\(\int \frac {\cot ^{-1}(x)}{x (1+x^2)} \, dx\) [27]

Optimal result

Integrand size = 13, antiderivative size = 49 \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \] Output:

1/2*I*arccot(x)^2+arccot(x)*ln(2-2/(1-I*x))+1/2*I*polylog(2,-1+2/(1-I*x))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=-\frac {1}{2} i \cot ^{-1}(x)^2+\cot ^{-1}(x) \log \left (1+e^{2 i \cot ^{-1}(x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(x)}\right ) \] Input:

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

Output:

(-1/2*I)*ArcCot[x]^2 + ArcCot[x]*Log[1 + E^((2*I)*ArcCot[x])] - (I/2)*Poly 
Log[2, -E^((2*I)*ArcCot[x])]
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5460, 5404, 2897}

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.

\(\displaystyle \int \frac {\cot ^{-1}(x)}{x \left (x^2+1\right )} \, dx\)

\(\Big \downarrow \) 5460

\(\displaystyle i \int \frac {\cot ^{-1}(x)}{x (x+i)}dx+\frac {1}{2} i \cot ^{-1}(x)^2\)

\(\Big \downarrow \) 5404

\(\displaystyle i \left (-i \int \frac {\log \left (2-\frac {2}{1-i x}\right )}{x^2+1}dx-i \log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x)\right )+\frac {1}{2} i \cot ^{-1}(x)^2\)

\(\Big \downarrow \) 2897

\(\displaystyle i \left (\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i x}-1\right )-i \log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x)\right )+\frac {1}{2} i \cot ^{-1}(x)^2\)

Input:

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

Output:

(I/2)*ArcCot[x]^2 + I*((-I)*ArcCot[x]*Log[2 - 2/(1 - I*x)] + PolyLog[2, -1 
 + 2/(1 - I*x)]/2)
 

Defintions of rubi rules used

rule 2897

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[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

rule 5404

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcCot[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Si 
mp[b*c*(p/d)   Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 
 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* 
d^2 + e^2, 0]
 

rule 5460

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (41 ) = 82\).

Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.39

method	result	size
risch	\(-\frac {\pi \ln \left (x^{2}+1\right )}{4}+\frac {\pi \ln \left (-i x \right )}{2}+\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {i \ln \left (-i x +1\right )^{2}}{8}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}-\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}-\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}\)	\(117\)
default	\(-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\operatorname {arccot}\left (x \right ) \ln \left (x \right )-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(161\)
parts	\(-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\operatorname {arccot}\left (x \right ) \ln \left (x \right )-\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}+\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(161\)

Input:

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

Output:

-1/4*Pi*ln(x^2+1)+1/2*Pi*ln(-I*x)+1/4*I*ln(1/2+1/2*I*x)*ln(1-I*x)-1/4*I*di 
log(1/2-1/2*I*x)+1/8*I*ln(1-I*x)^2+1/2*I*dilog(1-I*x)-1/4*I*ln(1/2-1/2*I*x 
)*ln(1+I*x)+1/4*I*dilog(1/2+1/2*I*x)-1/8*I*ln(1+I*x)^2-1/2*I*dilog(1+I*x)
 

Fricas [F]

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

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

Output:

integral(arccot(x)/(x^3 + x), x)
 

Sympy [F]

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

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

Output:

Integral(acot(x)/(x*(x**2 + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(acot(x)/(x*(x^2 + 1)),x)
 

Output:

int(acot(x)/(x*(x^2 + 1)), x)
 

Reduce [F]

\[ \int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx=\int \frac {\mathit {acot} \left (x \right )}{x^{3}+x}d x \] Input:

int(acot(x)/x/(x^2+1),x)
                                                                                    
                                                                                    
 

Output:

int(acot(x)/(x**3 + x),x)