Integrand size = 18, antiderivative size = 86 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {b x^2}{2}+x \cot ^{-1}(c-(1+i c) \cot (a+b x))+\frac {1}{2} i x \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b} \]
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
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 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=x \cot ^{-1}(c+(-1-i c) \cot (a+b x))-\frac {i x \csc (a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \log (1-i \tan (b x))-i \log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))+i \operatorname {PolyLog}\left (2,\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)) \left (-2 i b x-\log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right )-\frac {\log (1-i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log (1+i \tan (b x)) ((i+c) \cos (a+b x)+(1+i c) \sin (a+b x))}{(-i+c) \cos (a+b x)+i (i+c) \sin (a+b x)}+\frac {\log \left (\frac {1}{2} \sec (b x) (\cos (a)+i \sin (a)) ((1+i c) \cos (a+b x)-(i+c) \sin (a+b x))\right ) \sec ^2(b x)}{1+i \tan (b x)}-2 b x \tan (b x)-i \log \left (1-\frac {\sec (b x) ((i+c) \cos (a)+(1+i c) \sin (a)) (\cos (a+b x)-i \sin (a+b x))}{2 c}\right ) \tan (b x)+i \log (1-i \tan (b x)) \tan (b x)-i \log (1+i \tan (b x)) \tan (b x)+\frac {i \log \left (1-\frac {1}{2} (\cos (a)+i \sin (a)) ((i+c) \cos (a)+(1+i c) \sin (a)) (-i+\tan (b x))\right ) \sec ^2(b x)}{-i+\tan (b x)}+\frac {i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) ((-i+c) \cos (a+b x)+i (i+c) \sin (a+b x))}{2 c}\right ) \sec ^2(b x)}{i+\tan (b x)}\right )} \]
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*...
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 |
\(\displaystyle i b \left (-i c \left (\frac {\operatorname {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b^2 c}+\frac {i x \log \left (1+i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )+x \cot ^{-1}(c-(1+i c) \cot (a+b x))\) |
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)))
3.2.81.3.1 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[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]
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 | \(\text {Expression too large to display}\) | \(1244\) |
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)*co t(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 )))))
Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.35 \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\frac {2 \, b^{2} x^{2} + 4 \, \pi b x + 2 i \, b x \log \left (\frac {{\left (c - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 \, a^{2} - 2 \, {\left (-i \, b x - i \, a\right )} \log \left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} - i}{c}\right ) + {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]
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
Exception generated. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
Exception raised: CoercionFailed >> Cannot convert -_t0**2*I + 2*c*exp(2*I *a) - I*exp(2*I*a) of type <class 'sympy.core.add.Add'> to QQ_I[b,c,_t0,ex p(I*a)]
Exception generated. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\int { \pi - \operatorname {arccot}\left ({\left (i \, c + 1\right )} \cot \left (b x + a\right ) - c\right ) \,d x } \]
Timed out. \[ \int \cot ^{-1}(c-(1+i c) \cot (a+b x)) \, dx=\int \Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right ) \,d x \]