3.46 \(\int x \tan ^{-1}(x)^2 \log (1+x^2) \, dx\)
Optimal. Leaf size=77 \[ \frac{1}{4} \log ^2\left (x^2+1\right )-\frac{3}{2} \log \left (x^2+1\right )-\frac{1}{2} x^2 \tan ^{-1}(x)^2+\frac{1}{2} \left (x^2+1\right ) \log \left (x^2+1\right ) \tan ^{-1}(x)^2-x \log \left (x^2+1\right ) \tan ^{-1}(x)-\frac{3}{2} \tan ^{-1}(x)^2+3 x \tan ^{-1}(x) \]
[Out]
3*x*ArcTan[x] - (3*ArcTan[x]^2)/2 - (x^2*ArcTan[x]^2)/2 - (3*Log[1 + x^2])/2 - x*ArcTan[x]*Log[1 + x^2] + ((1
+ x^2)*ArcTan[x]^2*Log[1 + x^2])/2 + Log[1 + x^2]^2/4
________________________________________________________________________________________
Rubi [A] time = 0.219527, antiderivative size = 77, normalized size of antiderivative
= 1., number of steps used = 13, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\)
= 0.833, Rules used = {4852, 4916, 4846, 260, 4884, 5023, 5009, 2475, 2390, 2301}
\[ \frac{1}{4} \log ^2\left (x^2+1\right )-\frac{3}{2} \log \left (x^2+1\right )-\frac{1}{2} x^2 \tan ^{-1}(x)^2+\frac{1}{2} \left (x^2+1\right ) \log \left (x^2+1\right ) \tan ^{-1}(x)^2-x \log \left (x^2+1\right ) \tan ^{-1}(x)-\frac{3}{2} \tan ^{-1}(x)^2+3 x \tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
Int[x*ArcTan[x]^2*Log[1 + x^2],x]
[Out]
3*x*ArcTan[x] - (3*ArcTan[x]^2)/2 - (x^2*ArcTan[x]^2)/2 - (3*Log[1 + x^2])/2 - x*ArcTan[x]*Log[1 + x^2] + ((1
+ x^2)*ArcTan[x]^2*Log[1 + x^2])/2 + Log[1 + x^2]^2/4
Rule 4852
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]
Rule 4916
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, 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] && GtQ[m, 1]
Rule 4846
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 4884
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> 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]
Rule 5023
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2*((d_.) + Log[(f_) + (g_.)*(x_)^2]*(e_.))*(x_), x_Symbol] :> Simp[((f
+ g*x^2)*(d + e*Log[f + g*x^2])*(a + b*ArcTan[c*x])^2)/(2*g), x] + (-Dist[b/c, Int[(d + e*Log[f + g*x^2])*(a +
b*ArcTan[c*x]), x], x] + Dist[b*c*e, Int[(x^2*(a + b*ArcTan[c*x]))/(1 + c^2*x^2), x], x] - Simp[(e*x^2*(a + b
*ArcTan[c*x])^2)/2, x]) /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[g, c^2*f]
Rule 5009
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.)), x_Symbol] :> Simp[x*(d + e*L
og[f + g*x^2])*(a + b*ArcTan[c*x]), x] + (-Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Di
st[2*e*g, Int[(x^2*(a + b*ArcTan[c*x]))/(f + g*x^2), x], x]) /; FreeQ[{a, b, c, d, e, f, g}, x]
Rule 2475
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(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] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])
Rule 2390
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]
Rule 2301
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]
Rubi steps
\begin{align*} \int x \tan ^{-1}(x)^2 \log \left (1+x^2\right ) \, dx &=-\frac{1}{2} x^2 \tan ^{-1}(x)^2+\frac{1}{2} \left (1+x^2\right ) \tan ^{-1}(x)^2 \log \left (1+x^2\right )+\int \frac{x^2 \tan ^{-1}(x)}{1+x^2} \, dx-\int \tan ^{-1}(x) \log \left (1+x^2\right ) \, dx\\ &=-\frac{1}{2} x^2 \tan ^{-1}(x)^2-x \tan ^{-1}(x) \log \left (1+x^2\right )+\frac{1}{2} \left (1+x^2\right ) \tan ^{-1}(x)^2 \log \left (1+x^2\right )+2 \int \frac{x^2 \tan ^{-1}(x)}{1+x^2} \, dx+\int \tan ^{-1}(x) \, dx-\int \frac{\tan ^{-1}(x)}{1+x^2} \, dx+\int \frac{x \log \left (1+x^2\right )}{1+x^2} \, dx\\ &=x \tan ^{-1}(x)-\frac{1}{2} \tan ^{-1}(x)^2-\frac{1}{2} x^2 \tan ^{-1}(x)^2-x \tan ^{-1}(x) \log \left (1+x^2\right )+\frac{1}{2} \left (1+x^2\right ) \tan ^{-1}(x)^2 \log \left (1+x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1+x)}{1+x} \, dx,x,x^2\right )+2 \int \tan ^{-1}(x) \, dx-2 \int \frac{\tan ^{-1}(x)}{1+x^2} \, dx-\int \frac{x}{1+x^2} \, dx\\ &=3 x \tan ^{-1}(x)-\frac{3}{2} \tan ^{-1}(x)^2-\frac{1}{2} x^2 \tan ^{-1}(x)^2-\frac{1}{2} \log \left (1+x^2\right )-x \tan ^{-1}(x) \log \left (1+x^2\right )+\frac{1}{2} \left (1+x^2\right ) \tan ^{-1}(x)^2 \log \left (1+x^2\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+x^2\right )-2 \int \frac{x}{1+x^2} \, dx\\ &=3 x \tan ^{-1}(x)-\frac{3}{2} \tan ^{-1}(x)^2-\frac{1}{2} x^2 \tan ^{-1}(x)^2-\frac{3}{2} \log \left (1+x^2\right )-x \tan ^{-1}(x) \log \left (1+x^2\right )+\frac{1}{2} \left (1+x^2\right ) \tan ^{-1}(x)^2 \log \left (1+x^2\right )+\frac{1}{4} \log ^2\left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0188882, size = 58, normalized size = 0.75 \[ \frac{1}{4} \left (\left (\log \left (x^2+1\right )-6\right ) \log \left (x^2+1\right )+2 \left (-x^2+\left (x^2+1\right ) \log \left (x^2+1\right )-3\right ) \tan ^{-1}(x)^2-4 x \left (\log \left (x^2+1\right )-3\right ) \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x*ArcTan[x]^2*Log[1 + x^2],x]
[Out]
(-4*x*ArcTan[x]*(-3 + Log[1 + x^2]) + (-6 + Log[1 + x^2])*Log[1 + x^2] + 2*ArcTan[x]^2*(-3 - x^2 + (1 + x^2)*L
og[1 + x^2]))/4
________________________________________________________________________________________
Maple [C] time = 0.913, size = 3134, normalized size = 40.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*arctan(x)^2*ln(x^2+1),x)
[Out]
3*x*arctan(x)-1/2*x^2*arctan(x)^2-1/2*arctan(x)^2-2*arctan(x)*ln(2)*x+2*arctan(x)*ln((1+I*x)^2/(x^2+1)+1)*x+2*
I*arctan(x)*ln(2)-arctan(x)^2*ln((1+I*x)^2/(x^2+1)+1)*x^2-1/2*arctan(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3+1
/2*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))^3+1/2*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)
^3+arctan(x)^2*ln(2)*x^2+ln((1+I*x)^2/(x^2+1)+1)^2+(x^2*arctan(x)^2+arctan(x)^2+2*I*arctan(x)-2*x*arctan(x)-2*
ln((1+I*x)^2/(x^2+1)+1))*ln((1+I*x)/(x^2+1)^(1/2))-2*ln(2)*ln((1+I*x)^2/(x^2+1)+1)-1/2*I*Pi*ln((1+I*x)^2/(x^2+
1)+1)*csgn(I*((1+I*x)^2/(x^2+1)+1))^2*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)-1/2*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*
(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2+I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*((1
+I*x)^2/(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2-I*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+1)^
(1/2))*ln((1+I*x)^2/(x^2+1)+1)+1/2*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/
(x^2+1)+1)^2)*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)+1/4*I*arctan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3*x^2+1/4*I
*arctan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)*csgn(I*((1+I*x)^2/(x^2+1)+1))^2+1/4*I*arctan(x)^2*Pi*csgn(I*(1
+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2+1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^
2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)+3*ln((1+I*x)^2/(x^2+1)+1)-1/4*I*arctan(x)^2*Pi
*csgn(I*(1+I*x)^2/(x^2+1))^3*x^2-3*I*arctan(x)-arctan(x)^2*ln((1+I*x)^2/(x^2+1)+1)+arctan(x)^2*ln(2)-1/2*I*arc
tan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I*((1+I*x)^2/(x^2+1)+1))+1/2*I*arctan(x)*Pi*csgn(I*(1+I*x)^
2/(x^2+1))^3*x+1/2*I*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3*x+1/2*I*arctan(x)^2*Pi*c
sgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+1)^(1/2))+1/2*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*(1+I*x)/(x^2+
1)^(1/2))^2*csgn(I*(1+I*x)^2/(x^2+1))-1/2*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)*csgn(I*
(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)/(x^
2+1)^(1/2))^2-1/2*I*arctan(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3*x-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^
2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3*x^2-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))^3-1/4*I*arctan(x)^2*Pi*csgn
(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3+1/4*I*arctan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3+1/2*I*P
i*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*(1+I*x)^2/(x^2+1))^3+1/2*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*(1+I*x)^2/(x^2+1
)/((1+I*x)^2/(x^2+1)+1)^2)^3-1/2*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3+arctan(x)*Pi*c
sgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I*((1+I*x)^2/(x^2+1)+1))-1/2*arctan(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2
)*csgn(I*((1+I*x)^2/(x^2+1)+1))^2-arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+1)^(1/2))-1/2*a
rctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2+1/2*arctan(x)*Pi*csg
n(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2-1/2*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x
^2+1)+1)^2)^2*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/
(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)*x^2+1/2*I*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+
1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)*x-1/2*I*arctan(x)*Pi*csg
n(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2*x-1/2*I*arctan(x)*Pi*csgn(I*(1+I*x)
^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)*x-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/
(x^2+1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2*x^2-1/2*I*arctan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I*((1
+I*x)^2/(x^2+1)+1))*x^2+1/2*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+1)^(1/2))*x^2+1/2
*I*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2*x+1/2*I*csgn(I*(1+I*x)^2/(x^2+1)/((1
+I*x)^2/(x^2+1)+1)^2)*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)*ln((1+I*x)^2/(x^2+1)+1)*Pi+I*a
rctan(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I*((1+I*x)^2/(x^2+1)+1))*x-1/2*I*arctan(x)*Pi*csgn(I*((1+I*
x)^2/(x^2+1)+1)^2)*csgn(I*((1+I*x)^2/(x^2+1)+1))^2*x-I*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)
/(x^2+1)^(1/2))*x-1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1
)^2)*csgn(I/((1+I*x)^2/(x^2+1)+1)^2)+1/4*I*arctan(x)^2*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)*csgn(I*((1+I*x)^2/(x
^2+1)+1))^2*x^2+1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^
2)^2*x^2+1/4*I*arctan(x)^2*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^2*csgn(I/((1+I*x)^2/(x^2+1)+1)
^2)*x^2
________________________________________________________________________________________
Maxima [A] time = 1.45918, size = 90, normalized size = 1.17 \begin{align*} -\frac{1}{2} \,{\left (x^{2} -{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1\right )} \arctan \left (x\right )^{2} -{\left (x \log \left (x^{2} + 1\right ) - 3 \, x + 2 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) + \arctan \left (x\right )^{2} + \frac{1}{4} \, \log \left (x^{2} + 1\right )^{2} - \frac{3}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arctan(x)^2*log(x^2+1),x, algorithm="maxima")
[Out]
-1/2*(x^2 - (x^2 + 1)*log(x^2 + 1) + 1)*arctan(x)^2 - (x*log(x^2 + 1) - 3*x + 2*arctan(x))*arctan(x) + arctan(
x)^2 + 1/4*log(x^2 + 1)^2 - 3/2*log(x^2 + 1)
________________________________________________________________________________________
Fricas [A] time = 2.43965, size = 173, normalized size = 2.25 \begin{align*} -\frac{1}{2} \,{\left (x^{2} + 3\right )} \arctan \left (x\right )^{2} + 3 \, x \arctan \left (x\right ) + \frac{1}{2} \,{\left ({\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 2 \, x \arctan \left (x\right ) - 3\right )} \log \left (x^{2} + 1\right ) + \frac{1}{4} \, \log \left (x^{2} + 1\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arctan(x)^2*log(x^2+1),x, algorithm="fricas")
[Out]
-1/2*(x^2 + 3)*arctan(x)^2 + 3*x*arctan(x) + 1/2*((x^2 + 1)*arctan(x)^2 - 2*x*arctan(x) - 3)*log(x^2 + 1) + 1/
4*log(x^2 + 1)^2
________________________________________________________________________________________
Sympy [A] time = 2.85276, size = 87, normalized size = 1.13 \begin{align*} \frac{x^{2} \log{\left (x^{2} + 1 \right )} \operatorname{atan}^{2}{\left (x \right )}}{2} - \frac{x^{2} \operatorname{atan}^{2}{\left (x \right )}}{2} - x \log{\left (x^{2} + 1 \right )} \operatorname{atan}{\left (x \right )} + 3 x \operatorname{atan}{\left (x \right )} + \frac{\log{\left (x^{2} + 1 \right )}^{2}}{4} + \frac{\log{\left (x^{2} + 1 \right )} \operatorname{atan}^{2}{\left (x \right )}}{2} - \frac{3 \log{\left (x^{2} + 1 \right )}}{2} - \frac{3 \operatorname{atan}^{2}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*atan(x)**2*ln(x**2+1),x)
[Out]
x**2*log(x**2 + 1)*atan(x)**2/2 - x**2*atan(x)**2/2 - x*log(x**2 + 1)*atan(x) + 3*x*atan(x) + log(x**2 + 1)**2
/4 + log(x**2 + 1)*atan(x)**2/2 - 3*log(x**2 + 1)/2 - 3*atan(x)**2/2
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \arctan \left (x\right )^{2} \log \left (x^{2} + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*arctan(x)^2*log(x^2+1),x, algorithm="giac")
[Out]
integrate(x*arctan(x)^2*log(x^2 + 1), x)