Integrand size = 25, antiderivative size = 94 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a x}{2}+\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {3 a \cos (c+d x)}{2 d}-\frac {3 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^2(c+d x)}{2 d} \]
-3/2*a*x+3/2*a*arctanh(cos(d*x+c))/d-3/2*a*cos(d*x+c)/d-3/2*a*cot(d*x+c)/d +1/2*a*cos(d*x+c)^2*cot(d*x+c)/d-1/2*a*cos(d*x+c)*cot(d*x+c)^2/d
Time = 0.51 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \left (12 c+12 d x+8 \cos (c+d x)+8 \cot (c+d x)+\csc ^2\left (\frac {1}{2} (c+d x)\right )-12 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+12 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )+2 \sin (2 (c+d x))\right )}{8 d} \]
-1/8*(a*(12*c + 12*d*x + 8*Cos[c + d*x] + 8*Cot[c + d*x] + Csc[(c + d*x)/2 ]^2 - 12*Log[Cos[(c + d*x)/2]] + 12*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/ 2]^2 + 2*Sin[2*(c + d*x)]))/d
Time = 0.39 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3317, 3042, 25, 3071, 252, 262, 216, 3072, 252, 262, 219}
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.
| \(\displaystyle \int \cos (c+d x) \cot ^3(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^2(c+d x) \cot ^2(c+d x)dx+a \int \cos (c+d x) \cot ^3(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx+a \int -\sin \left (c+d x+\frac {\pi }{2}\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 3071 |
| \(\displaystyle -\frac {a \int \frac {\cot ^4(c+d x)}{\left (\cot ^2(c+d x)+1\right )^2}d\cot (c+d x)}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} \int \frac {\cot ^2(c+d x)}{\cot ^2(c+d x)+1}d\cot (c+d x)-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} \left (\cot (c+d x)-\int \frac {1}{\cot ^2(c+d x)+1}d\cot (c+d x)\right )-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -a \int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^3dx-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle -\frac {a \int \frac {\cos ^4(c+d x)}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)}{d}-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle -\frac {a \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \int \frac {\cos ^2(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)\right )}{d}-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {a \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)-\cos (c+d x)\right )\right )}{d}-\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a \left (\frac {3}{2} (\cot (c+d x)-\arctan (\cot (c+d x)))-\frac {\cot ^3(c+d x)}{2 \left (\cot ^2(c+d x)+1\right )}\right )}{d}-\frac {a \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{d}\) |
-((a*((-3*(ArcTanh[Cos[c + d*x]] - Cos[c + d*x]))/2 + Cos[c + d*x]^3/(2*(1 - Cos[c + d*x]^2))))/d) - (a*((3*(-ArcTan[Cot[c + d*x]] + Cot[c + d*x]))/ 2 - Cot[c + d*x]^3/(2*(1 + Cot[c + d*x]^2))))/d
3.4.71.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[I nt[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(116\) |
| default | \(\frac {a \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+a \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )}{d}\) | \(116\) |
| parallelrisch | \(\frac {\left (-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\left (\frac {\left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {1}{2}\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-\frac {\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{9}-\frac {\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{9}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}+\left (\cos \left (d x +c \right )-\frac {3}{4}\right ) \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 d x}{2}\right ) a}{d}\) | \(133\) |
| risch | \(-\frac {3 a x}{2}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a \left ({\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}-2 i {\mathrm e}^{2 i \left (d x +c \right )}+2 i\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}\) | \(155\) |
| norman | \(\frac {-\frac {a}{8 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {3 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {3 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {3 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-3 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {5 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {5 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\) | \(214\) |
1/d*(a*(-1/sin(d*x+c)*cos(d*x+c)^5-(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c )-3/2*d*x-3/2*c)+a*(-1/2/sin(d*x+c)^2*cos(d*x+c)^5-1/2*cos(d*x+c)^3-3/2*co s(d*x+c)-3/2*ln(csc(d*x+c)-cot(d*x+c))))
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.48 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {6 \, a d x \cos \left (d x + c\right )^{2} + 4 \, a \cos \left (d x + c\right )^{3} - 6 \, a d x - 6 \, a \cos \left (d x + c\right ) - 3 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
-1/4*(6*a*d*x*cos(d*x + c)^2 + 4*a*cos(d*x + c)^3 - 6*a*d*x - 6*a*cos(d*x + c) - 3*(a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 3*(a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2) + 2*(a*cos(d*x + c)^3 - 3*a*cos( d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {2 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a - a {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
-1/4*(2*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*a - a*(2*cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(c os(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d
Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.73 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, {\left (d x + c\right )} a - 12 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2}}}{8 \, d} \]
1/8*(a*tan(1/2*d*x + 1/2*c)^2 - 12*(d*x + c)*a - 12*a*log(abs(tan(1/2*d*x + 1/2*c))) + 4*a*tan(1/2*d*x + 1/2*c) + (6*a*tan(1/2*d*x + 1/2*c)^6 + 4*a* tan(1/2*d*x + 1/2*c)^5 - 5*a*tan(1/2*d*x + 1/2*c)^4 - 16*a*tan(1/2*d*x + 1 /2*c)^3 - 12*a*tan(1/2*d*x + 1/2*c)^2 - 4*a*tan(1/2*d*x + 1/2*c) - a)/(tan (1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))^2)/d
Time = 10.43 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.54 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+8\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {3\,a\,\mathrm {atan}\left (\frac {9\,a^2}{9\,a^2-9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9\,a^2-9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
(a*tan(c/2 + (d*x)/2))/(2*d) - (a/2 + 2*a*tan(c/2 + (d*x)/2) + 9*a*tan(c/2 + (d*x)/2)^2 + 8*a*tan(c/2 + (d*x)/2)^3 + (17*a*tan(c/2 + (d*x)/2)^4)/2 - 2*a*tan(c/2 + (d*x)/2)^5)/(d*(4*tan(c/2 + (d*x)/2)^2 + 8*tan(c/2 + (d*x)/ 2)^4 + 4*tan(c/2 + (d*x)/2)^6)) + (a*tan(c/2 + (d*x)/2)^2)/(8*d) - (3*a*lo g(tan(c/2 + (d*x)/2)))/(2*d) - (3*a*atan((9*a^2)/(9*a^2 - 9*a^2*tan(c/2 + (d*x)/2)) + (9*a^2*tan(c/2 + (d*x)/2))/(9*a^2 - 9*a^2*tan(c/2 + (d*x)/2))) )/d