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

Optimal. Leaf size=292 \[ -\frac{2 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2}-\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^3 d^2}+\frac{i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^3 d^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2 (-c x+i)}-\frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2 (-c x+i)}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^2}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2}+\frac{2 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (-c x+i)}+\frac{b^2 \tan ^{-1}(c x)}{2 c^3 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.494662, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.56, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4864, 4862, 627, 44, 203, 4884, 4994, 6610} \[ -\frac{2 b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2}-\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^3 d^2}+\frac{i b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{c^3 d^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2 (-c x+i)}-\frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2 (-c x+i)}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^2}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2}+\frac{2 i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (-c x+i)}+\frac{b^2 \tan ^{-1}(c x)}{2 c^3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.21438, size = 362, normalized size = 1.24 \[ -\frac{6 a b \left (-4 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-2 \log \left (c^2 x^2+1\right )-8 \tan ^{-1}(c x)^2-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )+2 \tan ^{-1}(c x) \left (2 c x-4 i \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\sin \left (2 \tan ^{-1}(c x)\right )+i \cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 \left (-12 \left (2 \tan ^{-1}(c x)+i\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-12 i \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )-16 \tan ^{-1}(c x)^3+12 c x \tan ^{-1}(c x)^2-12 i \tan ^{-1}(c x)^2-24 i \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+24 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+6 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-6 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-3 \sin \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+6 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-3 i \cos \left (2 \tan ^{-1}(c x)\right )\right )+12 i a^2 \log \left (c^2 x^2+1\right )+12 a^2 c x+\frac{12 a^2}{c x-i}-24 a^2 \tan ^{-1}(c x)}{12 c^3 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(12*a^2*c*x + (12*a^2)/(-I + c*x) - 24*a^2*ArcTan[c*x] + (12*I)*a^2*Log[1 + c^2*x^2] + b^2*((-12*I)*ArcTan[c*
x]^2 + 12*c*x*ArcTan[c*x]^2 - 16*ArcTan[c*x]^3 - (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*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*ArcTan[c*x]^2
*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*(I + 2*ArcTan[c*x])*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*(-8*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 2*Log[1 + c^2*x^2] - 4*PolyLog[2, -E^((2*I)*A
rcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(2*c*x + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + E^((2*I)*Arc
Tan[c*x])] + Sin[2*ArcTan[c*x]])))/(12*c^3*d^2)

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Maple [C]  time = 0.612, size = 4774, normalized size = 16.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arc
tan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+I/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(
c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I/c^3*b^2/d^2*Pi*csg
n(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1-
I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1
+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/
(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2
+1)^(1/2))+2/c^3*b^2/d^2/(8*c*x-8*I)+2/c^3*a^2/d^2*arctan(c*x)-1/c^3*a^2/d^2/(c*x-I)-1/c^2*b^2/d^2*arctan(c*x)
^2*x-3/2/c^3*b^2/d^2*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-2/c^3*a*b/d^2*dilog(-1/2*I*(c*x+I))+1/c^3*a*b/d
^2*ln(c*x-I)^2+1/8/c^3*a*b/d^2*ln(c^4*x^4+10*c^2*x^2+9)+3/4/c^3*a*b/d^2*ln(c^2*x^2+1)+2/c^3*b^2/d^2*arctan(c*x
)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-1/2/c^3*b^2/d^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2/c^3*
b^2/d^2*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/c^3*b^2/d^2*arctan(c*x)^2/(c*x-I)-1/c^3*b^2/d^2*Pi*p
olylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+2/c^3*b^2/d^2*Pi*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2/c^3*b^2/d^2*Pi*di
log(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I/c^3*b^2/d^2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))-2/c^3*b^2/d^2*Pi*arctan
(c*x)^2-1/c^2*a^2/d^2*x+4/3/c^3*b^2/d^2*arctan(c*x)^3+3/2*I/c^3*a*b/d^2*arctan(c*x)-2/c^3*a*b/d^2*arctan(c*x)/
(c*x-I)-2/c^3*a*b/d^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))-2/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/
(c^2*x^2+1)+1))^2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I
*c*x)^2/(c^2*x^2+1)+1))^3*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+2/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((
1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2
+1)+1))^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))-2/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*
x^2+1)+1))^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2
/(c^2*x^2+1)+1))^3*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*
c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)^2-1/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(
c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2+1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1
+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)^2-1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+
1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/c^3*b
^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*dilog(1-I*
(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/2/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+
1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2
*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+3/2*I
/c^3*b^2/d^2*arctan(c*x)^2-I/c^3*a^2/d^2*ln(c^2*x^2+1)+1/2*I/c^3*b^2/d^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2)
)+1/2*I/c^3*b^2/d^2*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/4*I/c^3*b^2/d^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1
))+1/2/c^2*b^2/d^2*arctan(c*x)/(c*x-I)*x-2/c^2*a*b/d^2*arctan(c*x)*x-1/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^
2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I/c^3*a*b/d^2/(c*x-I)+2*I/c^3*b^2/d
^2*arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))-2*I/c^3*b^2/d^2*arctan(c*x)^2*ln(c*x-I)+1/2*I/c^3*b^2/d^2*arc
tan(c*x)/(c*x-I)-2*I/c^2*b^2/d^2/(8*c*x-8*I)*x-1/4*I/c^3*a*b/d^2*arctan(1/6*c^3*x^3+7/6*c*x)+1/4*I/c^3*a*b/d^2
*arctan(1/2*c*x)-1/2*I/c^3*a*b/d^2*arctan(1/2*c*x-1/2*I)+1/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*
csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/c^3*b^2/d
^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*dilog(1-
I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-4*I/c^3*a*b/d^2*arctan(c*x)*ln(c*x-I)+2*I/c^3*b^2/d^2*Pi*arctan(c*x)*ln(1-I*(1+
I*c*x)/(c^2*x^2+1)^(1/2))-2*I/c^3*b^2/d^2*Pi*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)+2*I/c^3*b^2/d^2*Pi*arct
an(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+1/2/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c
*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1
))+I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)*ln((1+I*c*x)^2/(c^
2*x^2+1)+1)+1/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^
2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)^2-2*I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I
*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+2*I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/
(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-I/c^3*b^2/d^2*Pi*csgn((1+
I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c^3*b^2/
d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1
/2))-2*I/c^3*b^2/d^2*Pi*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln(1+I*(1+I*c*
x)/(c^2*x^2+1)^(1/2))-1/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn(
(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-1/c^3*b^2/d^2*Pi*c
sgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^
2*x^2+1)+1))*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(
(1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)*ln((1+I*c*x)^2/
(c^2*x^2+1)+1)-I/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*
x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c^3*b^2/d^2*Pi
*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(
c^2*x^2+1)+1))*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-I/c^3*b^2/d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1
)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2*arctan(c*x)*ln((1+I*c*x)^2/(c^2*x^2+1)+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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