3.88 \(\int \frac{(d+i c d x)^3 (a+b \tan ^{-1}(c x))^2}{x} \, dx\)

Optimal. Leaf size=385 \[ -i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b d^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{10}{3} b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 a b c d^3 x+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{20}{3} i b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d^3 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} b^2 d^3 \log \left (c^2 x^2+1\right )-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x) \]

[Out]

3*a*b*c*d^3*x - (I/3)*b^2*c*d^3*x + (I/3)*b^2*d^3*ArcTan[c*x] + 3*b^2*c*d^3*x*ArcTan[c*x] + (I/3)*b*c^2*d^3*x^
2*(a + b*ArcTan[c*x]) - (29*d^3*(a + b*ArcTan[c*x])^2)/6 + (3*I)*c*d^3*x*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^
2*(a + b*ArcTan[c*x])^2)/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x])^2 + 2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 -
2/(1 + I*c*x)] + ((20*I)/3)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (3*b^2*d^3*Log[1 + c^2*x^2])/2 - (
10*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/3 - I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*
d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^3*
PolyLog[3, -1 + 2/(1 + I*c*x)])/2

________________________________________________________________________________________

Rubi [A]  time = 0.769411, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 260, 321, 203} \[ -i b d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b d^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{10}{3} b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 a b c d^3 x+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{20}{3} i b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d^3 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} b^2 d^3 \log \left (c^2 x^2+1\right )-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*a*b*c*d^3*x - (I/3)*b^2*c*d^3*x + (I/3)*b^2*d^3*ArcTan[c*x] + 3*b^2*c*d^3*x*ArcTan[c*x] + (I/3)*b*c^2*d^3*x^
2*(a + b*ArcTan[c*x]) - (29*d^3*(a + b*ArcTan[c*x])^2)/6 + (3*I)*c*d^3*x*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^
2*(a + b*ArcTan[c*x])^2)/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x])^2 + 2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 -
2/(1 + I*c*x)] + ((20*I)/3)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (3*b^2*d^3*Log[1 + c^2*x^2])/2 - (
10*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/3 - I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*
d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^3*
PolyLog[3, -1 + 2/(1 + I*c*x)])/2

Rule 4876

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

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 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx &=\int \left (3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx+\left (3 i c d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (3 c^2 d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-\left (4 b c d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (6 i b c^2 d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\left (3 b c^3 d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (2 i b c^4 d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\left (6 i b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx+\left (2 b c d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 b c d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 b c d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (3 b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx+\frac{1}{3} \left (2 i b c^2 d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (2 i b c^2 d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=3 a b c d^3 x+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+6 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\frac{1}{3} \left (2 i b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx+\left (i b^2 c d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (i b^2 c d^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (6 i b^2 c d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 b^2 c d^3\right ) \int \tan ^{-1}(c x) \, dx-\frac{1}{3} \left (i b^2 c^3 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=3 a b c d^3 x-\frac{1}{3} i b^2 c d^3 x+3 b^2 c d^3 x \tan ^{-1}(c x)+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\frac{20}{3} i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )-\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )+\frac{1}{3} \left (i b^2 c d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx-\frac{1}{3} \left (2 i b^2 c d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=3 a b c d^3 x-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x)+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\frac{20}{3} i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 d^3 \log \left (1+c^2 x^2\right )-3 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{3} \left (2 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )\\ &=3 a b c d^3 x-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x)+\frac{1}{3} i b c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{29}{6} d^3 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{2} c^2 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i c^3 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\frac{20}{3} i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 d^3 \log \left (1+c^2 x^2\right )-\frac{10}{3} b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )+\frac{1}{2} b^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )\\ \end{align*}

Mathematica [A]  time = 0.88216, size = 465, normalized size = 1.21 \[ -\frac{1}{24} i d^3 \left (-24 a b \text{PolyLog}(2,-i c x)+24 a b \text{PolyLog}(2,i c x)-24 b^2 \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )-8 b^2 \left (3 \tan ^{-1}(c x)-10 i\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+12 i b^2 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )-12 i b^2 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+8 a^2 c^3 x^3-36 i a^2 c^2 x^2-72 a^2 c x+24 i a^2 \log (c x)-8 a b c^2 x^2+80 a b \log \left (c^2 x^2+1\right )+16 a b c^3 x^3 \tan ^{-1}(c x)-72 i a b c^2 x^2 \tan ^{-1}(c x)+72 i a b c x-144 a b c x \tan ^{-1}(c x)-72 i a b \tan ^{-1}(c x)-36 i b^2 \log \left (c^2 x^2+1\right )+8 b^2 c^3 x^3 \tan ^{-1}(c x)^2-36 i b^2 c^2 x^2 \tan ^{-1}(c x)^2-8 b^2 c^2 x^2 \tan ^{-1}(c x)+8 b^2 c x-72 b^2 c x \tan ^{-1}(c x)^2+72 i b^2 c x \tan ^{-1}(c x)-16 b^2 \tan ^{-1}(c x)^3+44 i b^2 \tan ^{-1}(c x)^2-8 b^2 \tan ^{-1}(c x)+24 i b^2 \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-24 i b^2 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-160 b^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\pi ^3 b^2\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-I/24)*d^3*(b^2*Pi^3 - 72*a^2*c*x + (72*I)*a*b*c*x + 8*b^2*c*x - (36*I)*a^2*c^2*x^2 - 8*a*b*c^2*x^2 + 8*a^2*c
^3*x^3 - (72*I)*a*b*ArcTan[c*x] - 8*b^2*ArcTan[c*x] - 144*a*b*c*x*ArcTan[c*x] + (72*I)*b^2*c*x*ArcTan[c*x] - (
72*I)*a*b*c^2*x^2*ArcTan[c*x] - 8*b^2*c^2*x^2*ArcTan[c*x] + 16*a*b*c^3*x^3*ArcTan[c*x] + (44*I)*b^2*ArcTan[c*x
]^2 - 72*b^2*c*x*ArcTan[c*x]^2 - (36*I)*b^2*c^2*x^2*ArcTan[c*x]^2 + 8*b^2*c^3*x^3*ArcTan[c*x]^2 - 16*b^2*ArcTa
n[c*x]^3 + (24*I)*b^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 160*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*Arc
Tan[c*x])] - (24*I)*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a^2*Log[c*x] + 80*a*b*Log[1 + c^
2*x^2] - (36*I)*b^2*Log[1 + c^2*x^2] - 24*b^2*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 8*b^2*(-10*I +
3*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 24*a*b*PolyLog[2, (-I)*c*x] + 24*a*b*PolyLog[2, I*c*x] + (
12*I)*b^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (12*I)*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])])

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Maple [C]  time = 2.737, size = 1651, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*d^3*b^2+11/6*d^3*b^2*arctan(c*x)^2-10/3*I*d^3*a*b*ln(c^2*x^2+1)+2*d^3*a*b*arctan(c*x)*ln(c*x)+3*I*d^3*a^2*
c*x-1/3*I*d^3*a^2*c^3*x^3+3*a*b*c*d^3*x+3*b^2*c*d^3*x*arctan(c*x)-1/3*I*b^2*c*d^3*x-3/2*d^3*b^2*arctan(c*x)^2*
c^2*x^2+I*d^3*b^2*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2*I*d^3*b^2*arctan(c*x)*polylog(2,-(1+I*c*x)
/(c^2*x^2+1)^(1/2))+20/3*I*d^3*b^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-2*I*d^3*b^2*arctan(c*x)*pol
ylog(2,(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2*I*d^3*b^2*Pi*arctan(c*x)^2+20/3*I*d^3*b^2*arctan(c*x)*ln(1+I*(1+I*c*x)
/(c^2*x^2+1)^(1/2))+I*d^3*a*b*dilog(1+I*c*x)-I*d^3*a*b*dilog(1-I*c*x)+1/2*I*d^3*b^2*Pi*csgn(I*((1+I*c*x)^2/(c^
2*x^2+1)-1))*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1
))*arctan(c*x)^2+2*d^3*b^2*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+20/3*d^3*b^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)
^(1/2))-1/2*d^3*b^2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+2*d^3*b^2*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+3*d^
3*b^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+20/3*d^3*b^2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+d^3*a^2*ln(c*x)+1/2*I*
d^3*b^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((
1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-1/2*I*d^3*b^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I*c*x
)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-1/2*I*d^3*b^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+
1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-1/2*I*d^3*b^2*Pi*csgn(I
*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x
^2+1)+1))^2*arctan(c*x)^2+6*I*d^3*a*b*arctan(c*x)*c*x-2/3*I*d^3*a*b*arctan(c*x)*c^3*x^3-3/2*d^3*a^2*c^2*x^2+d^
3*b^2*arctan(c*x)^2*ln(c*x)+d^3*b^2*arctan(c*x)^2*ln(1-(1+I*c*x)/(c^2*x^2+1)^(1/2))-d^3*b^2*arctan(c*x)^2*ln((
1+I*c*x)^2/(c^2*x^2+1)-1)+d^3*b^2*arctan(c*x)^2*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1/2))-8/3*I*d^3*b^2*arctan(c*x)-3*
d^3*a*b*arctan(c*x)-3*d^3*a*b*arctan(c*x)*c^2*x^2+1/3*I*d^3*b^2*arctan(c*x)*c^2*x^2+3*I*d^3*b^2*arctan(c*x)^2*
c*x-1/3*I*d^3*b^2*arctan(c*x)^2*c^3*x^3+1/3*I*d^3*a*b*c^2*x^2+I*d^3*a*b*ln(c*x)*ln(1+I*c*x)-I*d^3*a*b*ln(c*x)*
ln(1-I*c*x)-1/2*I*d^3*b^2*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/2
*I*d^3*b^2*Pi*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2+1/2*I*d^3*b^2*Pi*c
sgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*a^2*c^3*d^3*x^3 - 36*I*b^2*c^5*d^3*integrate(1/48*x^5*arctan(c*x)^2/(c^2*x^3 + x), x) - 12*b^2*c^5*d^3*
integrate(1/48*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 3*I*b^2*c^5*d^3*integrate(1/48*x^5*log(c^2
*x^2 + 1)^2/(c^2*x^3 + x), x) - 96*I*a*b*c^5*d^3*integrate(1/48*x^5*arctan(c*x)/(c^2*x^3 + x), x) - 8*b^2*c^5*
d^3*integrate(1/48*x^5*arctan(c*x)/(c^2*x^3 + x), x) - 4*I*b^2*c^5*d^3*integrate(1/48*x^5*log(c^2*x^2 + 1)/(c^
2*x^3 + x), x) - 108*b^2*c^4*d^3*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^3 + x), x) + 36*I*b^2*c^4*d^3*integra
te(1/48*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 9*b^2*c^4*d^3*integrate(1/48*x^4*log(c^2*x^2 + 1)
^2/(c^2*x^3 + x), x) - 288*a*b*c^4*d^3*integrate(1/48*x^4*arctan(c*x)/(c^2*x^3 + x), x) + 44*I*b^2*c^4*d^3*int
egrate(1/48*x^4*arctan(c*x)/(c^2*x^3 + x), x) - 22*b^2*c^4*d^3*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^3 +
x), x) - 3/2*a^2*c^2*d^3*x^2 + 72*I*b^2*c^3*d^3*integrate(1/48*x^3*arctan(c*x)^2/(c^2*x^3 + x), x) + 24*b^2*c^
3*d^3*integrate(1/48*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + 6*I*b^2*c^3*d^3*integrate(1/48*x^3*l
og(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) - 96*I*a*b*c^3*d^3*integrate(1/48*x^3*arctan(c*x)/(c^2*x^3 + x), x) + 108*
b^2*c^3*d^3*integrate(1/48*x^3*arctan(c*x)/(c^2*x^3 + x), x) + 54*I*b^2*c^3*d^3*integrate(1/48*x^3*log(c^2*x^2
+ 1)/(c^2*x^3 + x), x) + 3/4*I*b^2*d^3*arctan(c*x)^3 - 72*b^2*c^2*d^3*integrate(1/48*x^2*arctan(c*x)^2/(c^2*x
^3 + x), x) + 24*I*b^2*c^2*d^3*integrate(1/48*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) - 192*a*b*c^2
*d^3*integrate(1/48*x^2*arctan(c*x)/(c^2*x^3 + x), x) - 72*I*b^2*c^2*d^3*integrate(1/48*x^2*arctan(c*x)/(c^2*x
^3 + x), x) - 1/48*b^2*d^3*log(c^2*x^2 + 1)^3 + 3*I*a^2*c*d^3*x + 36*b^2*c*d^3*integrate(1/48*x*arctan(c*x)*lo
g(c^2*x^2 + 1)/(c^2*x^3 + x), x) + 9*I*b^2*c*d^3*integrate(1/48*x*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 3/16*
b^2*d^3*log(c^2*x^2 + 1)^2 + 3*I*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^3 + 36*b^2*d^3*integrate(1/48*ar
ctan(c*x)^2/(c^2*x^3 + x), x) - 12*I*b^2*d^3*integrate(1/48*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^3 + x), x) + 3
*b^2*d^3*integrate(1/48*log(c^2*x^2 + 1)^2/(c^2*x^3 + x), x) + 96*a*b*d^3*integrate(1/48*arctan(c*x)/(c^2*x^3
+ x), x) + a^2*d^3*log(x) - 1/96*(8*I*b^2*c^3*d^3*x^3 + 36*b^2*c^2*d^3*x^2 - 72*I*b^2*c*d^3*x)*arctan(c*x)^2 +
1/96*(8*b^2*c^3*d^3*x^3 - 36*I*b^2*c^2*d^3*x^2 - 72*b^2*c*d^3*x)*arctan(c*x)*log(c^2*x^2 + 1) - 1/96*(-2*I*b^
2*c^3*d^3*x^3 - 9*b^2*c^2*d^3*x^2 + 18*I*b^2*c*d^3*x)*log(c^2*x^2 + 1)^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-4 i \, a^{2} c^{3} d^{3} x^{3} - 12 \, a^{2} c^{2} d^{3} x^{2} + 12 i \, a^{2} c d^{3} x + 4 \, a^{2} d^{3} +{\left (i \, b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} - 3 i \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\left (4 \, a b c^{3} d^{3} x^{3} - 12 i \, a b c^{2} d^{3} x^{2} - 12 \, a b c d^{3} x + 4 i \, a b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{4 \, x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/4*(-4*I*a^2*c^3*d^3*x^3 - 12*a^2*c^2*d^3*x^2 + 12*I*a^2*c*d^3*x + 4*a^2*d^3 + (I*b^2*c^3*d^3*x^3 +
3*b^2*c^2*d^3*x^2 - 3*I*b^2*c*d^3*x - b^2*d^3)*log(-(c*x + I)/(c*x - I))^2 + (4*a*b*c^3*d^3*x^3 - 12*I*a*b*c^2
*d^3*x^2 - 12*a*b*c*d^3*x + 4*I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*(Integral(a**2/x, x) + Integral(3*I*a**2*c, x) + Integral(-3*a**2*c**2*x, x) + Integral(b**2*atan(c*x)**2
/x, x) + Integral(-I*a**2*c**3*x**2, x) + Integral(3*I*b**2*c*atan(c*x)**2, x) + Integral(2*a*b*atan(c*x)/x, x
) + Integral(-3*b**2*c**2*x*atan(c*x)**2, x) + Integral(6*I*a*b*c*atan(c*x), x) + Integral(-I*b**2*c**3*x**2*a
tan(c*x)**2, x) + Integral(-6*a*b*c**2*x*atan(c*x), x) + Integral(-2*I*a*b*c**3*x**2*atan(c*x), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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