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

Optimal. Leaf size=276 \[ -\frac{3}{2} i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{4} i c \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+\frac{1}{2} c \tan ^{-1}(a x)^3-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2-3 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.522815, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4950, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315} \[ -\frac{3}{2} i c \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{4} i c \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+\frac{1}{2} c \tan ^{-1}(a x)^3-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2-3 c \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4950

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

Rule 4850

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

Rule 4988

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[(
Log[1 + u]*(a + b*ArcTan[c*x])^p)/(d + e*x^2), x], x] - Dist[1/2, Int[(Log[1 - u]*(a + b*ArcTan[c*x])^p)/(d +
e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - (2*I)/(I - c*x))^
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 4998

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(I*(a
+ b*ArcTan[c*x])^p*PolyLog[k + 1, u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[k
+ 1, u])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[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 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 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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac{\tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-(6 a c) \int \frac{\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3 c\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{1}{2} (3 a c) \int \tan ^{-1}(a x)^2 \, dx+\frac{1}{2} (3 a c) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+(3 a c) \int \frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} a c x \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+(3 i a c) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 i a c) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (3 a^2 c\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{2} (3 a c) \int \frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} (3 a c) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(3 a c) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx\\ &=-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{4} i c \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )+(3 a c) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{4} i c \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )-(3 i c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )\\ &=-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^3+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-3 c \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )-\frac{3}{2} i c \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{4} i c \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0709174, size = 284, normalized size = 1.03 \[ -\frac{3}{4} i c \text{PolyLog}\left (4,\frac{-a x-i}{a x-i}\right )+\frac{3}{4} i c \text{PolyLog}\left (4,\frac{a x+i}{a x-i}\right )+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{-a x-i}{a x-i}\right )-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{a x+i}{a x-i}\right )+\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,\frac{-a x-i}{a x-i}\right )-\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left (3,\frac{a x+i}{a x-i}\right )-\frac{3}{2} c \left (-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \tan ^{-1}(a x)^2-i \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )+\frac{1}{2} c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2 i}{-a x+i}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(1 + a^2*x^2)*ArcTan[a*x]^3)/2 + 2*c*ArcTan[a*x]^3*ArcTanh[1 - (2*I)/(I - a*x)] - (3*c*((-I)*ArcTan[a*x]^2
+ a*x*ArcTan[a*x]^2 + 2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/2
+ ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, (-I - a*x)/(-I + a*x)] - ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, (I + a*x)
/(-I + a*x)] + (3*c*ArcTan[a*x]*PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (3*c*ArcTan[a*x]*PolyLog[3, (I + a*x)/(
-I + a*x)])/2 - ((3*I)/4)*c*PolyLog[4, (-I - a*x)/(-I + a*x)] + ((3*I)/4)*c*PolyLog[4, (I + a*x)/(-I + a*x)]

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Maple [A]  time = 1.325, size = 460, normalized size = 1.7 \begin{align*}{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{2} \left ( -i\arctan \left ( ax \right ) +\arctan \left ( ax \right ) xa-3 \right ) \left ( ax+i \right ) }{2}}+6\,ic{\it polylog} \left ( 4,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,c\arctan \left ( ax \right ) \ln \left ({\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}}+1 \right ) +{\frac{3\,i}{2}}c{\it polylog} \left ( 2,-{\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}} \right ) +c \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\,ic \left ( \arctan \left ( ax \right ) \right ) ^{2}+6\,c\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,ic{\it polylog} \left ( 4,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +c \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\,ic \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,c\arctan \left ( ax \right ){\it polylog} \left ( 3,-{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\,ic \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -c \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ({\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}}+1 \right ) +{\frac{3\,i}{2}}c \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-{\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}} \right ) -{\frac{3\,c\arctan \left ( ax \right ) }{2}{\it polylog} \left ( 3,-{\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,i}{4}}c{\it polylog} \left ( 4,-{\frac{ \left ( 1+iax \right ) ^{2}}{{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*arctan(a*x)^2*(-I*arctan(a*x)+arctan(a*x)*x*a-3)*(a*x+I)+6*I*c*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*
c*arctan(a*x)*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+3/2*I*c*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+c*arctan(a*x)^3*ln(1-(
1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*c*arctan(a*x)^2+6*c*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*c*p
olylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+c*arctan(a*x)^3*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c*arctan(a*x)^2*p
olylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*c*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c*arctan(a*x
)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-c*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+3/2*I*c*arctan(a*x)^
2*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-3/2*c*arctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-3/4*I*c*polylog(4,
-(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}{16} \, a^{2} c x^{2} \arctan \left (a x\right )^{3} - \frac{3}{64} \, a^{2} c x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{12 \, a^{4} c x^{4} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) - 12 \, a^{3} c x^{3} \arctan \left (a x\right )^{2} + 56 \,{\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3} + 3 \,{\left (a^{3} c x^{3} + 2 \,{\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{64 \,{\left (a^{2} x^{3} + x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*a^2*c*x^2*arctan(a*x)^3 - 3/64*a^2*c*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2 + integrate(1/64*(12*a^4*c*x^4*ar
ctan(a*x)*log(a^2*x^2 + 1) - 12*a^3*c*x^3*arctan(a*x)^2 + 56*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 3*(
a^3*c*x^3 + 2*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^3 + x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(atan(a*x)**3/x, x) + Integral(a**2*x*atan(a*x)**3, 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 )} \arctan \left (a x\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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