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

Optimal. Leaf size=306 \[ \frac{2 b c \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{4 i c \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.767048, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854} \[ \frac{2 b c \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{4 i c \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*c)/(2*d^2*(I - c*x)) + (b^2*c*ArcTan[c*x])/(2*d^2) - (I*b*c*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - ((I/2
)*c*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(d^2*x) + (c*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) - (
(4*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 - ((2*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*
x)])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d
^2 + (2*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])
/d^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 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 4864

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

Rule 4862

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

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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])

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2 (d+i c d x)^2} \, dx &=\int \left (\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-i+c x)^2}+\frac{2 i c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-i+c x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx}{d^2}-\frac{(2 i c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^2}+\frac{\left (2 i c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{d^2}+\frac{c^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{d^2}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{(2 b c) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac{\left (4 i b c^2\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}{d^2}+\frac{\left (8 i b c^2\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}{d^2}+\frac{\left (2 b c^2\right ) \int \left (-\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{(2 i b c) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx}{d^2}-\frac{\left (i b c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{d^2}+\frac{\left (i b c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{d^2}-\frac{\left (4 i b c^2\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}{d^2}+\frac{\left (4 i b c^2\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}{d^2}-\frac{\left (2 b^2 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac{\left (2 b^2 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac{\left (2 b^2 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac{\left (2 b^2 c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=-\frac{b^2 c}{2 d^2 (i-c x)}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (b^2 c^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac{b^2 c}{2 d^2 (i-c x)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 2.57128, size = 398, normalized size = 1.3 \[ -\frac{6 a b c \left (4 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )-4 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+8 \tan ^{-1}(c x)^2-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )+\tan ^{-1}(c x) \left (\frac{4}{c x}+8 i \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+2 \sin \left (2 \tan ^{-1}(c x)\right )+2 i \cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 c \left (-24 \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+\frac{12 \tan ^{-1}(c x)^2}{c x}+12 i \tan ^{-1}(c x)^2+24 i \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-24 \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+6 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-6 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-3 \sin \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+6 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-3 i \cos \left (2 \tan ^{-1}(c x)\right )+\pi ^3\right )-12 i a^2 c \log \left (c^2 x^2+1\right )+\frac{12 a^2 c}{c x-i}+24 i a^2 c \log (c x)+24 a^2 c \tan ^{-1}(c x)+\frac{12 a^2}{x}}{12 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((12*a^2)/x + (12*a^2*c)/(-I + c*x) + 24*a^2*c*ArcTan[c*x] + (24*I)*a^2*c*Log[c*x] - (12*I)*a^2*c*Log[1 + c^2
*x^2] + b^2*c*(Pi^3 + (12*I)*ArcTan[c*x]^2 + (12*ArcTan[c*x]^2)/(c*x) - (3*I)*Cos[2*ArcTan[c*x]] + 6*ArcTan[c*
x]*Cos[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + (24*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan
[c*x])] - 24*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] +
(12*I)*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 3*Sin[2*ArcTan[c*x]] -
(6*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + 6*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]) + 6*a*b*c*(8*ArcTan[c*x]^2 + Cos[2*
ArcTan[c*x]] - 4*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 4*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + A
rcTan[c*x]*(4/(c*x) + (2*I)*Cos[2*ArcTan[c*x]] + (8*I)*Log[1 - E^((2*I)*ArcTan[c*x])] + 2*Sin[2*ArcTan[c*x]]))
)/(12*d^2)

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Maple [C]  time = 0.996, size = 9420, normalized size = 30.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))^2/x^2/(d+I*c*d*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+b*arctan(c*x))^2/x^2/(d+I*c*d*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{b^{2} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 4 i \, a b \log \left (-\frac{c x + i}{c x - i}\right ) - 4 \, a^{2}}{4 \, c^{2} d^{2} x^{4} - 8 i \, c d^{2} x^{3} - 4 \, d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(-(c*x + I)/(c*x - I))^2 - 4*I*a*b*log(-(c*x + I)/(c*x - I)) - 4*a^2)/(4*c^2*d^2*x^4 - 8*I*c*
d^2*x^3 - 4*d^2*x^2), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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