\(\int \frac {\cot ^{-1}(c x)}{x (1+x^2)} \, dx\) [49]

Optimal result

Integrand size = 15, antiderivative size = 223 \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\cot ^{-1}(c x) \log \left (\frac {2}{1-i c x}\right )-\frac {1}{2} \cot ^{-1}(c x) \log \left (\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{2} \cot ^{-1}(c x) \log \left (-\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1+\frac {2 i c (i+x)}{(1+c) (1-i c x)}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5049, 4941, 2438, 4967, 2449, 2352, 2497} \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,-\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,1-\frac {2 i c (i-x)}{(1-c) (1-i c x)}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (2,\frac {2 i c (x+i)}{(c+1) (1-i c x)}+1\right )+\log \left (\frac {2}{1-i c x}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (\frac {2 i c (-x+i)}{(1-c) (1-i c x)}\right ) \cot ^{-1}(c x)-\frac {1}{2} \log \left (-\frac {2 i c (x+i)}{(c+1) (1-i c x)}\right ) \cot ^{-1}(c x) \]

[In]

Int[ArcCot[c*x]/(x*(1 + x^2)),x]

[Out]

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]*L
og[((-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)*Pol
yLog[2, 1 + ((2*I)*c*(I + x))/((1 + c)*(1 - I*c*x))]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 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 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4941

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Dist[I*(b/2), Int[Log[1 + I/(c
*x)]/x, x], x] + Dist[I*(b/2), Int[Log[1 - I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4967

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (-Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] + Dist[b*(c/e), Int[Log[2*c*(
(d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcCot[c*x])*(Log[2*c*((d + e*x)/((c
*d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 5049

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\cot ^{-1}(c x)}{x}-\frac {x \cot ^{-1}(c x)}{1+x^2}\right ) \, dx \\ & = \int \frac {\cot ^{-1}(c x)}{x} \, dx-\int \frac {x \cot ^{-1}(c x)}{1+x^2} \, dx \\ & = \frac {1}{2} i \int \frac {\log \left (1-\frac {i}{c x}\right )}{x} \, dx-\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{c x}\right )}{x} \, dx-\int \left (-\frac {\cot ^{-1}(c x)}{2 (i-x)}+\frac {\cot ^{-1}(c x)}{2 (i+x)}\right ) \, dx \\ & = -\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} \int \frac {\cot ^{-1}(c x)}{i-x} \, dx-\frac {1}{2} \int \frac {\cot ^{-1}(c x)}{i+x} \, 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 )+2 \left (\frac {1}{2} c \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\right )-\frac {1}{2} c \int \frac {\log \left (\frac {2 c (i-x)}{(-i+i c) (1-i c x)}\right )}{1+c^2 x^2} \, dx-\frac {1}{2} c \int \frac {\log \left (\frac {2 c (i+x)}{(i+i c) (1-i c x)}\right )}{1+c^2 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,-\frac {i}{c x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {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 )+2 \left (\frac {1}{2} i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 2.04 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.47 \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\frac {1}{2} \left (-i \left (\cot ^{-1}(c x) \left (\cot ^{-1}(c x)+2 i \log \left (1+e^{2 i \cot ^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(c x)}\right )\right )+\frac {(-1+c) (1+c) \left (i \cot ^{-1}(c x)^2+2 i \arcsin \left (\sqrt {\frac {1}{1-c^2}}\right ) \arctan \left (\frac {\sqrt {c^2}}{c x}\right )-\left (\cot ^{-1}(c x)-\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (\frac {-1+\left (-1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}-c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )-\left (\cot ^{-1}(c x)+\arcsin \left (\sqrt {\frac {1}{1-c^2}}\right )\right ) \log \left (-\frac {1+\left (1+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}+c^2 \left (-1+e^{2 i \cot ^{-1}(c x)}\right )}{-1+c^2}\right )+\frac {1}{2} i \left (\operatorname {PolyLog}\left (2,\frac {\left (1+c^2-2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+c^2+2 \sqrt {c^2}\right ) e^{2 i \cot ^{-1}(c x)}}{-1+c^2}\right )\right )\right )}{-1+c^2}\right ) \]

[In]

Integrate[ArcCot[c*x]/(x*(1 + x^2)),x]

[Out]

((-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)*ArcCot[c*x]) - c^2*(-1 + E^((2*I)*Arc
Cot[c*x])))/(-1 + c^2)] - (ArcCot[c*x] + ArcSin[Sqrt[(1 - c^2)^(-1)]])*Log[-((1 + (1 + 2*Sqrt[c^2])*E^((2*I)*A
rcCot[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

Maple [C] (verified)

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	\(-\frac {\operatorname {arccot}\left (c x \right ) \ln \left (c^{2} x^{2}+c^{2}\right )}{2}+\operatorname {arccot}\left (c x \right ) \ln \left (c x \right )+\frac {c^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{c^{2}}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{c^{2}}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{c^{2}}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{c^{2}}+\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x -i\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c x -i\right ) \ln \left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )-c -1}{-c -1}\right )}{2 c^{2}}-\frac {i \operatorname {dilog}\left (\frac {i \left (c x -i\right )+c -1}{c -1}\right )}{2 c^{2}}-\frac {i \ln \left (c^{2} x^{2}+c^{2}\right ) \ln \left (c x +i\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \ln \left (c x +i\right ) \ln \left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )-c -1}{-c -1}\right )}{2 c^{2}}+\frac {i \operatorname {dilog}\left (\frac {-i \left (c x +i\right )+c -1}{c -1}\right )}{2 c^{2}}\right )}{2}\)	\(393\)

[In]

int(arccot(c*x)/x/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

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)-ln(1-I*c*x))/c-I*(dilog(1+I*c*x)-dilo
g(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-_al
pha)*ln((_alpha*c-x)/_alpha/(c-1))-dilog((_alpha*c+x)/_alpha/(1+c))-dilog((_alpha*c-x)/_alpha/(c-1))),_alpha=R
ootOf(_Z^2*c^2+1)))

Fricas [F]

\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="fricas")

[Out]

integral(arccot(c*x)/(x^3 + x), x)

Sympy [F]

\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (c x \right )}}{x \left (x^{2} + 1\right )}\, dx \]

[In]

integrate(acot(c*x)/x/(x**2+1),x)

[Out]

Integral(acot(c*x)/(x*(x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="maxima")

[Out]

integrate(arccot(c*x)/((x^2 + 1)*x), x)

Giac [F]

\[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (c x\right )}{{\left (x^{2} + 1\right )} x} \,d x } \]

[In]

integrate(arccot(c*x)/x/(x^2+1),x, algorithm="giac")

[Out]

integrate(arccot(c*x)/((x^2 + 1)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(c x)}{x \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (c\,x\right )}{x\,\left (x^2+1\right )} \,d x \]

[In]

int(acot(c*x)/(x*(x^2 + 1)),x)

[Out]

int(acot(c*x)/(x*(x^2 + 1)), x)