Integrand size = 13, antiderivative size = 72 \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\frac {1}{2 x}-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2+\frac {\arctan (x)}{2}-\cot ^{-1}(x) \log \left (2-\frac {2}{1-i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i x}\right ) \]
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))
Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.76 \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\frac {1}{2} \left (\frac {1}{x}+i \cot ^{-1}(x)^2+\cot ^{-1}(x) \left (-1-\frac {1}{x^2}-2 \log \left (1+e^{2 i \cot ^{-1}(x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(x)}\right )\right ) \]
(x^(-1) + I*ArcCot[x]^2 + ArcCot[x]*(-1 - x^(-2) - 2*Log[1 + E^((2*I)*ArcC ot[x])]) + I*PolyLog[2, -E^((2*I)*ArcCot[x])])/2
Time = 0.47 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5454, 5362, 264, 216, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(x)}{x^3 \left (x^2+1\right )} \, dx\) |
| \(\Big \downarrow \) 5454 |
| \(\displaystyle \int \frac {\cot ^{-1}(x)}{x^3}dx-\int \frac {\cot ^{-1}(x)}{x \left (x^2+1\right )}dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{x^2 \left (x^2+1\right )}dx-\int \frac {\cot ^{-1}(x)}{x \left (x^2+1\right )}dx-\frac {\cot ^{-1}(x)}{2 x^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^2+1}dx+\frac {1}{x}\right )-\int \frac {\cot ^{-1}(x)}{x \left (x^2+1\right )}dx-\frac {\cot ^{-1}(x)}{2 x^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\int \frac {\cot ^{-1}(x)}{x \left (x^2+1\right )}dx+\frac {1}{2} \left (\arctan (x)+\frac {1}{x}\right )-\frac {\cot ^{-1}(x)}{2 x^2}\) |
| \(\Big \downarrow \) 5460 |
| \(\displaystyle -i \int \frac {\cot ^{-1}(x)}{x (x+i)}dx+\frac {1}{2} \left (\arctan (x)+\frac {1}{x}\right )-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2\) |
| \(\Big \downarrow \) 5404 |
| \(\displaystyle -i \left (-i \int \frac {\log \left (2-\frac {2}{1-i x}\right )}{x^2+1}dx-i \log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x)\right )+\frac {1}{2} \left (\arctan (x)+\frac {1}{x}\right )-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {1}{2} \left (\arctan (x)+\frac {1}{x}\right )-i \left (\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i x}-1\right )-i \log \left (2-\frac {2}{1-i x}\right ) \cot ^{-1}(x)\right )-\frac {\cot ^{-1}(x)}{2 x^2}-\frac {1}{2} i \cot ^{-1}(x)^2\) |
-1/2*ArcCot[x]/x^2 - (I/2)*ArcCot[x]^2 + (x^(-1) + ArcTan[x])/2 - I*((-I)* ArcCot[x]*Log[2 - 2/(1 - I*x)] + PolyLog[2, -1 + 2/(1 - I*x)]/2)
3.1.44.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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]]
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] + Simp[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]
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]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Simp[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]
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]
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 | \(\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {\pi }{4 x^{2}}-\frac {\pi \ln \left (-i x \right )}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {1}{2 x}+\frac {i \operatorname {dilog}\left (i x +1\right )}{2}-\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \operatorname {dilog}\left (-i x +1\right )}{2}+\frac {i \ln \left (-i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4}-\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}+\frac {i \ln \left (i x +1\right )^{2}}{8}-\frac {i \ln \left (-i x \right )}{4}+\frac {i \ln \left (-i x +1\right )}{4 x^{2}}+\frac {i \ln \left (i x \right )}{4}+\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \ln \left (i x +1\right )}{4 x^{2}}\) | \(190\) |
-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)
\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{x^{3} \left (x^{2} + 1\right )}\, dx \]
\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(x)}{x^3 \left (1+x^2\right )} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{x^3\,\left (x^2+1\right )} \,d x \]