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

Optimal. Leaf size=354 \[ \frac{3}{2} a c^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]

[Out]

-(a^3*c^3*x^2)/20 + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/20
- (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3*A
rcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (33*
a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/5 - a*c^3*Log[1 + a^2*x^2] + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*
x)] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(
1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10

________________________________________________________________________________________

Rubi [A]  time = 1.28021, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260, 266, 43} \[ \frac{3}{2} a c^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*c^3*x^2)/20 + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/20
- (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3*A
rcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (33*
a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/5 - a*c^3*Log[1 + a^2*x^2] + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*
x)] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(
1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10

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 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]

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 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 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 \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3}{x^2} \, dx &=\int \left (3 a^2 c^3 \tan ^{-1}(a x)^3+\frac{c^3 \tan ^{-1}(a x)^3}{x^2}+3 a^4 c^3 x^2 \tan ^{-1}(a x)^3+a^6 c^3 x^4 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (9 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (3 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^7 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=2 i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\left (3 i a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx+\left (9 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (3 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^5 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{5} \left (3 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+9 a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (18 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^3 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx-\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^6 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+6 a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+9 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx-\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{10} \left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=3 a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+6 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{9}{2} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{10} \left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{10} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{10} \left (a^5 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{3}{2} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+3 a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )-\frac{1}{5} \left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{20} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{21}{20} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )-\frac{1}{20} \left (a^5 c^3\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}{20} a^3 c^3 x^2+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.72241, size = 298, normalized size = 0.84 \[ \frac{c^3 \left (120 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-264 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+60 a x \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+132 a x \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-2 a^3 x^3-40 a x \log \left (a^2 x^2+1\right )+8 a^6 x^6 \tan ^{-1}(a x)^3-6 a^5 x^5 \tan ^{-1}(a x)^2+40 a^4 x^4 \tan ^{-1}(a x)^3+4 a^4 x^4 \tan ^{-1}(a x)-48 a^3 x^3 \tan ^{-1}(a x)^2+120 a^2 x^2 \tan ^{-1}(a x)^3+84 a^2 x^2 \tan ^{-1}(a x)-2 a x-5 i \pi ^3 a x-48 i a x \tan ^{-1}(a x)^3-42 a x \tan ^{-1}(a x)^2-40 \tan ^{-1}(a x)^3+120 a x \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+264 a x \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{40 x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(-2*a*x - (5*I)*a*Pi^3*x - 2*a^3*x^3 + 84*a^2*x^2*ArcTan[a*x] + 4*a^4*x^4*ArcTan[a*x] - 42*a*x*ArcTan[a*x
]^2 - 48*a^3*x^3*ArcTan[a*x]^2 - 6*a^5*x^5*ArcTan[a*x]^2 - 40*ArcTan[a*x]^3 - (48*I)*a*x*ArcTan[a*x]^3 + 120*a
^2*x^2*ArcTan[a*x]^3 + 40*a^4*x^4*ArcTan[a*x]^3 + 8*a^6*x^6*ArcTan[a*x]^3 + 120*a*x*ArcTan[a*x]^2*Log[1 - E^((
-2*I)*ArcTan[a*x])] + 264*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 40*a*x*Log[1 + a^2*x^2] + (120*I)
*a*x*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (264*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x]
)] + 60*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 132*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(40*x)

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Maple [C]  time = 10.748, size = 10139, normalized size = 28.6 \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)^3*arctan(a*x)^3/x^2,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)^3*arctan(a*x)^3/x^2,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^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

c**3*(Integral(3*a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x) + Integral(3*a**4*x**2*atan(a*x)**3, x
) + Integral(a**6*x**4*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 )}^{3} \arctan \left (a x\right )^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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