Integrand size = 17, antiderivative size = 85 \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {b x^2}{2}+x \arctan (c+(1-i c) \cot (a+b x))+\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )+\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b} \]
1/2*b*x^2-x*arctan(-c-(1-I*c)*cot(b*x+a))+1/2*I*x*ln(1-I*c*exp(2*I*a+2*I*b *x))+1/4*polylog(2,I*c*exp(2*I*a+2*I*b*x))/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(929\) vs. \(2(85)=170\).
Time = 11.67 (sec) , antiderivative size = 929, normalized size of antiderivative = 10.93 \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=x \arctan (c+(1-i c) \cot (a+b x))-\frac {i x \csc ^2(a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-i \log \left (\frac {\sec (b x) ((1-i c) \cos (a+b x)+(-i+c) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))+i \operatorname {PolyLog}\left (2,\frac {\sec (b x) ((-i+c) \cos (a)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} \sec (b x) ((1+i c) \cos (a)-(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) (1+i c+(i+c) \cot (a+b x)) \left (2 i b x+\log \left (1-\frac {\sec (b x) ((-i+c) \cos (a)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )+\log \left (1+\frac {1}{2} \sec (b x) ((-1-i c) \cos (a)+(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )+\frac {(-i+c) \cos (a+b x) (\log (1-i \tan (b x))-\log (1+i \tan (b x)))}{(i+c) \cos (a+b x)+(1+i c) \sin (a+b x)}+\frac {(i+c) (\log (1-i \tan (b x))-\log (1+i \tan (b x))) \sin (a+b x)}{(1-i c) \cos (a+b x)+(-i+c) \sin (a+b x)}+2 b x \tan (b x)+i \log \left (1-\frac {\sec (b x) ((-i+c) \cos (a)+i (i+c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)-i \log \left (1+\frac {1}{2} \sec (b x) ((-1-i c) \cos (a)+(i+c) \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right ) \tan (b x)-i \log (1-i \tan (b x)) \tan (b x)+i \cos ^2(a) \log (1+i \tan (b x)) \tan (b x)+i \log (1+i \tan (b x)) \sin ^2(a) \tan (b x)+\frac {i \log \left (\frac {\sec (b x) ((1-i c) \cos (a+b x)+(-i+c) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right ) \sec ^2(b x)}{-i+\tan (b x)}-\frac {i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right )} \]
x*ArcTan[c + (1 - I*c)*Cot[a + b*x]] - (I*x*Csc[a + b*x]^2*(2*b*x*Log[2*Co s[b*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - I*Log[(Sec[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a ] - (2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[ 2*b*x]] + I*PolyLog[2, (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)*Sin[a])*(Cos [a + b*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, (Sec[b*x]*((1 + I*c)*Co s[a] - (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2])*(Cos[b*x] - I* Sin[b*x])*(Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(1 + I*c + (I + c)* Cot[a + b*x])*((2*I)*b*x + Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c)* Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] + Log[1 + (Sec[b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2] + ((-I + c)*Cos[a + b*x]*(Log[1 - I*Tan[b*x]] - Log[1 + I*Tan[b*x]]))/((I + c)*Co s[a + b*x] + (1 + I*c)*Sin[a + b*x]) + ((I + c)*(Log[1 - I*Tan[b*x]] - Log [1 + I*Tan[b*x]])*Sin[a + b*x])/((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]) + 2*b*x*Tan[b*x] + I*Log[1 - (Sec[b*x]*((-I + c)*Cos[a] + I*(I + c) *Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)]*Tan[b*x] - I*Log[1 + (Sec [b*x]*((-1 - I*c)*Cos[a] + (I + c)*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]) )/2]*Tan[b*x] - I*Log[1 - I*Tan[b*x]]*Tan[b*x] + I*Cos[a]^2*Log[1 + I*Tan[ b*x]]*Tan[b*x] + I*Log[1 + I*Tan[b*x]]*Sin[a]^2*Tan[b*x] + (I*Log[(Sec[...
Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5688, 25, 2615, 2620, 2715, 2838}
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.
| \(\displaystyle \int \arctan (c+(1-i c) \cot (a+b x)) \, dx\) |
| \(\Big \downarrow \) 5688 |
| \(\displaystyle x \arctan (c+(1-i c) \cot (a+b x))-i b \int -\frac {x}{e^{2 i a+2 i b x} c+i}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i b \int \frac {x}{e^{2 i a+2 i b x} c+i}dx+x \arctan (c+(1-i c) \cot (a+b x))\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle i b \left (i c \int \frac {e^{2 i a+2 i b x} x}{e^{2 i a+2 i b x} c+i}dx-\frac {i x^2}{2}\right )+x \arctan (c+(1-i c) \cot (a+b x))\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i b \left (i c \left (\frac {i \int \log \left (1-i c e^{2 i a+2 i b x}\right )dx}{2 b c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )+x \arctan (c+(1-i c) \cot (a+b x))\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle i b \left (i c \left (\frac {\int e^{-2 i a-2 i b x} \log \left (1-i c e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2 c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )+x \arctan (c+(1-i c) \cot (a+b x))\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle x \arctan (c+(1-i c) \cot (a+b x))+i b \left (i c \left (-\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b^2 c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )\) |
x*ArcTan[c + (1 - I*c)*Cot[a + b*x]] + I*b*((-1/2*I)*x^2 + I*c*(((-1/2*I)* x*Log[1 - I*c*E^((2*I)*a + (2*I)*b*x)])/(b*c) - PolyLog[2, I*c*E^((2*I)*a + (2*I)*b*x)]/(4*b^2*c)))
3.1.67.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcT an[c + d*Cot[a + b*x]], x] - Simp[I*b Int[x/(c - I*d - c*E^(2*I*a + 2*I*b *x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (73 ) = 146\).
Time = 1.07 (sec) , antiderivative size = 584, normalized size of antiderivative = 6.87
| method | result | size |
| derivativedivides | \(-\frac {\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) c^{2}}{2 i+2 c}+\frac {2 i \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) c}{2 i+2 c}-\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )}{2 i+2 c}-\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) c^{2}}{2 i+2 c}-\frac {2 i \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) c}{2 i+2 c}+\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right )}{2 i+2 c}-\left (i c -1\right )^{2} \left (-\frac {\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (\cot \left (b x +a \right ) \left (i c -1\right )-c +i\right )}{2}\right )+\ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) \ln \left (-\frac {i \left (\cot \left (b x +a \right ) \left (i c -1\right )-c +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )^{2}}{4}}{2 \left (i+c \right )}+\frac {\frac {i \left (\operatorname {dilog}\left (-\frac {\cot \left (b x +a \right ) \left (i c -1\right )-c +i}{2 c}\right )+\ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) \ln \left (-\frac {\cot \left (b x +a \right ) \left (i c -1\right )-c +i}{2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {-i+\cot \left (b x +a \right ) \left (i c -1\right )-c}{-2 i-2 c}\right )+\ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) \ln \left (\frac {-i+\cot \left (b x +a \right ) \left (i c -1\right )-c}{-2 i-2 c}\right )\right )}{2}}{2 i+2 c}\right )}{b \left (i c -1\right )}\) | \(584\) |
| default | \(-\frac {\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) c^{2}}{2 i+2 c}+\frac {2 i \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) c}{2 i+2 c}-\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )}{2 i+2 c}-\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) c^{2}}{2 i+2 c}-\frac {2 i \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) c}{2 i+2 c}+\frac {\arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) \ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right )}{2 i+2 c}-\left (i c -1\right )^{2} \left (-\frac {\frac {i \left (\operatorname {dilog}\left (-\frac {i \left (\cot \left (b x +a \right ) \left (i c -1\right )-c +i\right )}{2}\right )+\ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right ) \ln \left (-\frac {i \left (\cot \left (b x +a \right ) \left (i c -1\right )-c +i\right )}{2}\right )\right )}{2}-\frac {i \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )^{2}}{4}}{2 \left (i+c \right )}+\frac {\frac {i \left (\operatorname {dilog}\left (-\frac {\cot \left (b x +a \right ) \left (i c -1\right )-c +i}{2 c}\right )+\ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) \ln \left (-\frac {\cot \left (b x +a \right ) \left (i c -1\right )-c +i}{2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {-i+\cot \left (b x +a \right ) \left (i c -1\right )-c}{-2 i-2 c}\right )+\ln \left (\cot \left (b x +a \right ) \left (i c -1\right )+c +i\right ) \ln \left (\frac {-i+\cot \left (b x +a \right ) \left (i c -1\right )-c}{-2 i-2 c}\right )\right )}{2}}{2 i+2 c}\right )}{b \left (i c -1\right )}\) | \(584\) |
| risch | \(\text {Expression too large to display}\) | \(1248\) |
-1/b/(-1+I*c)*(arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(-I+cot(b*x+a)*( -1+I*c)-c)*c^2+2*I*arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(-I+cot(b*x+ a)*(-1+I*c)-c)*c-arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(-I+cot(b*x+a) *(-1+I*c)-c)-arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(cot(b*x+a)*(-1+I* c)+c+I)*c^2-2*I*arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(cot(b*x+a)*(-1 +I*c)+c+I)*c+arctan(-c+cot(b*x+a)*(-1+I*c))/(2*I+2*c)*ln(cot(b*x+a)*(-1+I* c)+c+I)-(-1+I*c)^2*(-1/2/(I+c)*(1/2*I*(dilog(-1/2*I*(cot(b*x+a)*(-1+I*c)-c +I))+ln(-I+cot(b*x+a)*(-1+I*c)-c)*ln(-1/2*I*(cot(b*x+a)*(-1+I*c)-c+I)))-1/ 4*I*ln(-I+cot(b*x+a)*(-1+I*c)-c)^2)+1/2/(I+c)*(1/2*I*(dilog(-1/2*(cot(b*x+ a)*(-1+I*c)-c+I)/c)+ln(cot(b*x+a)*(-1+I*c)+c+I)*ln(-1/2*(cot(b*x+a)*(-1+I* c)-c+I)/c))-1/2*I*(dilog((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))+ln(cot(b*x +a)*(-1+I*c)+c+I)*ln((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))))))
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.32 \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=\frac {2 \, b^{2} x^{2} + 2 i \, b x \log \left (-\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}\right ) - 2 \, a^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]
1/4*(2*b^2*x^2 + 2*I*b*x*log(-(c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) + I)) - 2*a^2 - 2*(-I*b*x - I*a)*log(-I*c*e^(2*I*b*x + 2*I*a) + 1) - 2*I*a*log((c*e^(2*I*b*x + 2*I*a) + I)/c) + dilog(I*c*e^(2*I*b*x + 2*I*a) ))/b
Exception generated. \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
Exception raised: CoercionFailed >> Cannot convert _t0**2 - exp(2*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[b,_t0,exp(I*a)]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (64) = 128\).
Time = 0.36 (sec) , antiderivative size = 458, normalized size of antiderivative = 5.39 \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=-\frac {{\left (i \, c - 1\right )} {\left (\frac {4 i \, {\left (b x + a\right )} \log \left (-\frac {2 \, {\left (i \, c^{2} - {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) - 2 \, c - i\right )}}{-2 i \, c^{2} + 2 \, {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) - 2 i}\right )}{i \, c - 1} - \frac {i \, {\left (4 \, {\left (b x + a\right )} {\left (\log \left (-i \, c^{2} + {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) + 2 \, c + i\right ) - \log \left (-i \, c^{2} + {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) - i\right )\right )} - 2 i \, \log \left (-i \, c^{2} + {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) + 2 \, c + i\right ) \log \left (-\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c + i}{2 \, c} + 1\right ) + 2 i \, \log \left (-i \, c^{2} + {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) + 2 \, c + i\right ) \log \left (\tan \left (b x + a\right ) - i\right ) - 2 i \, \log \left (\frac {1}{2} \, {\left (c - i\right )} \tan \left (b x + a\right ) - \frac {1}{2} i \, c + \frac {1}{2}\right ) \log \left (\tan \left (b x + a\right ) - i\right ) - i \, \log \left (\tan \left (b x + a\right ) - i\right )^{2} - 2 i \, \log \left (c^{2} + 1\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) + 2 i \, \log \left (\tan \left (b x + a\right ) - i\right ) \log \left (-\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) + 2 i \, \log \left (c^{2} + 1\right ) \log \left (-i \, \tan \left (b x + a\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-\frac {1}{2} \, {\left (c - i\right )} \tan \left (b x + a\right ) + \frac {1}{2} i \, c + \frac {1}{2}\right ) - 2 i \, {\rm Li}_2\left (\frac {{\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c + i}{2 \, c}\right ) + 2 i \, {\rm Li}_2\left (\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{i \, c - 1}\right )} - 8 \, {\left (b x + a\right )} \arctan \left (c + \frac {-i \, c + 1}{\tan \left (b x + a\right )}\right ) + 4 \, {\left (-i \, b x - i \, a\right )} \log \left (-\frac {2 \, {\left (i \, c^{2} - {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) - 2 \, c - i\right )}}{-2 i \, c^{2} + 2 \, {\left (c^{2} + 1\right )} \tan \left (b x + a\right ) - 2 i}\right )}{8 \, b} \]
-1/8*((I*c - 1)*(4*I*(b*x + a)*log(-2*(I*c^2 - (c^2 + 1)*tan(b*x + a) - 2* c - I)/(-2*I*c^2 + 2*(c^2 + 1)*tan(b*x + a) - 2*I))/(I*c - 1) - I*(4*(b*x + a)*(log(-I*c^2 + (c^2 + 1)*tan(b*x + a) + 2*c + I) - log(-I*c^2 + (c^2 + 1)*tan(b*x + a) - I)) - 2*I*log(-I*c^2 + (c^2 + 1)*tan(b*x + a) + 2*c + I )*log(-1/2*((I*c + 1)*tan(b*x + a) + c + I)/c + 1) + 2*I*log(-I*c^2 + (c^2 + 1)*tan(b*x + a) + 2*c + I)*log(tan(b*x + a) - I) - 2*I*log(1/2*(c - I)* tan(b*x + a) - 1/2*I*c + 1/2)*log(tan(b*x + a) - I) - I*log(tan(b*x + a) - I)^2 - 2*I*log(c^2 + 1)*log(I*tan(b*x + a) + 1) + 2*I*log(tan(b*x + a) - I)*log(-1/2*I*tan(b*x + a) + 1/2) + 2*I*log(c^2 + 1)*log(-I*tan(b*x + a) + 1) - 2*I*dilog(-1/2*(c - I)*tan(b*x + a) + 1/2*I*c + 1/2) - 2*I*dilog(1/2 *((I*c + 1)*tan(b*x + a) + c + I)/c) + 2*I*dilog(1/2*I*tan(b*x + a) + 1/2) )/(I*c - 1)) - 8*(b*x + a)*arctan(c + (-I*c + 1)/tan(b*x + a)) + 4*(-I*b*x - I*a)*log(-2*(I*c^2 - (c^2 + 1)*tan(b*x + a) - 2*c - I)/(-2*I*c^2 + 2*(c ^2 + 1)*tan(b*x + a) - 2*I)))/b
\[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=\int { -\arctan \left (-{\left (-i \, c + 1\right )} \cot \left (b x + a\right ) - c\right ) \,d x } \]
Timed out. \[ \int \arctan (c+(1-i c) \cot (a+b x)) \, dx=\int \mathrm {atan}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right ) \,d x \]