Integrand size = 12, antiderivative size = 81 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=-\frac {2 \arctan (x)}{3 x}-\frac {\arctan (x)^2}{3}+\log (x)-\frac {1}{2} \log \left (1+x^2\right )-\frac {\log \left (1+x^2\right )}{6 x^2}-\frac {\arctan (x) \log \left (1+x^2\right )}{3 x^3}+\frac {1}{12} \log ^2\left (1+x^2\right )+\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{6} \]
-2/3*arctan(x)/x-1/3*arctan(x)^2+ln(x)-1/2*ln(x^2+1)-1/6*ln(x^2+1)/x^2-1/3 *arctan(x)*ln(x^2+1)/x^3+1/12*ln(x^2+1)^2+1/6*polylog(2,-x^2)
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=-\frac {2 \arctan (x)}{3 x}-\frac {\arctan (x)^2}{3}+\log (x)-\frac {1}{2} \log \left (1+x^2\right )-\frac {\log \left (1+x^2\right )}{6 x^2}-\frac {\arctan (x) \log \left (1+x^2\right )}{3 x^3}+\frac {1}{12} \log ^2\left (1+x^2\right )+\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{6} \]
(-2*ArcTan[x])/(3*x) - ArcTan[x]^2/3 + Log[x] - Log[1 + x^2]/2 - Log[1 + x ^2]/(6*x^2) - (ArcTan[x]*Log[1 + x^2])/(3*x^3) + Log[1 + x^2]^2/12 + PolyL og[2, -x^2]/6
Time = 0.65 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5552, 2925, 2857, 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.
| \(\displaystyle \int \frac {\arctan (x) \log \left (x^2+1\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 5552 |
| \(\displaystyle \frac {2}{3} \int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{3} \int \frac {\log \left (x^2+1\right )}{x^3 \left (x^2+1\right )}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {2}{3} \int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{6} \int \frac {\log \left (x^2+1\right )}{x^4 \left (x^2+1\right )}dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2857 |
| \(\displaystyle \frac {2}{3} \int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx+\frac {1}{6} \int \left (\frac {\log \left (x^2+1\right )}{x^2+1}-\frac {\log \left (x^2+1\right )}{x^2}+\frac {\log \left (x^2+1\right )}{x^4}\right )dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \int \frac {\arctan (x)}{x^2 \left (x^2+1\right )}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {2}{3} \left (\int \frac {\arctan (x)}{x^2}dx-\int \frac {\arctan (x)}{x^2+1}dx\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\arctan (x)}{x^2+1}dx+\int \frac {1}{x \left (x^2+1\right )}dx-\frac {\arctan (x)}{x}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \int \frac {1}{x^2 \left (x^2+1\right )}dx^2-\frac {\arctan (x)}{x}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \left (\int \frac {1}{x^2}dx^2-\int \frac {1}{x^2+1}dx^2\right )-\frac {\arctan (x)}{x}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \left (\log \left (x^2\right )-\int \frac {1}{x^2+1}dx^2\right )-\frac {\arctan (x)}{x}\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{3} \left (-\int \frac {\arctan (x)}{x^2+1}dx-\frac {\arctan (x)}{x}+\frac {1}{2} \left (\log \left (x^2\right )-\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {2}{3} \left (-\frac {1}{2} \arctan (x)^2-\frac {\arctan (x)}{x}+\frac {1}{2} \left (\log \left (x^2\right )-\log \left (x^2+1\right )\right )\right )-\frac {\arctan (x) \log \left (x^2+1\right )}{3 x^3}+\frac {1}{6} \left (\operatorname {PolyLog}\left (2,-x^2\right )+\frac {1}{2} \log ^2\left (x^2+1\right )-\frac {\log \left (x^2+1\right )}{x^2}-\log \left (x^2+1\right )+\log \left (x^2\right )\right )\) |
(2*(-(ArcTan[x]/x) - ArcTan[x]^2/2 + (Log[x^2] - Log[1 + x^2])/2))/3 - (Ar cTan[x]*Log[1 + x^2])/(3*x^3) + (Log[x^2] - Log[1 + x^2] - Log[1 + x^2]/x^ 2 + Log[1 + x^2]^2/2 + PolyLog[2, -x^2])/6
3.13.83.3.1 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[(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]
\[\int \frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{x^{4}}d x\]
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{4}} \,d x } \]
Result contains complex when optimal does not.
Time = 12.99 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=\frac {2 \log {\left (x \right )}}{3} + \frac {\log {\left (2 x^{2} \right )}}{6} + \frac {\log {\left (x^{2} + 1 \right )}^{2}}{12} - \frac {\log {\left (x^{2} + 1 \right )}}{3} - \frac {\log {\left (2 x^{2} + 2 \right )}}{6} - \frac {\operatorname {atan}^{2}{\left (x \right )}}{3} + \frac {\operatorname {Li}_{2}\left (x^{2} e^{i \pi }\right )}{6} - \frac {2 \operatorname {atan}{\left (x \right )}}{3 x} - \frac {\log {\left (x^{2} + 1 \right )}}{6 x^{2}} - \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{3 x^{3}} \]
2*log(x)/3 + log(2*x**2)/6 + log(x**2 + 1)**2/12 - log(x**2 + 1)/3 - log(2 *x**2 + 2)/6 - atan(x)**2/3 + polylog(2, x**2*exp_polar(I*pi))/6 - 2*atan( x)/(3*x) - log(x**2 + 1)/(6*x**2) - log(x**2 + 1)*atan(x)/(3*x**3)
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=-\frac {1}{3} \, {\left (\frac {2}{x} + \frac {\log \left (x^{2} + 1\right )}{x^{3}} + 2 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) + \frac {4 \, x^{2} \arctan \left (x\right )^{2} + x^{2} \log \left (x^{2} + 1\right )^{2} - 2 \, x^{2} {\rm Li}_2\left (x^{2} + 1\right ) + 12 \, x^{2} \log \left (x\right ) - 2 \, {\left (x^{2} \log \left (-x^{2}\right ) + 3 \, x^{2} + 1\right )} \log \left (x^{2} + 1\right )}{12 \, x^{2}} \]
-1/3*(2/x + log(x^2 + 1)/x^3 + 2*arctan(x))*arctan(x) + 1/12*(4*x^2*arctan (x)^2 + x^2*log(x^2 + 1)^2 - 2*x^2*dilog(x^2 + 1) + 12*x^2*log(x) - 2*(x^2 *log(-x^2) + 3*x^2 + 1)*log(x^2 + 1))/x^2
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^4} \, dx=\int \frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{x^4} \,d x \]