Integrand size = 15, antiderivative size = 223 \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )-\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1+\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right ) \]
arccot(c*x)*ln(2/(1-I*c*x))-1/2*arccot(c*x)*ln(2*I*c*(I-x)/(1-c)/(1-I*c*x) )-1/2*arccot(c*x)*ln(-2*I*c*(I+x)/(1+c)/(1-I*c*x))-1/2*I*polylog(2,-I/c/x) +1/2*I*polylog(2,I/c/x)+1/2*I*polylog(2,1-2/(1-I*c*x))-1/4*I*polylog(2,1-2 *I*c*(I-x)/(1-c)/(1-I*c*x))-1/4*I*polylog(2,1+2*I*c*(I+x)/(1+c)/(1-I*c*x))
Time = 2.04 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.47 \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\frac {1}{2} \left (-i \left (\cot ^{-1}(c x) \left (\cot ^{-1}(c x)+2 i \log \left (1+e^{2 i \cot ^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(c x)}\right )\right )+\frac {(-1+c) (1+c) \left (i \cot ^{-1}(c x)^2+2 i \arcsin \left (\sqrt {\frac {1}{1-c^2}}\right ) \arctan \left (\frac {\sqrt {c^2}}{c x}\right )-\left (\cot ^{-1}(c x)-\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (\frac {-1+\left (-1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}-c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )-\left (\cot ^{-1}(c x)+\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (-\frac {1+\left (1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}+c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )+\frac {1}{2} i \left (\operatorname {PolyLog}\left (2,\frac {\left (1+c^2-2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )\right )\right )}{-1+c^2}\right ) \]
((-I)*(ArcCot[c*x]*(ArcCot[c*x] + (2*I)*Log[1 + E^((2*I)*ArcCot[c*x])]) + PolyLog[2, -E^((2*I)*ArcCot[c*x])]) + ((-1 + c)*(1 + c)*(I*ArcCot[c*x]^2 + (2*I)*ArcSin[Sqrt[(1 - c^2)^(-1)]]*ArcTan[Sqrt[c^2]/(c*x)] - (ArcCot[c*x] - ArcSin[Sqrt[(1 - c^2)^(-1)]])*Log[(-1 + (-1 + 2*Sqrt[c^2])*E^((2*I)*Arc Cot[c*x]) - c^2*(-1 + E^((2*I)*ArcCot[c*x])))/(-1 + c^2)] - (ArcCot[c*x] + ArcSin[Sqrt[(1 - c^2)^(-1)]])*Log[-((1 + (1 + 2*Sqrt[c^2])*E^((2*I)*ArcCo t[c*x]) + c^2*(-1 + E^((2*I)*ArcCot[c*x])))/(-1 + c^2))] + (I/2)*(PolyLog[ 2, ((1 + c^2 - 2*Sqrt[c^2])*E^((2*I)*ArcCot[c*x]))/(-1 + c^2)] + PolyLog[2 , ((1 + c^2 + 2*Sqrt[c^2])*E^((2*I)*ArcCot[c*x]))/(-1 + c^2)])))/(-1 + c^2 ))/2
Time = 0.48 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5464, 2009}
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.
| \(\displaystyle \int \frac {\cot ^{-1}(c x)}{x \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 5464 |
| \(\displaystyle \int \left (\frac {\cot ^{-1}(c x)}{x}-\frac {x \cot ^{-1}(c x)}{x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 i c (x+i)}{(c+1) (1-i c x)}+1\right )+\log \left (\frac {2}{1-i c x}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x)\) |
ArcCot[c*x]*Log[2/(1 - I*c*x)] - (ArcCot[c*x]*Log[((2*I)*c*(I - x))/((1 - c)*(1 - I*c*x))])/2 - (ArcCot[c*x]*Log[((-2*I)*c*(I + x))/((1 + c)*(1 - I* c*x))])/2 - (I/2)*PolyLog[2, (-I)/(c*x)] + (I/2)*PolyLog[2, I/(c*x)] + (I/ 2)*PolyLog[2, 1 - 2/(1 - I*c*x)] - (I/4)*PolyLog[2, 1 - ((2*I)*c*(I - x))/ ((1 - c)*(1 - I*c*x))] - (I/4)*PolyLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*( 1 - I*c*x))]
3.1.49.3.1 Defintions of rubi rules used
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a + b*ArcCot[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] && !(EqQ[m, 1] && NeQ[a, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.82 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.88
| method | result | size |
| parts | \(\operatorname {arccot}\left (c x \right ) \ln \left (x \right )-\frac {\ln \left (x^{2}+1\right ) \operatorname {arccot}\left (c x \right )}{2}+\frac {c \left (-\frac {i \ln \left (x \right ) \left (\ln \left (i c x +1\right )-\ln \left (-i c x +1\right )\right )}{c}-\frac {i \left (\operatorname {dilog}\left (i c x +1\right )-\operatorname {dilog}\left (-i c x +1\right )\right )}{c}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2} c^{2}+1\right )}{\sum }\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (x^{2}+1\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \underline {\hspace {1.25 ex}}\alpha +x}{\underline {\hspace {1.25 ex}}\alpha \left (1+c \right )}\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {c \underline {\hspace {1.25 ex}}\alpha -x}{\underline {\hspace {1.25 ex}}\alpha \left (c -1\right )}\right )-\operatorname {dilog}\left (\frac {c \underline {\hspace {1.25 ex}}\alpha +x}{\underline {\hspace {1.25 ex}}\alpha \left (1+c \right )}\right )-\operatorname {dilog}\left (\frac {c \underline {\hspace {1.25 ex}}\alpha -x}{\underline {\hspace {1.25 ex}}\alpha \left (c -1\right )}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{2 c^{2}}\right )}{2}\) | \(197\) |
| risch | \(-\frac {\pi \ln \left (c^{2} x^{2}+c^{2}\right )}{4}+\frac {\pi \ln \left (-i c x \right )}{2}+\frac {i \ln \left (-i c x +1\right ) \ln \left (\frac {-i c x -c}{-c -1}\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {-i c x -c}{-c -1}\right )}{4}+\frac {i \ln \left (-i c x +1\right ) \ln \left (\frac {-i c x +c}{c -1}\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {-i c x +c}{c -1}\right )}{4}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (\frac {i c x -c}{-c -1}\right )}{4}-\frac {i \ln \left (i c x +1\right ) \ln \left (\frac {i c x -c}{-c -1}\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {i c x +c}{c -1}\right )}{4}-\frac {i \ln \left (i c x +1\right ) \ln \left (\frac {i c x +c}{c -1}\right )}{4}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}\) | \(232\) |
| derivativedivides | \(-\frac {\operatorname {arccot}\left (c x \right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{2}+\operatorname {arccot}\left (c x \right ) \ln \left (c x \right )+\frac {c^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2}}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2}}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{c^{2}}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{c^{2}}+\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x -i\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x +i\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}\right )}{2}\) | \(393\) |
| default | \(-\frac {\operatorname {arccot}\left (c x \right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{2}+\operatorname {arccot}\left (c x \right ) \ln \left (c x \right )+\frac {c^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2}}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2}}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{c^{2}}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{c^{2}}+\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x -i\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x +i\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}\right )}{2}\) | \(393\) |
arccot(c*x)*ln(x)-1/2*ln(x^2+1)*arccot(c*x)+1/2*c*(-I*ln(x)*(ln(1+I*c*x)-l n(1-I*c*x))/c-I*(dilog(1+I*c*x)-dilog(1-I*c*x))/c-1/2/c^2*sum(1/_alpha*(ln (x-_alpha)*ln(x^2+1)-ln(x-_alpha)*ln((_alpha*c+x)/_alpha/(1+c))-ln(x-_alph a)*ln((_alpha*c-x)/_alpha/(c-1))-dilog((_alpha*c+x)/_alpha/(1+c))-dilog((_ alpha*c-x)/_alpha/(c-1))),_alpha=RootOf(_Z^2*c^2+1)))
\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (c x \right )}}{x \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (c\,x\right )}{x\,\left (x^2+1\right )} \,d x \]