\(\int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx\) [181]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5274, 2215, 2221, 2317, 2438} \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {b x^2}{2} \]

[In]

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

[Out]

(b*x^2)/2 + x*ArcCot[c - (1 + I*c)*Cot[a + b*x]] + (I/2)*x*Log[1 + I*c*E^((2*I)*a + (2*I)*b*x)] + PolyLog[2, (
-I)*c*E^((2*I)*a + (2*I)*b*x)]/(4*b)

Rule 2215

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] - Dist[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 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 5274

Int[ArcCot[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcCot[c + d*Cot[a + b*x]], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = x \cot ^{-1}(c-(1+i c) \cot (a+b x))+(i b) \int \frac {x}{-i (-1-i c)+c-c e^{2 i a+2 i b x}} \, dx \\ & = \frac {b x^2}{2}+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+(b c) \int \frac {e^{2 i a+2 i b x} x}{-i (-1-i c)+c-c e^{2 i a+2 i 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 {1}{2} i \int \log \left (1-\frac {c e^{2 i a+2 i b x}}{-i (-1-i c)+c}\right ) \, 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 {\text {Subst}\left (\int \frac {\log \left (1-\frac {c x}{-i (-1-i c)+c}\right )}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{4 b} \\ & = \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} \\ \end{align*}

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 = 2.40 (sec) , antiderivative size = 872, normalized size of antiderivative = 10.14 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=x \cot ^{-1}(c+(-1-i c) \cot (a+b x))-\frac {i x \csc (a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-i \log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))+i \operatorname {PolyLog}\left (2,\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)) \left (-2 i b x-\log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-\frac {\log (1-i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log (1+i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \sec ^2(b x)}{1+i \tan (b x)}-2 b x \tan (b x)-i \log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (1+i \tan (b x)) \tan (b x)+\frac {i \log \left (1-\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right ) \sec ^2(b x)}{-i+\tan (b x)}+\frac {i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right )} \]

[In]

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

[Out]

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*L
og[(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 - (Sec[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)*Sin[a])*(-I + Tan[b*x]))/
2]*Sec[b*x]^2)/(-I + Tan[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)]*Sec[b*x]^2)/(I + Tan[b*x])))

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 = 1.37 (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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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+\left (i c +1\right ) \cot \left (b x +a \right )-c \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+\left (i c +1\right ) \cot \left (b x +a \right )-c \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+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2 i-2 c}-\left (i c +1\right )^{2} \left (-\frac {\frac {i \ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )^{2}}{4}-\frac {i \left (\left (\ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )-\ln \left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right ) \ln \left (-\frac {i \left (i-\left (i c +1\right ) \cot \left (b x +a \right )+c \right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right )}{2}}{2 \left (i-c \right )}+\frac {\frac {i \left (\operatorname {dilog}\left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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 (-\left (i c +1\right ) \cot \left (b x +a \right )-c +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+\left (i c +1\right ) \cot \left (b x +a \right )-c \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+\left (i c +1\right ) \cot \left (b x +a \right )-c \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+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2 i-2 c}-\left (i c +1\right )^{2} \left (-\frac {\frac {i \ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )^{2}}{4}-\frac {i \left (\left (\ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )-\ln \left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right ) \ln \left (-\frac {i \left (i-\left (i c +1\right ) \cot \left (b x +a \right )+c \right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right )}{2}}{2 \left (i-c \right )}+\frac {\frac {i \left (\operatorname {dilog}\left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{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 (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2 i-2 c}+\frac {\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right ) \ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right )}{2 i-2 c}-\frac {\frac {i \ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )^{2}}{4}-\frac {i \left (\left (\ln \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )-\ln \left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right ) \ln \left (-\frac {i \left (i-\left (i c +1\right ) \cot \left (b x +a \right )+c \right )}{2}\right )-\operatorname {dilog}\left (-\frac {i \left (i+\left (i c +1\right ) \cot \left (b x +a \right )-c \right )}{2}\right )\right )}{2}}{2 \left (i-c \right )}+\frac {\frac {i \left (\operatorname {dilog}\left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {-i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{-2 i+2 c}\right )\right )}{2}-\frac {i \left (\operatorname {dilog}\left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{2 c}\right )+\ln \left (-\left (i c +1\right ) \cot \left (b x +a \right )-c +i\right ) \ln \left (\frac {i-\left (i c +1\right ) \cot \left (b x +a \right )+c}{2 c}\right )\right )}{2}}{2 i-2 c}\right )-\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}+\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}}{b \left (i c +1\right )}\)	\(884\)
risch	\(\text {Expression too large to display}\)	\(1244\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.35 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {2 \, b^{2} x^{2} + 4 \, \pi b x + 2 i \, b x \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 \, a^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]

[In]

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

[Out]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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