3.372 \(\int x^2 (c+a^2 c x^2)^2 \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=321 \[ -\frac{4 c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{35 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a^3}+\frac{c^2 \log \left (a^2 x^2+1\right )}{30 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a^3}-\frac{1}{140} a c^2 x^4-\frac{11 c^2 x^2}{420 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a} \]

[Out]

(-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + (17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c
^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a*x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcT
an[a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a*x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3
)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)
])/(35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3
- (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a^3)

________________________________________________________________________________________

Rubi [A]  time = 1.79533, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 73, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43} \[ -\frac{4 c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{35 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{35 a^3}+\frac{c^2 \log \left (a^2 x^2+1\right )}{30 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{35 a^3}-\frac{1}{140} a c^2 x^4-\frac{11 c^2 x^2}{420 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]

[Out]

(-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + (17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c
^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a*x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcT
an[a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a*x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3
)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)
])/(35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3
- (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a^3)

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

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 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

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])

Rubi steps

\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)^3+2 a^2 c^2 x^4 \tan ^{-1}(a x)^3+a^4 c^2 x^6 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\left (a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (6 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{7} \left (3 a^5 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{c^2 \int x \tan ^{-1}(a x)^2 \, dx}{a}+\frac{c^2 \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}-\frac{1}{5} \left (6 a c^2\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{5} \left (6 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{7} \left (3 a^3 c^2\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\frac{1}{7} \left (3 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x^2 \tan ^{-1}(a x)^2}{2 a}-\frac{3}{10} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{i c^2 \tan ^{-1}(a x)^3}{3 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3+c^2 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{c^2 \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\frac{\left (6 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx}{5 a}-\frac{\left (6 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (3 a c^2\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx-\frac{1}{7} \left (3 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{7} \left (a^4 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x^2 \tan ^{-1}(a x)^2}{10 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac{i c^2 \tan ^{-1}(a x)^3}{15 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3}+\frac{1}{5} \left (3 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{5} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (6 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{c^2 \int \tan ^{-1}(a x) \, dx}{a^2}-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac{\left (6 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{5 a^2}+\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac{\left (3 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx}{7 a}+\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{7 a}+\frac{1}{7} \left (a^2 c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac{1}{7} \left (a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{14} \left (3 a^2 c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x \tan ^{-1}(a x)}{a^2}+\frac{1}{5} c^2 x^3 \tan ^{-1}(a x)+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{2 a^3}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3+\frac{c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{5 a^3}-\frac{i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3}-\frac{1}{7} c^2 \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{7} c^2 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{14} \left (3 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac{1}{14} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{7} \left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{\left (i c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{7 a^2}-\frac{\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (6 c^2\right ) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{\left (6 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (12 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac{c^2 \int \frac{x}{1+a^2 x^2} \, dx}{a}-\frac{1}{5} \left (a c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{35} \left (a^3 c^2\right ) \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=-\frac{4 c^2 x \tan ^{-1}(a x)}{5 a^2}+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{2 c^2 \tan ^{-1}(a x)^2}{5 a^3}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a^3}-\frac{c^2 \log \left (1+a^2 x^2\right )}{2 a^3}+\frac{i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{5 a^3}-\frac{c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^3}-\frac{\left (6 i c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac{c^2 \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{14 a^2}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{14 a^2}+\frac{\left (3 c^2\right ) \int \tan ^{-1}(a x) \, dx}{7 a^2}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (6 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (3 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx}{5 a}+\frac{\left (6 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{21} \left (a c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{14} \left (a c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{10} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{70} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a^3}+\frac{2 c^2 \log \left (1+a^2 x^2\right )}{5 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a^3}+\frac{c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{10 a^3}+\frac{\left (3 i c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}-\frac{c^2 \int \frac{x}{1+a^2 x^2} \, dx}{7 a}-\frac{\left (3 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx}{14 a}-\frac{\left (3 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx}{7 a}+\frac{1}{42} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{28} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{70} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 c^2 x^2}{35 a}-\frac{1}{140} a c^2 x^4-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a^3}+\frac{13 c^2 \log \left (1+a^2 x^2\right )}{140 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a^3}-\frac{4 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{35 a^3}+\frac{1}{42} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{28} \left (a c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{11 c^2 x^2}{420 a}-\frac{1}{140} a c^2 x^4-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{35 a^3}+\frac{c^2 \log \left (1+a^2 x^2\right )}{30 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{35 a^3}-\frac{4 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{35 a^3}\\ \end{align*}

Mathematica [A]  time = 1.08935, size = 233, normalized size = 0.73 \[ \frac{c^2 \left (96 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-48 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-3 a^4 x^4-11 a^2 x^2+14 \log \left (a^2 x^2+1\right )+60 a^7 x^7 \tan ^{-1}(a x)^3-30 a^6 x^6 \tan ^{-1}(a x)^2+168 a^5 x^5 \tan ^{-1}(a x)^3+12 a^5 x^5 \tan ^{-1}(a x)-81 a^4 x^4 \tan ^{-1}(a x)^2+140 a^3 x^3 \tan ^{-1}(a x)^3+34 a^3 x^3 \tan ^{-1}(a x)-48 a^2 x^2 \tan ^{-1}(a x)^2-6 a x \tan ^{-1}(a x)+32 i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-96 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-8\right )}{420 a^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]

[Out]

(c^2*(-8 - 11*a^2*x^2 - 3*a^4*x^4 - 6*a*x*ArcTan[a*x] + 34*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] + 3*Ar
cTan[a*x]^2 - 48*a^2*x^2*ArcTan[a*x]^2 - 81*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 + (32*I)*ArcTan[a
*x]^3 + 140*a^3*x^3*ArcTan[a*x]^3 + 168*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*ArcTan[a*x]^3 - 96*ArcTan[a*x]^2*Lo
g[1 + E^((2*I)*ArcTan[a*x])] + 14*Log[1 + a^2*x^2] + (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 4
8*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a^3)

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Maple [C]  time = 1.699, size = 1121, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^3,x)

[Out]

-1/140*a*c^2*x^4+1/35*a^2*c^2*x^5*arctan(a*x)-11/420*c^2*x^2/a+17/210*c^2*x^3*arctan(a*x)+1/3*c^2*x^3*arctan(a
*x)^3+1/140*c^2*arctan(a*x)^2/a^3+2/35*I/a^3*c^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(
a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2-1/70*c^2*x*arctan(a*x)
/a^2-4/35*c^2*x^2*arctan(a*x)^2/a-27/140*a*c^2*x^4*arctan(a*x)^2-1/14*a^3*c^2*x^6*arctan(a*x)^2+2/5*a^2*c^2*x^
5*arctan(a*x)^3+1/7*a^4*c^2*x^7*arctan(a*x)^3-4/35/a^3*c^2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/15/a^3*c^2*ln
((1+I*a*x)^2/(a^2*x^2+1)+1)+2/35*I/a^3*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*
arctan(a*x)^2-2/35*I/a^3*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2+2/35*I/a^3*c^2*Pi*csgn(I
*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2+4/35/a^3*c^2*arctan(a*x)^2*ln(a^2*x^2+1)-8/35/a^3*c^2*arctan(a*x)^2*
ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-8/35/a^3*c^2*arctan(a*x)^2*ln(2)+1/15*I/a^3*c^2*arctan(a*x)+8/105*I/a^3*c^2*ar
ctan(a*x)^3-2/105/a^3*c^2-4/35*I/a^3*c^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)
)^2*arctan(a*x)^2+4/35*I/a^3*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^
2*arctan(a*x)^2+2/35*I/a^3*c^2*Pi*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*arctan
(a*x)^2-2/35*I/a^3*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2
+1)+1)^2)^2*arctan(a*x)^2-2/35*I/a^3*c^2*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I*(1+I*a*x)^2/(a^2*x^2+
1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2-2/35*I/a^3*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn
(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2+8/35*I/a^3*c^2*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{840} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right )^{3} - \frac{1}{1120} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{980 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3} - 4 \,{\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \,{\left (15 \, a^{6} c^{2} x^{8} + 42 \, a^{4} c^{2} x^{6} + 35 \, a^{2} c^{2} x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) +{\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3} + 105 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{1120 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="maxima")

[Out]

1/840*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^3 - 1/1120*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 +
35*c^2*x^3)*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/1120*(980*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x
^4 + c^2*x^2)*arctan(a*x)^3 - 4*(15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2*x^3)*arctan(a*x)^2 + 4*(15*a^6*c^2
*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)*arctan(a*x)*log(a^2*x^2 + 1) + (15*a^5*c^2*x^7 + 42*a^3*c^2*x^5 + 35*a
*c^2*x^3 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x
^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a**2*c*x**2+c)**2*atan(a*x)**3,x)

[Out]

c**2*(Integral(x**2*atan(a*x)**3, x) + Integral(2*a**2*x**4*atan(a*x)**3, x) + Integral(a**4*x**6*atan(a*x)**3
, x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^2*x^2*arctan(a*x)^3, x)