrm e}^{3 i \left (d x +c \right )}-154 \,{\mathrm e}^{4 i \left (d x +c \right )}+123 i {\mathrm e}^{i \left (d x +c \right )}+161 \,{\mathrm e}^{2 i \left (d x +c \right )}-52}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d \,a^{3}}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}+\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}\) | \(148\) |
| norman | \(\frac {\frac {33 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1}{8 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {7 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d a}+\frac {\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {215 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {151 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {865 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}+\frac {557 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}\) | \(207\) |
1/4/d/a^3*(1/2*tan(1/2*d*x+1/2*c)^2-6*tan(1/2*d*x+1/2*c)-1/2/tan(1/2*d*x+1 /2*c)^2+6/tan(1/2*d*x+1/2*c)+22*ln(tan(1/2*d*x+1/2*c))+32/3/(tan(1/2*d*x+1 /2*c)+1)^3-16/(tan(1/2*d*x+1/2*c)+1)^2+56/(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (98) = 196\).
Time = 0.27 (sec) , antiderivative size = 365, normalized size of antiderivative = 3.44 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {104 \, \cos \left (d x + c\right )^{4} + 142 \, \cos \left (d x + c\right )^{3} - 90 \, \cos \left (d x + c\right )^{2} + 33 \, {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (\cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \cos \left (d x + c\right ) + 2\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (52 \, \cos \left (d x + c\right )^{3} - 19 \, \cos \left (d x + c\right )^{2} - 64 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) - 136 \, \cos \left (d x + c\right ) - 8}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - a^{3} d \cos \left (d x + c\right )^{3} - 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right ) + 2 \, a^{3} d - {\left (a^{3} d \cos \left (d x + c\right )^{3} + 2 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - 2 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
-1/12*(104*cos(d*x + c)^4 + 142*cos(d*x + c)^3 - 90*cos(d*x + c)^2 + 33*(c os(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 - (cos(d*x + c)^3 + 2*co s(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(1/2* cos(d*x + c) + 1/2) - 33*(cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c) ^2 - (cos(d*x + c)^3 + 2*cos(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(-1/2*cos(d*x + c) + 1/2) + 2*(52*cos(d*x + c)^3 - 1 9*cos(d*x + c)^2 - 64*cos(d*x + c) + 4)*sin(d*x + c) - 136*cos(d*x + c) - 8)/(a^3*d*cos(d*x + c)^4 - a^3*d*cos(d*x + c)^3 - 3*a^3*d*cos(d*x + c)^2 + a^3*d*cos(d*x + c) + 2*a^3*d - (a^3*d*cos(d*x + c)^3 + 2*a^3*d*cos(d*x + c)^2 - a^3*d*cos(d*x + c) - 2*a^3*d)*sin(d*x + c))
\[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Integral(cos(c + d*x)**2*csc(c + d*x)**3/(sin(c + d*x)**3 + 3*sin(c + d*x) **2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (98) = 196\).
Time = 0.26 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.33 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {403 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {681 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {372 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3}{\frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {3 \, {\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{a^{3}} + \frac {132 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{24 \, d} \]
1/24*((27*sin(d*x + c)/(cos(d*x + c) + 1) + 403*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 681*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 372*sin(d*x + c)^4/( cos(d*x + c) + 1)^4 - 3)/(a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^3* sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a^3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5) - 3*(12*sin(d*x + c)/(cos (d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a^3 + 132*log(sin(d* x + c)/(cos(d*x + c) + 1))/a^3)/d
Time = 0.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {132 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, {\left (66 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {3 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6}} + \frac {16 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}}}{24 \, d} \]
1/24*(132*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 3*(66*tan(1/2*d*x + 1/2*c)^ 2 - 12*tan(1/2*d*x + 1/2*c) + 1)/(a^3*tan(1/2*d*x + 1/2*c)^2) + 3*(a^3*tan (1/2*d*x + 1/2*c)^2 - 12*a^3*tan(1/2*d*x + 1/2*c))/a^6 + 16*(21*tan(1/2*d* x + 1/2*c)^2 + 36*tan(1/2*d*x + 1/2*c) + 19)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^3))/d
Time = 9.57 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}+\frac {62\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {227\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {403\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {1}{2}}{d\,\left (4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+12\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^3\,d} \]
tan(c/2 + (d*x)/2)^2/(8*a^3*d) + (11*log(tan(c/2 + (d*x)/2)))/(2*a^3*d) + ((9*tan(c/2 + (d*x)/2))/2 + (403*tan(c/2 + (d*x)/2)^2)/6 + (227*tan(c/2 + (d*x)/2)^3)/2 + 62*tan(c/2 + (d*x)/2)^4 - 1/2)/(d*(4*a^3*tan(c/2 + (d*x)/2 )^2 + 12*a^3*tan(c/2 + (d*x)/2)^3 + 12*a^3*tan(c/2 + (d*x)/2)^4 + 4*a^3*ta n(c/2 + (d*x)/2)^5)) - (3*tan(c/2 + (d*x)/2))/(2*a^3*d)