3.71 \(\int \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx\)
Optimal. Leaf size=86 \[ -\frac{\text{PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+x \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac{b x^2}{2} \]
[Out]
-(b*x^2)/2 + x*ArcTan[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)
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Rubi [A] time = 0.133581, antiderivative size = 86, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used =
{5165, 2184, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+x \tan ^{-1}(c-(1+i c) \cot (a+b x))-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
[In]
Int[ArcTan[c + (-1 - I*c)*Cot[a + b*x]],x]
[Out]
-(b*x^2)/2 + x*ArcTan[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 5165
Int[ArcTan[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*ArcTan[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]
Rule 2184
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 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279
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 2391
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]
Rubi steps
\begin{align*} \int \tan ^{-1}(c+(-1-i c) \cot (a+b x)) \, dx &=x \tan ^{-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 \tan ^{-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 \tan ^{-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 \tan ^{-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{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 \tan ^{-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{Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}\\ \end{align*}
Mathematica [B] time = 12.7368, size = 872, normalized size = 10.14 \[ x \tan ^{-1}(c+(-i c-1) \cot (a+b x))+\frac{i x \csc (a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-i \log \left (\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right ) \log (i \tan (b x)+1)+i \text{PolyLog}(2,i \sin (2 b x)-\cos (2 b x))+i \text{PolyLog}\left (2,\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \text{PolyLog}\left (2,\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(\cot (a+b x)+i) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x)) \left (\frac{\log \left (\frac{1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((i c+1) \cos (a+b x)-(c+i) \sin (a+b x))\right ) \sec ^2(b x)}{i \tan (b x)+1}+\frac{i \log \left (1-\frac{1}{2} (\cos (a)+i \sin (a)) ((c+i) \cos (a)+(i c+1) \sin (a)) (\tan (b x)-i)\right ) \sec ^2(b x)}{\tan (b x)-i}+\frac{i \log \left (\frac{\sec (b x) (\cos (a)-i \sin (a)) ((c-i) \cos (a+b x)+i (c+i) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{\tan (b x)+i}-2 i b x-\log \left (1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-2 b x \tan (b x)-i \log \left (1-\frac{\sec (b x) ((c+i) \cos (a)+(i c+1) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (i \tan (b x)+1) \tan (b x)-\frac{\log (1-i \tan (b x)) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}+\frac{\log (i \tan (b x)+1) ((c+i) \cos (a+b x)+(i c+1) \sin (a+b x))}{(c-i) \cos (a+b x)+i (c+i) \sin (a+b x)}\right )} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcTan[c + (-1 - I*c)*Cot[a + b*x]],x]
[Out]
x*ArcTan[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])))
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Maple [B] time = 0.136, size = 1753, normalized size = 20.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-arctan(-c-(-1-I*c)*cot(b*x+a)),x)
[Out]
1/2/b/(1+I*c)/(I-c)*dilog((-(1+I*c)*cot(b*x+a)+c-I)/(-2*I+2*c))*c-1/2/b/(1+I*c)/(I-c)*dilog(1/2*(-(1+I*c)*cot(
b*x+a)+c+I)/c)*c+1/8*I/b/(1+I*c)/(I-c)*ln((1+I*c)*cot(b*x+a)-c+I)^2-1/4*I/b/(1+I*c)/(I-c)*dilog((-(1+I*c)*cot(
b*x+a)+c-I)/(-2*I+2*c))+1/4*I/b/(1+I*c)/(I-c)*dilog(1/2*(-(1+I*c)*cot(b*x+a)+c+I)/c)+1/4*I/b/(1+I*c)/(I-c)*dil
og(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))+1/b/(1+I*c)*arctan(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(-(1+I*c)*cot(b*x+a)
-c+I)-1/2/b/(1+I*c)/(I-c)*dilog(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))*c-1/4/b/(1+I*c)/(I-c)*ln((1+I*c)*cot(b*x+a)-c
+I)^2*c+1/4*I/b/(1+I*c)/(I-c)*dilog((-(1+I*c)*cot(b*x+a)+c-I)/(-2*I+2*c))*c^2-1/b/(1+I*c)*arctan(-c+(1+I*c)*co
t(b*x+a))/(2*I-2*c)*ln((1+I*c)*cot(b*x+a)-c+I)+1/4*I/b/(1+I*c)/(I-c)*ln(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))*ln(-1
/2*I*(-(1+I*c)*cot(b*x+a)+c+I))-1/4*I/b/(1+I*c)/(I-c)*ln(-1/2*I*(-(1+I*c)*cot(b*x+a)+c+I))*ln((1+I*c)*cot(b*x+
a)-c+I)-1/4*I/b/(1+I*c)/(I-c)*dilog(1/2*(-(1+I*c)*cot(b*x+a)+c+I)/c)*c^2-1/4*I/b/(1+I*c)/(I-c)*ln((-(1+I*c)*co
t(b*x+a)+c-I)/(-2*I+2*c))*ln(-(1+I*c)*cot(b*x+a)-c+I)+1/4*I/b/(1+I*c)/(I-c)*ln(1/2*(-(1+I*c)*cot(b*x+a)+c+I)/c
)*ln(-(1+I*c)*cot(b*x+a)-c+I)-1/b/(1+I*c)*arctan(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln(-(1+I*c)*cot(b*x+a)-c+I)*
c^2+1/b/(1+I*c)*arctan(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln((1+I*c)*cot(b*x+a)-c+I)*c^2+1/2/b/(1+I*c)/(I-c)*ln(
-1/2*I*(-(1+I*c)*cot(b*x+a)+c+I))*ln((1+I*c)*cot(b*x+a)-c+I)*c+1/2/b/(1+I*c)/(I-c)*ln((-(1+I*c)*cot(b*x+a)+c-I
)/(-2*I+2*c))*ln(-(1+I*c)*cot(b*x+a)-c+I)*c-1/2/b/(1+I*c)/(I-c)*ln(1/2*(-(1+I*c)*cot(b*x+a)+c+I)/c)*ln(-(1+I*c
)*cot(b*x+a)-c+I)*c-1/4*I/b/(1+I*c)/(I-c)*dilog(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))*c^2-1/8*I/b/(1+I*c)/(I-c)*ln(
(1+I*c)*cot(b*x+a)-c+I)^2*c^2-1/2/b/(1+I*c)/(I-c)*ln(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))*ln(-1/2*I*(-(1+I*c)*cot(
b*x+a)+c+I))*c-1/4*I/b/(1+I*c)/(I-c)*ln(-1/2*I*((1+I*c)*cot(b*x+a)-c+I))*ln(-1/2*I*(-(1+I*c)*cot(b*x+a)+c+I))*
c^2+1/4*I/b/(1+I*c)/(I-c)*ln(-1/2*I*(-(1+I*c)*cot(b*x+a)+c+I))*ln((1+I*c)*cot(b*x+a)-c+I)*c^2+1/4*I/b/(1+I*c)/
(I-c)*ln((-(1+I*c)*cot(b*x+a)+c-I)/(-2*I+2*c))*ln(-(1+I*c)*cot(b*x+a)-c+I)*c^2-1/4*I/b/(1+I*c)/(I-c)*ln(1/2*(-
(1+I*c)*cot(b*x+a)+c+I)/c)*ln(-(1+I*c)*cot(b*x+a)-c+I)*c^2+2*I/b/(1+I*c)*arctan(-c+(1+I*c)*cot(b*x+a))/(2*I-2*
c)*ln(-(1+I*c)*cot(b*x+a)-c+I)*c-2*I/b/(1+I*c)*arctan(-c+(1+I*c)*cot(b*x+a))/(2*I-2*c)*ln((1+I*c)*cot(b*x+a)-c
+I)*c
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Maxima [B] time = 1.59481, size = 616, normalized size = 7.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-arctan(-c-(-1-I*c)*cot(b*x+a)),x, algorithm="maxima")
[Out]
-1/8*((I*c + 1)*(4*I*(b*x + a)*log((-2*I*c^2 + 2*(c^2 + 1)*tan(b*x + a) - 2*I)/(-2*I*c^2 + 2*(c^2 + 1)*tan(b*x
+ a) - 4*c + 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)*lo
g(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 + 2*(c^2 + 1)*tan(b*x + a) - 2*I)/(-2*I*c^2 + 2*(c^2 + 1)*tan(b*x + a) - 4*c
+ 2*I)))/b
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Fricas [A] time = 1.98425, size = 313, normalized size = 3.64 \begin{align*} -\frac{2 \, b^{2} x^{2} - 2 i \, b x \log \left (-\frac{{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} - i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c - i}\right ) - 2 \, a^{2} -{\left (-2 i \, b x - 2 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-arctan(-c-(-1-I*c)*cot(b*x+a)),x, algorithm="fricas")
[Out]
-1/4*(2*b^2*x^2 - 2*I*b*x*log(-(c*e^(2*I*b*x + 2*I*a) - I)*e^(-2*I*b*x - 2*I*a)/(c - I)) - 2*a^2 - (-2*I*b*x -
2*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
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} i b \int \frac{x}{c e^{2 i a} e^{2 i b x} - i}\, dx + \frac{i x \log{\left (- i c - \frac{i c}{e^{2 i a} e^{2 i b x} - 1} - \frac{i c e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} + 1 - \frac{1}{e^{2 i a} e^{2 i b x} - 1} - \frac{e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} \right )}}{2} - \frac{i x \log{\left (i c + \frac{i c}{e^{2 i a} e^{2 i b x} - 1} + \frac{i c e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} + 1 + \frac{1}{e^{2 i a} e^{2 i b x} - 1} + \frac{e^{i a} e^{i b x}}{e^{i a} e^{i b x} - e^{- i a} e^{- i b x}} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-atan(-c-(-1-I*c)*cot(b*x+a)),x)
[Out]
I*b*Integral(x/(c*exp(2*I*a)*exp(2*I*b*x) - I), x) + I*x*log(-I*c - I*c/(exp(2*I*a)*exp(2*I*b*x) - 1) - I*c*ex
p(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)) + 1 - 1/(exp(2*I*a)*exp(2*I*b*x) - 1) - exp(I*
a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)))/2 - I*x*log(I*c + I*c/(exp(2*I*a)*exp(2*I*b*x) -
1) + I*c*exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)) + 1 + 1/(exp(2*I*a)*exp(2*I*b*x) -
1) + exp(I*a)*exp(I*b*x)/(exp(I*a)*exp(I*b*x) - exp(-I*a)*exp(-I*b*x)))/2
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\arctan \left (-{\left (-i \, c - 1\right )} \cot \left (b x + a\right ) - c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-arctan(-c-(-1-I*c)*cot(b*x+a)),x, algorithm="giac")
[Out]
integrate(-arctan(-(-I*c - 1)*cot(b*x + a) - c), x)