\(\int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx\) [40]

Optimal result

Integrand size = 11, antiderivative size = 48 \[ \int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{2} i \cot ^{-1}(x)^2-\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5041, 4965, 2449, 2352} \[ \int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i x+1}\right )+\frac {1}{2} i \cot ^{-1}(x)^2-\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x) \]

[In]

Int[(x*ArcCot[x])/(1 + x^2),x]

[Out]

(I/2)*ArcCot[x]^2 - ArcCot[x]*Log[2/(1 + I*x)] + (I/2)*PolyLog[2, 1 - 2/(1 + I*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 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 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] - Dist[b*c*(p/e), Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[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 5041

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {x \cot ^{-1}(x)}{1+x^2} \, dx=-\cot ^{-1}(x) \log \left (1-e^{2 i \cot ^{-1}(x)}\right )+\frac {1}{2} i \left (\cot ^{-1}(x)^2+\operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(x)}\right )\right ) \]

[In]

Integrate[(x*ArcCot[x])/(1 + x^2),x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (40 ) = 80\).

Time = 0.35 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {\pi \ln \left (-2+\left (-i x +1\right )^{2}+2 i x \right )}{4}-\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {i \ln \left (i x +1\right )^{2}}{8}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}\)	\(98\)
default	\(\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(112\)
parts	\(\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\)	\(112\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x*arccot(x)/(x^2 + 1), x)

Sympy [F]

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

[In]

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

[Out]

Integral(x*acot(x)/(x**2 + 1), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x*arccot(x)/(x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(x*arccot(x)/(x^2 + 1), x)

Mupad [F(-1)]

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

[In]

int((x*acot(x))/(x^2 + 1),x)

[Out]

int((x*acot(x))/(x^2 + 1), x)