Integrand size = 12, antiderivative size = 41 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=\arctan (x)^2-\frac {\arctan (x) \log \left (1+x^2\right )}{x}-\frac {1}{4} \log ^2\left (1+x^2\right )-\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{2} \] Output:
arctan(x)^2-arctan(x)*ln(x^2+1)/x-1/4*ln(x^2+1)^2-1/2*polylog(2,-x^2)
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=\arctan (x)^2-\frac {\arctan (x) \log \left (1+x^2\right )}{x}-\frac {1}{4} \log ^2\left (1+x^2\right )-\frac {\operatorname {PolyLog}\left (2,-x^2\right )}{2} \] Input:
Integrate[(ArcTan[x]*Log[1 + x^2])/x^2,x]
Output:
ArcTan[x]^2 - (ArcTan[x]*Log[1 + x^2])/x - Log[1 + x^2]^2/4 - PolyLog[2, - x^2]/2
Time = 0.42 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5552, 2925, 2857, 2009, 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.
\(\displaystyle \int \frac {\arctan (x) \log \left (x^2+1\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 5552 |
\(\displaystyle 2 \int \frac {\arctan (x)}{x^2+1}dx+\int \frac {\log \left (x^2+1\right )}{x \left (x^2+1\right )}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{x}\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle 2 \int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \int \frac {\log \left (x^2+1\right )}{x^2 \left (x^2+1\right )}dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{x}\) |
\(\Big \downarrow \) 2857 |
\(\displaystyle 2 \int \frac {\arctan (x)}{x^2+1}dx+\frac {1}{2} \int \left (\frac {\log \left (x^2+1\right )}{-x^2-1}+\frac {\log \left (x^2+1\right )}{x^2}\right )dx^2-\frac {\arctan (x) \log \left (x^2+1\right )}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {\arctan (x)}{x^2+1}dx-\frac {\arctan (x) \log \left (x^2+1\right )}{x}+\frac {1}{2} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2} \log ^2\left (x^2+1\right )\right )\) |
\(\Big \downarrow \) 5419 |
\(\displaystyle -\frac {\arctan (x) \log \left (x^2+1\right )}{x}+\arctan (x)^2+\frac {1}{2} \left (-\operatorname {PolyLog}\left (2,-x^2\right )-\frac {1}{2} \log ^2\left (x^2+1\right )\right )\) |
Input:
Int[(ArcTan[x]*Log[1 + x^2])/x^2,x]
Output:
ArcTan[x]^2 - (ArcTan[x]*Log[1 + x^2])/x + (-1/2*Log[1 + x^2]^2 - PolyLog[ 2, -x^2])/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_)]*(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_.))*((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]
\[\int \frac {\arctan \left (x \right ) \ln \left (x^{2}+1\right )}{x^{2}}d x\]
Input:
int(arctan(x)*ln(x^2+1)/x^2,x)
Output:
int(arctan(x)*ln(x^2+1)/x^2,x)
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{2}} \,d x } \] Input:
integrate(arctan(x)*log(x^2+1)/x^2,x, algorithm="fricas")
Output:
integral(arctan(x)*log(x^2 + 1)/x^2, x)
Result contains complex when optimal does not.
Time = 38.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=- \frac {\log {\left (x^{2} + 1 \right )}^{2}}{4} + \operatorname {atan}^{2}{\left (x \right )} - \frac {\operatorname {Li}_{2}\left (x^{2} e^{i \pi }\right )}{2} - \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\left (x \right )}}{x} \] Input:
integrate(atan(x)*ln(x**2+1)/x**2,x)
Output:
-log(x**2 + 1)**2/4 + atan(x)**2 - polylog(2, x**2*exp_polar(I*pi))/2 - lo g(x**2 + 1)*atan(x)/x
Time = 0.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=-{\left (\frac {\log \left (x^{2} + 1\right )}{x} - 2 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) - \arctan \left (x\right )^{2} + \frac {1}{2} \, \log \left (-x^{2}\right ) \log \left (x^{2} + 1\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right )^{2} + \frac {1}{2} \, {\rm Li}_2\left (x^{2} + 1\right ) \] Input:
integrate(arctan(x)*log(x^2+1)/x^2,x, algorithm="maxima")
Output:
-(log(x^2 + 1)/x - 2*arctan(x))*arctan(x) - arctan(x)^2 + 1/2*log(-x^2)*lo g(x^2 + 1) - 1/4*log(x^2 + 1)^2 + 1/2*dilog(x^2 + 1)
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=\int { \frac {\arctan \left (x\right ) \log \left (x^{2} + 1\right )}{x^{2}} \,d x } \] Input:
integrate(arctan(x)*log(x^2+1)/x^2,x, algorithm="giac")
Output:
integrate(arctan(x)*log(x^2 + 1)/x^2, x)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx={\mathrm {atan}\left (x\right )}^2-\frac {{\ln \left (x^2+1\right )}^2}{4}-\frac {{\mathrm {Li}}_{\mathrm {2}}\left (x^2+1\right )}{2}-\frac {\ln \left (x^2+1\right )\,\mathrm {atan}\left (x\right )}{x} \] Input:
int((log(x^2 + 1)*atan(x))/x^2,x)
Output:
atan(x)^2 - log(x^2 + 1)^2/4 - dilog(x^2 + 1)/2 - (log(x^2 + 1)*atan(x))/x
\[ \int \frac {\arctan (x) \log \left (1+x^2\right )}{x^2} \, dx=\frac {\mathit {atan} \left (x \right )^{2} x -\mathit {atan} \left (x \right ) \mathrm {log}\left (x^{2}+1\right )+\left (\int \frac {\mathrm {log}\left (x^{2}+1\right )}{x^{3}+x}d x \right ) x}{x} \] Input:
int(atan(x)*log(x^2+1)/x^2,x)
Output:
(atan(x)**2*x - atan(x)*log(x**2 + 1) + int(log(x**2 + 1)/(x**3 + x),x)*x) /x