\(\int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx\) [33]

Optimal result

Integrand size = 18, antiderivative size = 86 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {b x^2}{2}+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b} \] Output:

1/2*b*x^2+x*(Pi-arccot(-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
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(872\) vs. \(2(86)=172\).

Time = 2.04 (sec) , antiderivative size = 872, normalized size of antiderivative = 10.14 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=x \cot ^{-1}(c+(-1-i c) \cot (a+b x))-\frac {i x \csc (a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-i \log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))+i \operatorname {PolyLog}\left (2,\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)) \left (-2 i b x-\log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-\frac {\log (1-i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log (1+i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \sec ^2(b x)}{1+i \tan (b x)}-2 b x \tan (b x)-i \log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (1+i \tan (b x)) \tan (b x)+\frac {i \log \left (1-\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right ) \sec ^2(b x)}{-i+\tan (b x)}+\frac {i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right )} \] Input:

Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]
 

Output:

x*ArcCot[c + (-1 - I*c)*Cot[a + b*x]] - (I*x*Csc[a + b*x]*(2*b*x*Log[2*Cos 
[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] + I*(I + c)*Sin[a + b*x]))/(2*c)]*Log[1 - I*Tan[b*x]] - 
I*Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[ 
a + b*x]))/2]*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] + (1 + I*c)*Sin[a])*(Cos[a + b 
*x] - I*Sin[a + b*x]))/(2*c)] - I*PolyLog[2, ((Cos[a] + I*Sin[a])*((I + c) 
*Cos[a] + (1 + I*c)*Sin[a])*(-I + Tan[b*x]))/2])*(Cos[b*x] - I*Sin[b*x])*( 
Cos[b*x] + I*Sin[b*x]))/((I + Cot[a + b*x])*((-I + c)*Cos[a + b*x] + I*(I 
+ c)*Sin[a + b*x])*((-2*I)*b*x - Log[1 - (Sec[b*x]*((I + c)*Cos[a] + (1 + 
I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x]))/(2*c)] - (Log[1 - I*Tan[b*x] 
]*((I + c)*Cos[a + b*x] + (1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] 
+ I*(I + c)*Sin[a + b*x]) + (Log[1 + I*Tan[b*x]]*((I + c)*Cos[a + b*x] + ( 
1 + I*c)*Sin[a + b*x]))/((-I + c)*Cos[a + b*x] + I*(I + c)*Sin[a + b*x]) + 
 (Log[(Sec[b*x]*(Cos[a] + I*Sin[a])*((1 + I*c)*Cos[a + b*x] - (I + c)*Sin[ 
a + b*x]))/2]*Sec[b*x]^2)/(1 + I*Tan[b*x]) - 2*b*x*Tan[b*x] - I*Log[1 - (S 
ec[b*x]*((I + c)*Cos[a] + (1 + I*c)*Sin[a])*(Cos[a + b*x] - I*Sin[a + b*x] 
))/(2*c)]*Tan[b*x] + I*Log[1 - I*Tan[b*x]]*Tan[b*x] - I*Log[1 + I*Tan[b*x] 
]*Tan[b*x] + (I*Log[1 - ((Cos[a] + I*Sin[a])*((I + c)*Cos[a] + (1 + I*c)*S 
in[a])*(-I + Tan[b*x]))/2]*Sec[b*x]^2)/(-I + Tan[b*x]) + (I*Log[(Sec[b*...
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5689, 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.

\(\displaystyle \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx\)

\(\Big \downarrow \) 5689

\(\displaystyle i b \int \frac {x}{i-c e^{2 i a+2 i b x}}dx+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\)

\(\Big \downarrow \) 2615

\(\displaystyle i b \left (-i c \int \frac {e^{2 i a+2 i b x} x}{i-c e^{2 i a+2 i b x}}dx-\frac {i x^2}{2}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\)

\(\Big \downarrow \) 2620

\(\displaystyle i b \left (-i c \left (\frac {i x \log \left (1+i c e^{2 i a+2 i b x}\right )}{2 b c}-\frac {i \int \log \left (i e^{2 i a+2 i b x} c+1\right )dx}{2 b c}\right )-\frac {i x^2}{2}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\)

\(\Big \downarrow \) 2715

\(\displaystyle i b \left (-i c \left (\frac {i x \log \left (1+i c e^{2 i a+2 i b x}\right )}{2 b c}-\frac {\int e^{-2 i a-2 i b x} \log \left (i e^{2 i a+2 i b x} c+1\right )de^{2 i a+2 i b x}}{4 b^2 c}\right )-\frac {i x^2}{2}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\)

\(\Big \downarrow \) 2838

\(\displaystyle i b \left (-i c \left (\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b^2 c}+\frac {i x \log \left (1+i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\)

Input:

Int[ArcCot[c - (1 + I*c)*Cot[a + b*x]],x]
 

Output:

x*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + I*b*((-1/2*I)*x^2 - I*c*(((I/2)*x*L 
og[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)))
 

Defintions of rubi rules used

rule 2615

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]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 5689

Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcC 
ot[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (75 ) = 150\).

Time = 1.13 (sec) , antiderivative size = 630, normalized size of antiderivative = 7.33

method	result	size
default	\(\pi x -\frac {\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) c^{2}}{2 i-2 c}-\frac {2 i \operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) c}{2 i-2 c}-\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right )}{2 i-2 c}-\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c^{2}}{2 i-2 c}+\frac {2 i \operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c}{2 i-2 c}+\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2 i-2 c}-\left (i c +1\right )^{2} \left (-\frac {-\frac {i \left (\left (\ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )-\ln \left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right ) \ln \left (-\frac {i \left (i+c -\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right )}{2}+\frac {i \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}}{4}}{2 \left (i-c \right )}+\frac {-\frac {i \left (\operatorname {dilog}\left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )\right )}{2}}{2 i-2 c}\right )}{b \left (i c +1\right )}\)	\(630\)
parts	\(\pi x -\frac {\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) c^{2}}{2 i-2 c}-\frac {2 i \operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) c}{2 i-2 c}-\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right )}{2 i-2 c}-\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c^{2}}{2 i-2 c}+\frac {2 i \operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c}{2 i-2 c}+\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2 i-2 c}-\left (i c +1\right )^{2} \left (-\frac {-\frac {i \left (\left (\ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )-\ln \left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right ) \ln \left (-\frac {i \left (i+c -\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right )}{2}+\frac {i \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}}{4}}{2 \left (i-c \right )}+\frac {-\frac {i \left (\operatorname {dilog}\left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )\right )}{2}}{2 i-2 c}\right )}{b \left (i c +1\right )}\)	\(630\)
derivativedivides	\(\frac {\left (i c +1\right )^{2} \left (\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right )}{2 i-2 c}-\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2 i-2 c}-\frac {-\frac {i \left (\left (\ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )-\ln \left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right ) \ln \left (-\frac {i \left (i+c -\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2}\right )\right )}{2}+\frac {i \ln \left (i-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}}{4}}{2 \left (i-c \right )}+\frac {-\frac {i \left (\operatorname {dilog}\left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{2 c}\right )\right )}{2}+\frac {i \left (\operatorname {dilog}\left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )+\ln \left (-c -\left (i c +1\right ) \cot \left (b x +a \right )+i\right ) \ln \left (\frac {-i+c -\left (i c +1\right ) \cot \left (b x +a \right )}{-2 i+2 c}\right )\right )}{2}}{2 i-2 c}\right )+\frac {\pi \ln \left (4 c^{2}+4 \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c +\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right ) c^{2}}{4 i-4 c}-\frac {i \pi \ln \left (4 c^{2}+4 \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c +\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right ) c}{2 i-2 c}-\frac {\pi \ln \left (4 c^{2}+4 \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c +\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right )}{2 \left (2 i-2 c \right )}+\frac {i \pi \arctan \left (c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c^{2}}{2 i-2 c}+\frac {2 \pi \arctan \left (c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c}{2 i-2 c}-\frac {i \pi \arctan \left (c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2 i-2 c}-\frac {\pi \ln \left (\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right ) c^{2}}{2 \left (2 i-2 c \right )}+\frac {i \pi \ln \left (\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right ) c}{2 i-2 c}+\frac {\pi \ln \left (\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )^{2}+1\right )}{4 i-4 c}+\frac {i \pi \arctan \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c^{2}}{2 i-2 c}+\frac {2 \pi \arctan \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) c}{2 i-2 c}-\frac {i \pi \arctan \left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{2 i-2 c}}{b \left (i c +1\right )}\)	\(884\)
risch	\(\text {Expression too large to display}\)	\(1244\)

Input:

int(Pi-arccot(-c+(I*c+1)*cot(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

Pi*x-1/b/(I*c+1)*(arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(-c-(I*c+1)*co 
t(b*x+a)+I)*c^2-2*I*arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(-c-(I*c+1)* 
cot(b*x+a)+I)*c-arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(-c-(I*c+1)*cot( 
b*x+a)+I)-arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(I*c+1)*cot(b*x+a 
))*c^2+2*I*arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(I*c+1)*cot(b*x+ 
a))*c+arccot(-c+(I*c+1)*cot(b*x+a))/(2*I-2*c)*ln(I-c+(I*c+1)*cot(b*x+a))-( 
I*c+1)^2*(-1/2/(I-c)*(-1/2*I*((ln(I-c+(I*c+1)*cot(b*x+a))-ln(-1/2*I*(I-c+( 
I*c+1)*cot(b*x+a))))*ln(-1/2*I*(I+c-(I*c+1)*cot(b*x+a)))-dilog(-1/2*I*(I-c 
+(I*c+1)*cot(b*x+a))))+1/4*I*ln(I-c+(I*c+1)*cot(b*x+a))^2)+1/2/(I-c)*(-1/2 
*I*(dilog(1/2*(I+c-(I*c+1)*cot(b*x+a))/c)+ln(-c-(I*c+1)*cot(b*x+a)+I)*ln(1 
/2*(I+c-(I*c+1)*cot(b*x+a))/c))+1/2*I*(dilog((-I+c-(I*c+1)*cot(b*x+a))/(-2 
*I+2*c))+ln(-c-(I*c+1)*cot(b*x+a)+I)*ln((-I+c-(I*c+1)*cot(b*x+a))/(-2*I+2* 
c))))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.35 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {2 \, b^{2} x^{2} + 4 \, \pi b x + 2 i \, b x \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 \, a^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \] Input:

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="fricas")
 

Output:

1/4*(2*b^2*x^2 + 4*pi*b*x + 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
 

Sympy [F(-2)]

Exception generated. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \] Input:

integrate(pi-acot(-c+(1+I*c)*cot(b*x+a)),x)
 

Output:

Exception raised: CoercionFailed >> Cannot convert 2*_t0**2*c*exp(2*I*a) - 
 _t0**2*I*exp(2*I*a) - I of type <class 'sympy.core.add.Add'> to QQ_I[b,c, 
_t0,exp(I*a)]
 

Maxima [F(-2)]

Exception generated. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\text {Exception raised: ValueError} \] Input:

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c-1>0)', see `assume?` for more 
details)Is
 

Giac [F]

\[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\int { \pi - \operatorname {arccot}\left ({\left (i \, c + 1\right )} \cot \left (b x + a\right ) - c\right ) \,d x } \] Input:

integrate(pi-arccot(-c+(1+I*c)*cot(b*x+a)),x, algorithm="giac")
 

Output:

integrate(pi - arccot((I*c + 1)*cot(b*x + a) - c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\int \Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right ) \,d x \] Input:

int(Pi + acot(c - cot(a + b*x)*(c*1i + 1)),x)
 

Output:

int(Pi + acot(c - cot(a + b*x)*(c*1i + 1)), x)
 

Reduce [F]

\[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=-\left (\int \mathit {acot} \left (\cot \left (b x +a \right ) c i +\cot \left (b x +a \right )-c \right )d x \right )+\pi x \] Input:

int(Pi-acot(-c+(1+I*c)*cot(b*x+a)),x)
 

Output:

 - int(acot(cot(a + b*x)*c*i + cot(a + b*x) - c),x) + pi*x