Integrand size = 13, antiderivative size = 188 \[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=-\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,1+\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right ) \] Output:
-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(2,1-2/(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 = 1.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.64 \[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\frac {1}{2} \left (-i \cot ^{-1}(c x)^2-2 i \arcsin \left (\sqrt {\frac {1}{1-c^2}}\right ) \arctan \left (\frac {\sqrt {c^2}}{c x}\right )-2 \cot ^{-1}(c x) \log \left (1-e^{2 i \cot ^{-1}(c x)}\right )+\left (\cot ^{-1}(c x)-\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (\frac {-1+\left (-1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}-c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )+\left (\cot ^{-1}(c x)+\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (-\frac {1+\left (1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}+c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )+i \left (\cot ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c x)}\right )\right )-\frac {1}{2} i \left (\operatorname {PolyLog}\left (2,\frac {\left (1+c^2-2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )\right )\right ) \] Input:
Integrate[(x*ArcCot[c*x])/(1 + x^2),x]
Output:
((-I)*ArcCot[c*x]^2 - (2*I)*ArcSin[Sqrt[(1 - c^2)^(-1)]]*ArcTan[Sqrt[c^2]/ (c*x)] - 2*ArcCot[c*x]*Log[1 - E^((2*I)*ArcCot[c*x])] + (ArcCot[c*x] - Arc Sin[Sqrt[(1 - c^2)^(-1)]])*Log[(-1 + (-1 + 2*Sqrt[c^2])*E^((2*I)*ArcCot[c* x]) - c^2*(-1 + E^((2*I)*ArcCot[c*x])))/(-1 + c^2)] + (ArcCot[c*x] + ArcSi n[Sqrt[(1 - c^2)^(-1)]])*Log[-((1 + (1 + 2*Sqrt[c^2])*E^((2*I)*ArcCot[c*x] ) + c^2*(-1 + E^((2*I)*ArcCot[c*x])))/(-1 + c^2))] + I*(ArcCot[c*x]^2 + Po lyLog[2, E^((2*I)*ArcCot[c*x])]) - (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)]))/2
Time = 0.42 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.
| \(\displaystyle \int \frac {x \cot ^{-1}(c x)}{x^2+1} \, dx\) |
| \(\Big \downarrow \) 5464 |
| \(\displaystyle \int \left (\frac {\cot ^{-1}(c x)}{2 (x+i)}-\frac {\cot ^{-1}(c x)}{2 (-x+i)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )+\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 i c (x+i)}{(c+1) (1-i c x)}+1\right )+\log \left (\frac {2}{1-i c x}\right ) \left (-\cot ^{-1}(c x)\right )+\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)+\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x)\) |
Input:
Int[(x*ArcCot[c*x])/(1 + x^2),x]
Output:
-(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, 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))]
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 = 1.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(\frac {\ln \left (x^{2}+1\right ) \operatorname {arccot}\left (c x \right )}{2}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (x^{2}+1\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha c +x}{\underline {\hspace {1.25 ex}}\alpha \left (1+c \right )}\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\underline {\hspace {1.25 ex}}\alpha c -x}{\underline {\hspace {1.25 ex}}\alpha \left (c -1\right )}\right )-\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha c +x}{\underline {\hspace {1.25 ex}}\alpha \left (1+c \right )}\right )-\operatorname {dilog}\left (\frac {\underline {\hspace {1.25 ex}}\alpha c -x}{\underline {\hspace {1.25 ex}}\alpha \left (c -1\right )}\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 c}\) | \(134\) |
| risch | \(\frac {\pi \ln \left (-c^{2}+\left (-i c x +1\right )^{2}-1+2 i c x \right )}{4}-\frac {i \operatorname {dilog}\left (\frac {-i c x -c}{-c -1}\right )}{4}-\frac {i \ln \left (-i c x +1\right ) \ln \left (\frac {-i c x -c}{-c -1}\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {-i c x +c}{c -1}\right )}{4}-\frac {i \ln \left (-i c x +1\right ) \ln \left (\frac {-i c x +c}{c -1}\right )}{4}+\frac {i \ln \left (i c x +1\right ) \ln \left (\frac {i c x -c}{-c -1}\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {i c x -c}{-c -1}\right )}{4}+\frac {i \ln \left (i c x +1\right ) \ln \left (\frac {i c x +c}{c -1}\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {i c x +c}{c -1}\right )}{4}\) | \(211\) |
| derivativedivides | \(\frac {\frac {c^{2} \ln \left (c^{2} x^{2}+c^{2}\right ) \operatorname {arccot}\left (c x \right )}{2}+\frac {c^{2} \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )-i \left (-i \operatorname {dilog}\left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )-i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )\right )-i \left (-i \operatorname {dilog}\left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )-i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )+i \left (i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )+i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )\right )+i \left (i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )+i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )\right )\right )}{2}\right )}{2}}{c^{2}}\) | \(317\) |
| default | \(\frac {\frac {c^{2} \ln \left (c^{2} x^{2}+c^{2}\right ) \operatorname {arccot}\left (c x \right )}{2}+\frac {c^{2} \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )-i \left (-i \operatorname {dilog}\left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )-i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )\right )-i \left (-i \operatorname {dilog}\left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )-i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+c^{2}\right )+i \left (i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )+i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )\right )+i \left (i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )+i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )\right )\right )}{2}\right )}{2}}{c^{2}}\) | \(317\) |
Input:
int(x*arccot(c*x)/(x^2+1),x,method=_RETURNVERBOSE)
Output:
1/2*ln(x^2+1)*arccot(c*x)+1/4/c*sum(1/_alpha*(ln(x-_alpha)*ln(x^2+1)-ln(x- _alpha)*ln((_alpha*c+x)/_alpha/(1+c))-ln(x-_alpha)*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))
\[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int { \frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1} \,d x } \] Input:
integrate(x*arccot(c*x)/(x^2+1),x, algorithm="fricas")
Output:
integral(x*arccot(c*x)/(x^2 + 1), x)
\[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int \frac {x \operatorname {acot}{\left (c x \right )}}{x^{2} + 1}\, dx \] Input:
integrate(x*acot(c*x)/(x**2+1),x)
Output:
Integral(x*acot(c*x)/(x**2 + 1), x)
\[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int { \frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1} \,d x } \] Input:
integrate(x*arccot(c*x)/(x^2+1),x, algorithm="maxima")
Output:
integrate(x*arccot(c*x)/(x^2 + 1), x)
\[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int { \frac {x \operatorname {arccot}\left (c x\right )}{x^{2} + 1} \,d x } \] Input:
integrate(x*arccot(c*x)/(x^2+1),x, algorithm="giac")
Output:
integrate(x*arccot(c*x)/(x^2 + 1), x)
Timed out. \[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int \frac {x\,\mathrm {acot}\left (c\,x\right )}{x^2+1} \,d x \] Input:
int((x*acot(c*x))/(x^2 + 1),x)
Output:
int((x*acot(c*x))/(x^2 + 1), x)
\[ \int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx=\int \frac {\mathit {acot} \left (c x \right ) x}{x^{2}+1}d x \] Input:
int(x*acot(c*x)/(x^2+1),x)
Output:
int((acot(c*x)*x)/(x**2 + 1),x)