Integrand size = 12, antiderivative size = 114 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=-\frac {7}{60 x^2}-\frac {2 \arctan (x)}{15 x^3}+\frac {2 \arctan (x)}{5 x}+\frac {\arctan (x)^2}{5}-\frac {5 \log (x)}{6}+\frac {5}{12} \log \left (1+x^2\right )-\frac {\log \left (1+x^2\right )}{20 x^4}+\frac {\log \left (1+x^2\right )}{10 x^2}-\frac {\arctan (x) \log \left (1+x^2\right )}{5 x^5}-\frac {1}{20} \log ^2\left (1+x^2\right )-\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{10} \]
-7/60/x^2-2/15*arctan(x)/x^3+2/5*arctan(x)/x+1/5*arctan(x)^2-5/6*ln(x)+5/1 2*ln(x^2+1)-1/20*ln(x^2+1)/x^4+1/10*ln(x^2+1)/x^2-1/5*arctan(x)*ln(x^2+1)/ x^5-1/20*ln(x^2+1)^2-1/10*polylog(2,-x^2)
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=-\frac {7}{60 x^2}-\frac {2 \arctan (x)}{15 x^3}+\frac {2 \arctan (x)}{5 x}+\frac {\arctan (x)^2}{5}-\frac {5 \log (x)}{6}+\frac {5}{12} \log \left (1+x^2\right )-\frac {\log \left (1+x^2\right )}{20 x^4}+\frac {\log \left (1+x^2\right )}{10 x^2}-\frac {\arctan (x) \log \left (1+x^2\right )}{5 x^5}-\frac {1}{20} \log ^2\left (1+x^2\right )-\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{10} \]
-7/(60*x^2) - (2*ArcTan[x])/(15*x^3) + (2*ArcTan[x])/(5*x) + ArcTan[x]^2/5 - (5*Log[x])/6 + (5*Log[1 + x^2])/12 - Log[1 + x^2]/(20*x^4) + Log[1 + x^ 2]/(10*x^2) - (ArcTan[x]*Log[1 + x^2])/(5*x^5) - Log[1 + x^2]^2/20 - PolyL og[2, -x^2]/10
Time = 0.97 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {5552, 2925, 2857, 2009, 5453, 5361, 243, 54, 2009, 5453, 5361, 243, 47, 14, 16, 5419}
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.
| \(\displaystyle \int \frac {\arctan (x) \log \left (x^2+1\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 5552 |
| \(\displaystyle \frac {2}{5} \int \frac {\arctan (x)}{x^4 \left (x^2+1\right )}dx+\frac {1}{5} \int \frac {\log \left (x^2+1\right )}{x^5 \left (x^2+1\right )}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {2}{5} \int \frac {\arctan (x)}{x^4 \left (x^2+1\right )}dx+\frac {1}{10} \int \frac {\log \left (x^2+1\right )}{x^6 \left (x^2+1\right )}dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2857 |
| \(\displaystyle \frac {2}{5} \int \frac {\arctan (x)}{x^4 \left (x^2+1\right )}dx+\frac {1}{10} \int \left (\frac {\log \left (x^2+1\right )}{-x^2-1}+\frac {\log \left (x^2+1\right )}{x^2}-\frac {\log \left (x^2+1\right )}{x^4}+\frac {\log \left (x^2+1\right )}{x^6}\right )dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} \int \frac {\arctan (x)}{x^4 \left (x^2+1\right )}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^4}dx-\int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {2}{5} \left (-\int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{3} \int \frac {1}{x^3 \left (x^2+1\right )}dx-\frac {\arctan (x)}{3 x^3}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{5} \left (-\int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{6} \int \frac {1}{x^4 \left (x^2+1\right )}dx^2-\frac {\arctan (x)}{3 x^3}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {2}{5} \left (-\int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{6} \int \left (-\frac {1}{x^2}+\frac {1}{x^4}+\frac {1}{x^2+1}\right )dx^2-\frac {\arctan (x)}{3 x^3}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} \left (-\int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx-\frac {\arctan (x)}{3 x^3}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {2}{5} \left (-\int \frac {\arctan (x)}{x^2}dx+\int \frac {\arctan (x)}{x^2+1}dx-\frac {\arctan (x)}{3 x^3}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^2+1}dx-\int \frac {1}{x \left (x^2+1\right )}dx-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)}{x}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^2+1}dx-\frac {1}{2} \int \frac {1}{x^2 \left (x^2+1\right )}dx^2-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)}{x}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \left (\int \frac {1}{x^2+1}dx^2-\int \frac {1}{x^2}dx^2\right )-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)}{x}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \left (\int \frac {1}{x^2+1}dx^2-\log \left (x^2\right )\right )-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)}{x}+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{5} \left (\int \frac {\arctan (x)}{x^2+1}dx-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)}{x}+\frac {1}{2} \left (\log \left (x^2+1\right )-\log \left (x^2\right )\right )+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle -\frac {\arctan (x) \log \left (x^2+1\right )}{5 x^5}+\frac {2}{5} \left (-\frac {\arctan (x)}{3 x^3}+\frac {\arctan (x)^2}{2}+\frac {\arctan (x)}{x}+\frac {1}{2} \left (\log \left (x^2+1\right )-\log \left (x^2\right )\right )+\frac {1}{6} \left (-\frac {1}{x^2}-\log \left (x^2\right )+\log \left (x^2+1\right )\right )\right )+\frac {1}{10} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2 x^2}-\frac {1}{2} \log ^2\left (x^2+1\right )+\frac {\log \left (x^2+1\right )}{x^2}+\frac {3}{2} \log \left (x^2+1\right )-\frac {3 \log \left (x^2\right )}{2}-\frac {\log \left (x^2+1\right )}{2 x^4}\right )\) |
-1/5*(ArcTan[x]*Log[1 + x^2])/x^5 + (2*(-1/3*ArcTan[x]/x^3 + ArcTan[x]/x + ArcTan[x]^2/2 + (-Log[x^2] + Log[1 + x^2])/2 + (-x^(-2) - Log[x^2] + Log[ 1 + x^2])/6))/5 + (-1/2*1/x^2 - (3*Log[x^2])/2 + (3*Log[1 + x^2])/2 - Log[ 1 + x^2]/(2*x^4) + Log[1 + x^2]/x^2 - Log[1 + x^2]^2/2 - PolyLog[2, -x^2]) /10
3.13.85.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(Log[(c_.)*((d_) + (e_.)*(x_))]*(x_)^(m_.))/((f_) + (g_.)*(x_)), x_Symb ol] :> Int[ExpandIntegrand[Log[c*(d + e*x)], x^m/(f + g*x), x], x] /; FreeQ [{c, d, e, f, g}, x] && EqQ[e*f - d*g, 0] && EqQ[c*d, 1] && IntegerQ[m]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[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_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*( e_.))*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(d + e*Log[f + g*x^2])*((a + b*ArcTan[c*x])/(m + 1)), x] + (-Simp[b*(c/(m + 1)) Int[x^(m + 1)*((d + e* Log[f + g*x^2])/(1 + c^2*x^2)), x], x] - Simp[2*e*(g/(m + 1)) Int[x^(m + 2)*((a + b*ArcTan[c*x])/(f + g*x^2)), x], x]) /; FreeQ[{a, b, c, d, e, f, g }, x] && ILtQ[m/2, 0]
\[\int \frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{x^{6}}d x\]
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{6}} \,d x } \]
Result contains complex when optimal does not.
Time = 17.97 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.18 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=- \frac {8 \log {\left (x \right )}}{15} - \frac {\log {\left (x^{2} \right )}}{20} - \frac {\log {\left (2 x^{2} \right )}}{10} - \frac {\log {\left (x^{2} + 1 \right )}^{2}}{20} + \frac {19 \log {\left (x^{2} + 1 \right )}}{60} + \frac {\log {\left (2 x^{2} + 2 \right )}}{10} + \frac {\operatorname {atan}^{2}{\left (x \right )}}{5} - \frac {\operatorname {Li}_{2}\left (x^{2} e^{i \pi }\right )}{10} + \frac {2 \operatorname {atan}{\left (x \right )}}{5 x} + \frac {\log {\left (x^{2} + 1 \right )}}{10 x^{2}} - \frac {7}{60 x^{2}} - \frac {2 \operatorname {atan}{\left (x \right )}}{15 x^{3}} - \frac {\log {\left (x^{2} + 1 \right )}}{20 x^{4}} - \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{5 x^{5}} \]
-8*log(x)/15 - log(x**2)/20 - log(2*x**2)/10 - log(x**2 + 1)**2/20 + 19*lo g(x**2 + 1)/60 + log(2*x**2 + 2)/10 + atan(x)**2/5 - polylog(2, x**2*exp_p olar(I*pi))/10 + 2*atan(x)/(5*x) + log(x**2 + 1)/(10*x**2) - 7/(60*x**2) - 2*atan(x)/(15*x**3) - log(x**2 + 1)/(20*x**4) - log(x**2 + 1)*atan(x)/(5* x**5)
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=\frac {1}{15} \, {\left (\frac {2 \, {\left (3 \, x^{2} - 1\right )}}{x^{3}} - \frac {3 \, \log \left (x^{2} + 1\right )}{x^{5}} + 6 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) - \frac {12 \, x^{4} \arctan \left (x\right )^{2} + 3 \, x^{4} \log \left (x^{2} + 1\right )^{2} - 6 \, x^{4} {\rm Li}_2\left (x^{2} + 1\right ) + 50 \, x^{4} \log \left (x\right ) + 7 \, x^{2} - {\left (6 \, x^{4} \log \left (-x^{2}\right ) + 25 \, x^{4} + 6 \, x^{2} - 3\right )} \log \left (x^{2} + 1\right )}{60 \, x^{4}} \]
1/15*(2*(3*x^2 - 1)/x^3 - 3*log(x^2 + 1)/x^5 + 6*arctan(x))*arctan(x) - 1/ 60*(12*x^4*arctan(x)^2 + 3*x^4*log(x^2 + 1)^2 - 6*x^4*dilog(x^2 + 1) + 50* x^4*log(x) + 7*x^2 - (6*x^4*log(-x^2) + 25*x^4 + 6*x^2 - 3)*log(x^2 + 1))/ x^4
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^6} \, dx=\int \frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{x^6} \,d x \]