3.1275 \(\int x^4 \tan ^{-1}(x) \log (1+x^2) \, dx\)

Optimal. Leaf size=111 \[ \frac{9 x^4}{200}-\frac{77 x^2}{300}-\frac{1}{20} \log ^2\left (x^2+1\right )-\frac{1}{20} x^4 \log \left (x^2+1\right )+\frac{1}{10} x^2 \log \left (x^2+1\right )+\frac{137}{300} \log \left (x^2+1\right )-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)+\frac{1}{5} x^5 \log \left (x^2+1\right ) \tan ^{-1}(x)-\frac{2}{5} x \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2 \]

[Out]

(-77*x^2)/300 + (9*x^4)/200 - (2*x*ArcTan[x])/5 + (2*x^3*ArcTan[x])/15 - (2*x^5*ArcTan[x])/25 + ArcTan[x]^2/5
+ (137*Log[1 + x^2])/300 + (x^2*Log[1 + x^2])/10 - (x^4*Log[1 + x^2])/20 + (x^5*ArcTan[x]*Log[1 + x^2])/5 - Lo
g[1 + x^2]^2/20

________________________________________________________________________________________

Rubi [A]  time = 0.444091, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.167, Rules used = {4852, 266, 43, 5021, 6725, 446, 77, 4916, 4846, 260, 4884, 2475, 2390, 2301} \[ \frac{9 x^4}{200}-\frac{77 x^2}{300}-\frac{1}{20} \log ^2\left (x^2+1\right )-\frac{1}{20} x^4 \log \left (x^2+1\right )+\frac{1}{10} x^2 \log \left (x^2+1\right )+\frac{137}{300} \log \left (x^2+1\right )-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)+\frac{1}{5} x^5 \log \left (x^2+1\right ) \tan ^{-1}(x)-\frac{2}{5} x \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcTan[x]*Log[1 + x^2],x]

[Out]

(-77*x^2)/300 + (9*x^4)/200 - (2*x*ArcTan[x])/5 + (2*x^3*ArcTan[x])/15 - (2*x^5*ArcTan[x])/25 + ArcTan[x]^2/5
+ (137*Log[1 + x^2])/300 + (x^2*Log[1 + x^2])/10 - (x^4*Log[1 + x^2])/20 + (x^5*ArcTan[x]*Log[1 + x^2])/5 - Lo
g[1 + x^2]^2/20

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5021

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> With
[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegrand
[(x*u)/(f + g*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 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^4 \tan ^{-1}(x) \log \left (1+x^2\right ) \, dx &=\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{10} \log ^2\left (1+x^2\right )-2 \int \left (\frac{x^3 \left (2-x^2+4 x^3 \tan ^{-1}(x)\right )}{20 \left (1+x^2\right )}-\frac{x \log \left (1+x^2\right )}{10 \left (1+x^2\right )}\right ) \, dx\\ &=\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{10} \log ^2\left (1+x^2\right )-\frac{1}{10} \int \frac{x^3 \left (2-x^2+4 x^3 \tan ^{-1}(x)\right )}{1+x^2} \, dx+\frac{1}{5} \int \frac{x \log \left (1+x^2\right )}{1+x^2} \, dx\\ &=\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{10} \log ^2\left (1+x^2\right )-\frac{1}{10} \int \left (-\frac{x^3 \left (-2+x^2\right )}{1+x^2}+\frac{4 x^6 \tan ^{-1}(x)}{1+x^2}\right ) \, dx+\frac{1}{10} \operatorname{Subst}\left (\int \frac{\log (1+x)}{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{10} \log ^2\left (1+x^2\right )+\frac{1}{10} \int \frac{x^3 \left (-2+x^2\right )}{1+x^2} \, dx+\frac{1}{10} \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+x^2\right )-\frac{2}{5} \int \frac{x^6 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )+\frac{1}{20} \operatorname{Subst}\left (\int \frac{(-2+x) x}{1+x} \, dx,x,x^2\right )-\frac{2}{5} \int x^4 \tan ^{-1}(x) \, dx+\frac{2}{5} \int \frac{x^4 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )+\frac{1}{20} \operatorname{Subst}\left (\int \left (-3+x+\frac{3}{1+x}\right ) \, dx,x,x^2\right )+\frac{2}{25} \int \frac{x^5}{1+x^2} \, dx+\frac{2}{5} \int x^2 \tan ^{-1}(x) \, dx-\frac{2}{5} \int \frac{x^2 \tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac{3 x^2}{20}+\frac{x^4}{40}+\frac{2}{15} x^3 \tan ^{-1}(x)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{3}{20} \log \left (1+x^2\right )+\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )+\frac{1}{25} \operatorname{Subst}\left (\int \frac{x^2}{1+x} \, dx,x,x^2\right )-\frac{2}{15} \int \frac{x^3}{1+x^2} \, dx-\frac{2}{5} \int \tan ^{-1}(x) \, dx+\frac{2}{5} \int \frac{\tan ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac{3 x^2}{20}+\frac{x^4}{40}-\frac{2}{5} x \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2+\frac{3}{20} \log \left (1+x^2\right )+\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )+\frac{1}{25} \operatorname{Subst}\left (\int \left (-1+x+\frac{1}{1+x}\right ) \, dx,x,x^2\right )-\frac{1}{15} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^2\right )+\frac{2}{5} \int \frac{x}{1+x^2} \, dx\\ &=-\frac{19 x^2}{100}+\frac{9 x^4}{200}-\frac{2}{5} x \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2+\frac{39}{100} \log \left (1+x^2\right )+\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )-\frac{1}{15} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=-\frac{77 x^2}{300}+\frac{9 x^4}{200}-\frac{2}{5} x \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2+\frac{137}{300} \log \left (1+x^2\right )+\frac{1}{10} x^2 \log \left (1+x^2\right )-\frac{1}{20} x^4 \log \left (1+x^2\right )+\frac{1}{5} x^5 \tan ^{-1}(x) \log \left (1+x^2\right )-\frac{1}{20} \log ^2\left (1+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0245255, size = 79, normalized size = 0.71 \[ \frac{1}{600} \left (\left (27 x^2-154\right ) x^2-30 \log ^2\left (x^2+1\right )+\left (-30 x^4+60 x^2+274\right ) \log \left (x^2+1\right )+8 x \left (-6 x^4+10 x^2+15 x^4 \log \left (x^2+1\right )-30\right ) \tan ^{-1}(x)+120 \tan ^{-1}(x)^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcTan[x]*Log[1 + x^2],x]

[Out]

(x^2*(-154 + 27*x^2) + 120*ArcTan[x]^2 + (274 + 60*x^2 - 30*x^4)*Log[1 + x^2] - 30*Log[1 + x^2]^2 + 8*x*ArcTan
[x]*(-30 + 10*x^2 - 6*x^4 + 15*x^4*Log[1 + x^2]))/600

________________________________________________________________________________________

Maple [C]  time = 1.822, size = 3626, normalized size = 32.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arctan(x)*ln(x^2+1),x)

[Out]

1/5*x^2*ln(2)-1/5*ln((1+I*x)^2/(x^2+1)+1)*x^2+1/10*ln((1+I*x)^2/(x^2+1)+1)*x^4+46/75*I*arctan(x)+3/10*ln(2)+9/
200*x^4-77/300*x^2+3/20*I*Pi*csgn(I*(1+I*x)/(x^2+1)^(1/2))*csgn(I*(1+I*x)^2/(x^2+1))^2+3/40*I*Pi*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+3/40*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))
^2*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)-2/5*x*arctan(x)+2/15*x^3*arctan(x)-2/25*x^5*arctan(x)-3/40*I*Pi*csgn(I*(1+I
*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3-2/5*I*ln(2)*arctan(x)-1/10*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))*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)+1/10*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^5-1/5*I*arctan(x)*Pi*csgn(I*((1+I*x)^2/
(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*x^5+1/40*I*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^4-1/20*I*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+1/5*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^5+1/10*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)^2*x^5-1/10*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^5+1/10*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^5-1/10*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^5+1/10*(-4*I*arctan(x)+4*x^5*arctan(x)+4*ln((1+I*x)^2/(x^2+1)+1)+3+2*x^2-x^4)*ln((1
+I*x)/(x^2+1)^(1/2))-3/20*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2-3/40*I*Pi*csgn(
I*(1+I*x)/(x^2+1)^(1/2))^2*csgn(I*(1+I*x)^2/(x^2+1))+3/40*I*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+1/40*I*Pi*csgn(I*(1+I*x)^2/(x^2+1))^3*x^4+1/40*I*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((
1+I*x)^2/(x^2+1)+1)^2)^3*x^4-1/40*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3*x^4-1/20*I*Pi*csgn(I*(1+I*x)^2/(x^2+1
))^3*x^2+1/5*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+1)^(1/2))+1/10*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)^2-1/10*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+
1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2+1/10*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)^2-1/10*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))+1/10*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)-1/5*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
+1/5*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*(1+I*x)/(x^2+1)^(1/2))*csgn(I*(1+I*x)^2/(x^2+1))^2+1/10*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)^2-1/20*I*Pi*csgn(I
*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2*x^2+1/20*I*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+1/20*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))^2*csgn(I*((1+I*x)^2/(x^2+1
)+1)^2)*x^2-1/10*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*x^2-1/10*I*arctan(x)*Pi*
csgn(I*(1+I*x)^2/(x^2+1))^3*x^5-1/10*I*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3*x^5+1/
10*I*arctan(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3*x^5-1/20*I*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(
x^2+1)^(1/2))*x^4-1/40*I*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^4-1/40*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))^2*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)*x^4+1/20*I*Pi*csgn(I*((1+I*x)^
2/(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2*x^4+1/10*I*Pi*csgn(I*(1+I*x)^2/(x^2+1))^2*csgn(I*(1+I*x)/(x^2+
1)^(1/2))*x^2+1/20*I*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^2-3/40
*I*Pi*csgn(I/((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+I*x)^2/(x^2+1)+1)
^2)-1/10*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)-1/40*I*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^
4+1/40*I*Pi*csgn(I*(1+I*x)^2/(x^2+1))*csgn(I*(1+I*x)/(x^2+1)^(1/2))^2*x^4-1/10*ln(2)*x^4-137/150*ln((1+I*x)^2/
(x^2+1)+1)-1/5*ln((1+I*x)^2/(x^2+1)+1)^2-3/40*I*Pi*csgn(I*(1+I*x)^2/(x^2+1))^3-1/10*arctan(x)*Pi*csgn(I*(1+I*x
)^2/(x^2+1))^3-1/10*arctan(x)*Pi*csgn(I*(1+I*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3+1/10*arctan(x)*Pi*csgn(I*
((1+I*x)^2/(x^2+1)+1)^2)^3-2/5*arctan(x)*ln((1+I*x)^2/(x^2+1)+1)*x^5+2/5*arctan(x)*ln(2)*x^5-1/10*I*Pi*ln((1+I
*x)^2/(x^2+1)+1)*csgn(I*(1+I*x)^2/(x^2+1))^3-1/10*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+1/10*I*Pi*ln((1+I*x)^2/(x^2+1)+1)*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3-1/20*I*Pi*csgn(I*(1+I
*x)^2/(x^2+1)/((1+I*x)^2/(x^2+1)+1)^2)^3*x^2+1/20*I*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^3*x^2+3/40*I*Pi*csgn(I*
((1+I*x)^2/(x^2+1)+1)^2)^3+1/10*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)+1/10*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)-1/5*arcta
n(x)*Pi*csgn(I*((1+I*x)^2/(x^2+1)+1))*csgn(I*((1+I*x)^2/(x^2+1)+1)^2)^2-181/600+2/5*ln(2)*ln((1+I*x)^2/(x^2+1)
+1)

________________________________________________________________________________________

Maxima [A]  time = 1.64656, size = 108, normalized size = 0.97 \begin{align*} \frac{9}{200} \, x^{4} - \frac{77}{300} \, x^{2} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (x^{2} + 1\right ) - 6 \, x^{5} + 10 \, x^{3} - 30 \, x + 30 \, \arctan \left (x\right )\right )} \arctan \left (x\right ) - \frac{1}{5} \, \arctan \left (x\right )^{2} - \frac{1}{300} \,{\left (15 \, x^{4} - 30 \, x^{2} - 137\right )} \log \left (x^{2} + 1\right ) - \frac{1}{20} \, \log \left (x^{2} + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(x)*log(x^2+1),x, algorithm="maxima")

[Out]

9/200*x^4 - 77/300*x^2 + 1/75*(15*x^5*log(x^2 + 1) - 6*x^5 + 10*x^3 - 30*x + 30*arctan(x))*arctan(x) - 1/5*arc
tan(x)^2 - 1/300*(15*x^4 - 30*x^2 - 137)*log(x^2 + 1) - 1/20*log(x^2 + 1)^2

________________________________________________________________________________________

Fricas [A]  time = 1.18781, size = 227, normalized size = 2.05 \begin{align*} \frac{9}{200} \, x^{4} - \frac{77}{300} \, x^{2} - \frac{2}{75} \,{\left (3 \, x^{5} - 5 \, x^{3} + 15 \, x\right )} \arctan \left (x\right ) + \frac{1}{5} \, \arctan \left (x\right )^{2} + \frac{1}{300} \,{\left (60 \, x^{5} \arctan \left (x\right ) - 15 \, x^{4} + 30 \, x^{2} + 137\right )} \log \left (x^{2} + 1\right ) - \frac{1}{20} \, \log \left (x^{2} + 1\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(x)*log(x^2+1),x, algorithm="fricas")

[Out]

9/200*x^4 - 77/300*x^2 - 2/75*(3*x^5 - 5*x^3 + 15*x)*arctan(x) + 1/5*arctan(x)^2 + 1/300*(60*x^5*arctan(x) - 1
5*x^4 + 30*x^2 + 137)*log(x^2 + 1) - 1/20*log(x^2 + 1)^2

________________________________________________________________________________________

Sympy [A]  time = 8.40477, size = 107, normalized size = 0.96 \begin{align*} \frac{x^{5} \log{\left (x^{2} + 1 \right )} \operatorname{atan}{\left (x \right )}}{5} - \frac{2 x^{5} \operatorname{atan}{\left (x \right )}}{25} - \frac{x^{4} \log{\left (x^{2} + 1 \right )}}{20} + \frac{9 x^{4}}{200} + \frac{2 x^{3} \operatorname{atan}{\left (x \right )}}{15} + \frac{x^{2} \log{\left (x^{2} + 1 \right )}}{10} - \frac{77 x^{2}}{300} - \frac{2 x \operatorname{atan}{\left (x \right )}}{5} - \frac{\log{\left (x^{2} + 1 \right )}^{2}}{20} + \frac{137 \log{\left (x^{2} + 1 \right )}}{300} + \frac{\operatorname{atan}^{2}{\left (x \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*atan(x)*ln(x**2+1),x)

[Out]

x**5*log(x**2 + 1)*atan(x)/5 - 2*x**5*atan(x)/25 - x**4*log(x**2 + 1)/20 + 9*x**4/200 + 2*x**3*atan(x)/15 + x*
*2*log(x**2 + 1)/10 - 77*x**2/300 - 2*x*atan(x)/5 - log(x**2 + 1)**2/20 + 137*log(x**2 + 1)/300 + atan(x)**2/5

________________________________________________________________________________________

Giac [A]  time = 1.10394, size = 227, normalized size = 2.05 \begin{align*} \frac{1}{10} \, \pi x^{5} \log \left (x^{2} + 1\right ) \mathrm{sgn}\left (x\right ) - \frac{1}{5} \, x^{5} \arctan \left (\frac{1}{x}\right ) \log \left (x^{2} + 1\right ) - \frac{1}{25} \, \pi x^{5} \mathrm{sgn}\left (x\right ) + \frac{2}{25} \, x^{5} \arctan \left (\frac{1}{x}\right ) - \frac{1}{20} \, x^{4} \log \left (x^{2} + 1\right ) + \frac{1}{15} \, \pi x^{3} \mathrm{sgn}\left (x\right ) + \frac{9}{200} \, x^{4} - \frac{2}{15} \, x^{3} \arctan \left (\frac{1}{x}\right ) + \frac{1}{10} \, x^{2} \log \left (x^{2} + 1\right ) - \frac{3}{10} \, \pi ^{2} \mathrm{sgn}\left (x\right ) - \frac{1}{5} \, \pi x \mathrm{sgn}\left (x\right ) - \frac{1}{5} \, \pi \arctan \left (\frac{1}{x}\right ) \mathrm{sgn}\left (x\right ) + \frac{1}{10} \, \pi ^{2} - \frac{77}{300} \, x^{2} + \frac{1}{5} \, \pi \arctan \left (x\right ) + \frac{1}{5} \, \pi \arctan \left (\frac{1}{x}\right ) + \frac{2}{5} \, x \arctan \left (\frac{1}{x}\right ) + \frac{1}{5} \, \arctan \left (\frac{1}{x}\right )^{2} - \frac{1}{20} \, \log \left (x^{2} + 1\right )^{2} + \frac{137}{300} \, \log \left (x^{2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arctan(x)*log(x^2+1),x, algorithm="giac")

[Out]

1/10*pi*x^5*log(x^2 + 1)*sgn(x) - 1/5*x^5*arctan(1/x)*log(x^2 + 1) - 1/25*pi*x^5*sgn(x) + 2/25*x^5*arctan(1/x)
- 1/20*x^4*log(x^2 + 1) + 1/15*pi*x^3*sgn(x) + 9/200*x^4 - 2/15*x^3*arctan(1/x) + 1/10*x^2*log(x^2 + 1) - 3/1
0*pi^2*sgn(x) - 1/5*pi*x*sgn(x) - 1/5*pi*arctan(1/x)*sgn(x) + 1/10*pi^2 - 77/300*x^2 + 1/5*pi*arctan(x) + 1/5*
pi*arctan(1/x) + 2/5*x*arctan(1/x) + 1/5*arctan(1/x)^2 - 1/20*log(x^2 + 1)^2 + 137/300*log(x^2 + 1)