\(\int \frac {\cot ^{-1}(x)}{x^3 (1+x^2)} \, dx\) [44]

Optimal result

Integrand size = 13, antiderivative size = 72 \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {\arctan (x)}{2}-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5039, 4947, 331, 209, 5045, 4989, 2497} \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\frac {\arctan (x)}{2}-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2}{1-i x}-1\right )-\frac {\cot ^{-1}(x)}{2 x^2}+\frac {1}{2 x}-\frac {1}{2} i \cot ^{-1}(x)^2-\log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x) \]

[In]

Int[ArcCot[x]/(x^3*(1 + x^2)),x]

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 4989

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] + Dist[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 5039

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcCot[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 5045

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{-1}(x)}{x^3} \, dx-\int \frac {\cot ^{-1}(x)}{x \left (1+x^2\right )} \, dx \\ & = -\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2-i \int \frac {\cot ^{-1}(x)}{x (i+x)} \, dx-\frac {1}{2} \int \frac {1}{x^2 \left (1+x^2\right )} \, dx \\ & = \frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )+\frac {1}{2} \int \frac {1}{1+x^2} \, dx-\int \frac {\log \left (2-\frac {2}{1-i x}\right )}{1+x^2} \, dx \\ & = \frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {\arctan (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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\frac {1}{2} \left (\frac {1}{x}+i \cot ^{-1}(x)^2+\cot ^{-1}(x) \left (-1-\frac {1}{x^2}-2 \log \left (1+e^{2 i \cot ^{-1}(x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(x)}\right )\right ) \]

[In]

Integrate[ArcCot[x]/(x^3*(1 + x^2)),x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (58 ) = 116\).

Time = 0.75 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.47

method	result	size
default	\(-\frac {\operatorname {arccot}\left (x \right )}{2 x^{2}}-\operatorname {arccot}\left (x \right ) \ln \left (x \right )+\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}+\frac {1}{2 x}+\frac {\arctan \left (x \right )}{2}+\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}\)	\(178\)
parts	\(-\frac {\operatorname {arccot}\left (x \right )}{2 x^{2}}-\operatorname {arccot}\left (x \right ) \ln \left (x \right )+\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}-\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}+\frac {1}{2 x}+\frac {\arctan \left (x \right )}{2}+\frac {i \ln \left (x \right ) \ln \left (i x +1\right )}{2}-\frac {i \ln \left (x \right ) \ln \left (-i x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}\)	\(178\)
risch	\(\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {\pi }{4 x^{2}}-\frac {\pi \ln \left (-i x \right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {1}{2 x}+\frac {i \operatorname {dilog}\left (i x +1\right )}{2}-\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}+\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i \ln \left (-i x \right )}{4}+\frac {i \ln \left (-i x +1\right )}{4 x^{2}}+\frac {i \ln \left (i x \right )}{4}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4 x^{2}}\)	\(190\)

[In]

int(arccot(x)/x^3/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

-1/2*arccot(x)/x^2-arccot(x)*ln(x)+1/2*arccot(x)*ln(x^2+1)-1/4*I*(ln(x-I)*ln(x^2+1)-1/2*ln(x-I)^2-dilog(-1/2*I
*(I+x))-ln(x-I)*ln(-1/2*I*(I+x)))+1/4*I*(ln(I+x)*ln(x^2+1)-1/2*ln(I+x)^2-dilog(1/2*I*(x-I))-ln(I+x)*ln(1/2*I*(
x-I)))+1/2/x+1/2*arctan(x)+1/2*I*ln(x)*ln(1+I*x)-1/2*I*ln(x)*ln(1-I*x)+1/2*I*dilog(1+I*x)-1/2*I*dilog(1-I*x)

Fricas [F]

\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]

[In]

integrate(arccot(x)/x^3/(x^2+1),x, algorithm="fricas")

[Out]

integral(arccot(x)/(x^5 + x^3), x)

Sympy [F]

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

[In]

integrate(acot(x)/x**3/(x**2+1),x)

[Out]

Integral(acot(x)/(x**3*(x**2 + 1)), x)

Maxima [F]

\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]

[In]

integrate(arccot(x)/x^3/(x^2+1),x, algorithm="maxima")

[Out]

integrate(arccot(x)/((x^2 + 1)*x^3), x)

Giac [F]

\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]

[In]

integrate(arccot(x)/x^3/(x^2+1),x, algorithm="giac")

[Out]

integrate(arccot(x)/((x^2 + 1)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{x^3\,\left (x^2+1\right )} \,d x \]

[In]

int(acot(x)/(x^3*(x^2 + 1)),x)

[Out]

int(acot(x)/(x^3*(x^2 + 1)), x)