\(\int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx\) [177]

Optimal result

Integrand size = 17, antiderivative size = 85 \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {b x^2}{2}+x \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b} \]

[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 = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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]

-1/2*(b*x^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. \(929\) vs. \(2(85)=170\).

Time = 4.89 (sec) , antiderivative size = 929, normalized size of antiderivative = 10.93 \[ \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 ^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 )} \]

[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*(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] + (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)*Cos[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)*Cos[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*S
in[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[b*x]*((1 - I*c)*Cos[a + b*x] + (-I + c)*Sin[a + b*x]))/(2*Cos[a
] - (2*I)*Sin[a])]*Sec[b*x]^2)/(-I + Tan[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)]*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. 586 vs. \(2 (76 ) = 152\).

Time = 1.57 (sec) , antiderivative size = 587, normalized size of antiderivative = 6.91

method	result	size
default	\(\pi x -\frac {-\frac {\operatorname {arccot}\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 \operatorname {arccot}\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 {\operatorname {arccot}\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}+\frac {\operatorname {arccot}\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 \operatorname {arccot}\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 {\operatorname {arccot}\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}+\left (i c -1\right )^{2} \left (-\frac {-\frac {i \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )^{2}}{4}+\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}}{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 )}\)	\(587\)
parts	\(\pi x -\frac {-\frac {\operatorname {arccot}\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 \operatorname {arccot}\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 {\operatorname {arccot}\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}+\frac {\operatorname {arccot}\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 \operatorname {arccot}\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 {\operatorname {arccot}\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}+\left (i c -1\right )^{2} \left (-\frac {-\frac {i \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )^{2}}{4}+\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}}{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 )}\)	\(587\)
derivativedivides	\(\frac {-\frac {\pi \ln \left (4 c^{2}+4 \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) c +\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right ) c^{2}}{2 \left (2 i+2 c \right )}-\frac {i \pi \ln \left (4 c^{2}+4 \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) c +\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right ) c}{2 i+2 c}+\frac {\pi \ln \left (4 c^{2}+4 \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) c +\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right )}{4 i+4 c}+\frac {i \pi \arctan \left (\cot \left (b x +a \right ) \left (i c -1\right )+c \right ) c^{2}}{2 i+2 c}-\frac {2 \pi \arctan \left (\cot \left (b x +a \right ) \left (i c -1\right )+c \right ) c}{2 i+2 c}-\frac {i \pi \arctan \left (\cot \left (b x +a \right ) \left (i c -1\right )+c \right )}{2 i+2 c}+\frac {\pi \ln \left (\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right ) c^{2}}{4 i+4 c}+\frac {i \pi \ln \left (\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right ) c}{2 i+2 c}-\frac {\pi \ln \left (\left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )^{2}+1\right )}{2 \left (2 i+2 c \right )}+\frac {i \pi \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) c^{2}}{2 i+2 c}-\frac {2 \pi \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right ) c}{2 i+2 c}-\frac {i \pi \arctan \left (-c +\cot \left (b x +a \right ) \left (i c -1\right )\right )}{2 i+2 c}-\left (i c -1\right )^{2} \left (\frac {\operatorname {arccot}\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}-\frac {\operatorname {arccot}\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 {-\frac {i \ln \left (-i+\cot \left (b x +a \right ) \left (i c -1\right )-c \right )^{2}}{4}+\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}}{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 )}\)	\(849\)
risch	\(\text {Expression too large to display}\)	\(1244\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.36 \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {2 \, b^{2} x^{2} - 4 \, \pi b x - 2 i \, b x \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + i}\right ) - 2 \, a^{2} + 2 \, {\left (i \, b x + i \, a\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]

[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*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 + 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**4 + 3*_t0**2*I*c*exp(2*I*a) - _t0**2*exp(2*I*a) + 2*c
**2*exp(4*I*a) + I*c*exp(4*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[b,c,_t0,exp(I*a)]

Maxima [F(-2)]

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

[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)