Integrand size = 13, antiderivative size = 67 \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2-\frac {\arctan (x)}{2}+\cot ^{-1}(x) \log \left (\frac {2}{1+i x}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right ) \]
1/2*x+1/2*x^2*arccot(x)-1/2*I*arccot(x)^2-1/2*arctan(x)+arccot(x)*ln(2/(1+ I*x))-1/2*I*polylog(2,1-2/(1+I*x))
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\frac {1}{2} \left (x-i \cot ^{-1}(x)^2+\cot ^{-1}(x) \left (1+x^2+2 \log \left (1-e^{2 i \cot ^{-1}(x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(x)}\right )\right ) \]
(x - I*ArcCot[x]^2 + ArcCot[x]*(1 + x^2 + 2*Log[1 - E^((2*I)*ArcCot[x])]) - I*PolyLog[2, E^((2*I)*ArcCot[x])])/2
Time = 0.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5452, 5362, 262, 216, 5456, 5380, 2849, 2752}
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.
\(\displaystyle \int \frac {x^3 \cot ^{-1}(x)}{x^2+1} \, dx\) |
\(\Big \downarrow \) 5452 |
\(\displaystyle \int x \cot ^{-1}(x)dx-\int \frac {x \cot ^{-1}(x)}{x^2+1}dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{x^2+1}dx-\int \frac {x \cot ^{-1}(x)}{x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(x)\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (x-\int \frac {1}{x^2+1}dx\right )-\int \frac {x \cot ^{-1}(x)}{x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(x)\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\int \frac {x \cot ^{-1}(x)}{x^2+1}dx+\frac {1}{2} (x-\arctan (x))+\frac {1}{2} x^2 \cot ^{-1}(x)\) |
\(\Big \downarrow \) 5456 |
\(\displaystyle \int \frac {\cot ^{-1}(x)}{i-x}dx+\frac {1}{2} (x-\arctan (x))+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2\) |
\(\Big \downarrow \) 5380 |
\(\displaystyle \int \frac {\log \left (\frac {2}{i x+1}\right )}{x^2+1}dx+\frac {1}{2} (x-\arctan (x))+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x)\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle -i \int \frac {\log \left (\frac {2}{i x+1}\right )}{1-\frac {2}{i x+1}}d\frac {1}{i x+1}+\frac {1}{2} (x-\arctan (x))+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x)\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {1}{2} (x-\arctan (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i x+1}\right )+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} i \cot ^{-1}(x)^2+\log \left (\frac {2}{1+i x}\right ) \cot ^{-1}(x)\) |
(x^2*ArcCot[x])/2 - (I/2)*ArcCot[x]^2 + (x - ArcTan[x])/2 + ArcCot[x]*Log[ 2/(1 + I*x)] - (I/2)*PolyLog[2, 1 - 2/(1 + I*x)]
3.1.38.3.1 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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[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 ]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[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*e*(p + 1))), x] - Simp[ 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (53 ) = 106\).
Time = 0.74 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {x^{2} \operatorname {arccot}\left (x \right )}{2}-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\) | \(126\) |
parts | \(\frac {x^{2} \operatorname {arccot}\left (x \right )}{2}-\frac {\operatorname {arccot}\left (x \right ) \ln \left (x^{2}+1\right )}{2}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}+\frac {i \left (\ln \left (x -i\right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (i+x \right )}{2}\right )-\ln \left (x -i\right ) \ln \left (-\frac {i \left (i+x \right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (i+x \right ) \ln \left (x^{2}+1\right )-\frac {\ln \left (i+x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (x -i\right )}{2}\right )-\ln \left (i+x \right ) \ln \left (\frac {i \left (x -i\right )}{2}\right )\right )}{4}\) | \(126\) |
risch | \(\frac {\pi \,x^{2}}{4}+\frac {\pi }{4}-\frac {\pi \ln \left (x^{2}+1\right )}{4}-\frac {i \ln \left (-i x +1\right ) x^{2}}{4}+\frac {i \ln \left (\frac {1}{2}+\frac {i x}{2}\right ) \ln \left (-i x +1\right )}{4}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i x}{2}\right )}{4}+\frac {x}{2}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i x}{2}\right )}{4}+\frac {i \ln \left (-i x +1\right )^{2}}{8}-\frac {i \ln \left (\frac {1}{2}-\frac {i x}{2}\right ) \ln \left (i x +1\right )}{4}-\frac {i \ln \left (-i x +1\right )}{4}+\frac {i \ln \left (i x +1\right )}{4}+\frac {i \ln \left (i x +1\right ) x^{2}}{4}-\frac {i \ln \left (i x +1\right )^{2}}{8}\) | \(147\) |
1/2*x^2*arccot(x)-1/2*arccot(x)*ln(x^2+1)+1/2*x-1/2*arctan(x)+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)))
\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]
\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int \frac {x^{3} \operatorname {acot}{\left (x \right )}}{x^{2} + 1}\, dx \]
\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]
\[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int { \frac {x^{3} \operatorname {arccot}\left (x\right )}{x^{2} + 1} \,d x } \]
Timed out. \[ \int \frac {x^3 \cot ^{-1}(x)}{1+x^2} \, dx=\int \frac {x^3\,\mathrm {acot}\left (x\right )}{x^2+1} \,d x \]