Integrand size = 17, antiderivative size = 85 \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {b x^2}{2}+x \cot ^{-1}(c+(1-i c) \cot (a+b x))-\frac {1}{2} i x \log \left (1-i c e^{2 i a+2 i b x}\right )-\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b} \]
-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. \(929\) 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 )} \]
x*ArcCot[c + (1 - I*c)*Cot[a + b*x]] + (I*x*Csc[a + b*x]^2*(2*b*x*Log[2*Co s[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)*Co s[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)*Co s[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*Sin[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[...
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.26, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5689, 25, 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 |
| \(\displaystyle i b \int -\frac {x}{e^{2 i a+2 i b x} c+i}dx+x \cot ^{-1}(c+(1-i c) \cot (a+b x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \cot ^{-1}(c+(1-i c) \cot (a+b x))-i b \int \frac {x}{e^{2 i a+2 i b x} c+i}dx\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle x \cot ^{-1}(c+(1-i c) \cot (a+b x))-i b \left (i c \int \frac {e^{2 i a+2 i b x} x}{e^{2 i a+2 i b x} c+i}dx-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle x \cot ^{-1}(c+(1-i c) \cot (a+b x))-i b \left (i c \left (\frac {i \int \log \left (1-i c e^{2 i a+2 i b x}\right )dx}{2 b c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle x \cot ^{-1}(c+(1-i c) \cot (a+b x))-i b \left (i c \left (\frac {\int e^{-2 i a-2 i b x} \log \left (1-i c e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2 c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle x \cot ^{-1}(c+(1-i c) \cot (a+b x))-i b \left (i c \left (-\frac {\operatorname {PolyLog}\left (2,i c e^{2 i a+2 i b x}\right )}{4 b^2 c}-\frac {i x \log \left (1-i c e^{2 i a+2 i b x}\right )}{2 b c}\right )-\frac {i x^2}{2}\right )\) |
x*ArcCot[c + (1 - I*c)*Cot[a + b*x]] - I*b*((-1/2*I)*x^2 + I*c*(((-1/2*I)* x*Log[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)))
3.2.77.3.1 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. 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 | \(\text {Expression too large to display}\) | \(1244\) |
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*(co t(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(c ot(b*x+a)*(-1+I*c)+c+I)*ln((-I+cot(b*x+a)*(-1+I*c)-c)/(-2*I-2*c))))))
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.36 \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=-\frac {2 \, b^{2} x^{2} - 4 \, \pi b x - 2 i \, b x \log \left (\frac {{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c + i}\right ) - 2 \, a^{2} + 2 \, {\left (i \, b x + i \, a\right )} \log \left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 2 i \, a \log \left (\frac {c e^{\left (2 i \, b x + 2 i \, a\right )} + i}{c}\right ) + {\rm Li}_2\left (i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \]
-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
Exception generated. \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\text {Exception raised: CoercionFailed} \]
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 typ e <class 'sympy.core.add.Add'> to QQ_I[b,c,_t0,exp(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. \[ \int \cot ^{-1}(c+(1-i c) \cot (a+b x)) \, dx=\int \Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right ) \,d x \]