3.378 \(\int \frac{(c+a^2 c x^2)^2 \tan ^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=311 \[ \frac{5}{2} a^3 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{2} a^3 c^2 \log \left (a^2 x^2+1\right )+a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)^3-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}+3 a^3 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+5 a^3 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3} \]

[Out]

-((a^2*c^2*ArcTan[a*x])/x) - (a^3*c^2*ArcTan[a*x]^2)/2 - (a*c^2*ArcTan[a*x]^2)/(2*x^2) - ((2*I)/3)*a^3*c^2*Arc
Tan[a*x]^3 - (c^2*ArcTan[a*x]^3)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^3)/x + a^4*c^2*x*ArcTan[a*x]^3 + a^3*c^2*Log
[x] + 3*a^3*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - (a^3*c^2*Log[1 + a^2*x^2])/2 + 5*a^3*c^2*ArcTan[a*x]^2*Log[
2 - 2/(1 - I*a*x)] - (5*I)*a^3*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a^3*c^2*ArcTan[a*x]*Poly
Log[2, 1 - 2/(1 + I*a*x)] + (5*a^3*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a^3*c^2*PolyLog[3, 1 - 2/(1 + I*
a*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.789185, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4918, 266, 36, 29, 31, 4924, 4868, 4992} \[ \frac{5}{2} a^3 c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{2} a^3 c^2 \log \left (a^2 x^2+1\right )+a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)^3-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}+3 a^3 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+5 a^3 c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^4,x]

[Out]

-((a^2*c^2*ArcTan[a*x])/x) - (a^3*c^2*ArcTan[a*x]^2)/2 - (a*c^2*ArcTan[a*x]^2)/(2*x^2) - ((2*I)/3)*a^3*c^2*Arc
Tan[a*x]^3 - (c^2*ArcTan[a*x]^3)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^3)/x + a^4*c^2*x*ArcTan[a*x]^3 + a^3*c^2*Log
[x] + 3*a^3*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - (a^3*c^2*Log[1 + a^2*x^2])/2 + 5*a^3*c^2*ArcTan[a*x]^2*Log[
2 - 2/(1 - I*a*x)] - (5*I)*a^3*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a^3*c^2*ArcTan[a*x]*Poly
Log[2, 1 - 2/(1 + I*a*x)] + (5*a^3*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a^3*c^2*PolyLog[3, 1 - 2/(1 + I*
a*x)])/2

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 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 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 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 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 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 4918

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 4924

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

Rule 4868

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTan[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2 - 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 4992

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a + b*ArcT
an[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]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3}{x^4} \, dx &=\int \left (a^4 c^2 \tan ^{-1}(a x)^3+\frac{c^2 \tan ^{-1}(a x)^3}{x^4}+\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x^2}\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^3}{x^4} \, dx+\left (2 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^4 c^2\right ) \int \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+\left (a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (6 a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (3 a^5 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+\left (a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x^3} \, dx+\left (6 i a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx-\left (a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx+\left (3 a^4 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx\\ &=-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+6 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (i a^3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx-\left (6 a^4 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (12 a^4 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+5 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-6 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{x^2} \, dx-\left (3 i a^4 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (6 i a^4 c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (a^4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^4 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+5 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+3 a^3 c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\left (a^3 c^2\right ) \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx-\left (i a^4 c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+5 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{5}{2} a^3 c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+5 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{5}{2} a^3 c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (a^5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}+a^4 c^2 x \tan ^{-1}(a x)^3+a^3 c^2 \log (x)+3 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{1}{2} a^3 c^2 \log \left (1+a^2 x^2\right )+5 a^3 c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-5 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+3 i a^3 c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{5}{2} a^3 c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{3}{2} a^3 c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.502387, size = 289, normalized size = 0.93 \[ \frac{c^2 \left (120 i a^3 x^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-72 i a^3 x^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+60 a^3 x^3 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+36 a^3 x^3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-5 i \pi ^3 a^3 x^3+24 a^3 x^3 \log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+24 a^4 x^4 \tan ^{-1}(a x)^3+16 i a^3 x^3 \tan ^{-1}(a x)^3-12 a^3 x^3 \tan ^{-1}(a x)^2-48 a^2 x^2 \tan ^{-1}(a x)^3-24 a^2 x^2 \tan ^{-1}(a x)+120 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+72 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-12 a x \tan ^{-1}(a x)^2-8 \tan ^{-1}(a x)^3\right )}{24 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^4,x]

[Out]

(c^2*((-5*I)*a^3*Pi^3*x^3 - 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTan[a*x]^2 - 8*ArcTa
n[a*x]^3 - 48*a^2*x^2*ArcTan[a*x]^3 + (16*I)*a^3*x^3*ArcTan[a*x]^3 + 24*a^4*x^4*ArcTan[a*x]^3 + 120*a^3*x^3*Ar
cTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 72*a^3*x^3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^3
*x^3*Log[(a*x)/Sqrt[1 + a^2*x^2]] + (120*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (72*I)*a^
3*x^3*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 60*a^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 36*a^3*
x^3*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(24*x^3)

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Maple [C]  time = 2.881, size = 5651, normalized size = 18.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^2*arctan(a*x)^3/x^4,x)

[Out]

result too large to display

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)^3/x^4,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)^3/x^4,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**2*atan(a*x)**3/x**4,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^2*arctan(a*x)^3/x^4,x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^2*arctan(a*x)^3/x^4, x)