3.1269 \(\int \frac {x^2 (a+b \tan ^{-1}(c x))^2}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=1033 \[ -\frac {i c \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) b^2}{2 \left (c^2 d-e\right ) e}-\frac {i c \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) b^2}{2 \left (c^2 d-e\right ) e}+\frac {i c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e}+\frac {i c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e}+\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 \sqrt {-d} e^{3/2}}-\frac {\text {Li}_3\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 \sqrt {-d} e^{3/2}}+\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right ) b}{\left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{i c x+1}\right ) b}{\left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 \sqrt {-d} e^{3/2}}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 \sqrt {-d} e^{3/2}}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}} \]

[Out]

-1/2*I*b^2*c*polylog(2,1-2/(1+I*c*x))/(c^2*d-e)/e+b*c*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/(c^2*d-e)/e-b*c*(a+b*a
rctan(c*x))*ln(2/(1+I*c*x))/(c^2*d-e)/e-1/2*b*c*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(
-d)^(1/2)-I*e^(1/2)))/(c^2*d-e)/e-1/2*b*c*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1
/2)+I*e^(1/2)))/(c^2*d-e)/e-1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)
^(1/2)-I*e^(1/2)))/e^(3/2)/(-d)^(1/2)-1/2*I*c*(a+b*arctan(c*x))^2/(c^2*d-e)/e-1/2*I*b^2*c*polylog(2,1-2/(1-I*c
*x))/(c^2*d-e)/e+1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^
(1/2)))/e^(3/2)/(-d)^(1/2)+1/4*(a+b*arctan(c*x))^2*ln(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(
1/2)))/e^(3/2)/(-d)^(1/2)-1/4*(a+b*arctan(c*x))^2*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1
/2)))/e^(3/2)/(-d)^(1/2)+1/4*I*b^2*c*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2))
)/(c^2*d-e)/e+1/4*I*b^2*c*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/(c^2*d-e)
/e+1/8*b^2*polylog(3,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/e^(3/2)/(-d)^(1/2)-1/8*b
^2*polylog(3,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/e^(3/2)/(-d)^(1/2)+1/4*(a+b*arct
an(c*x))^2/e^(3/2)/((-d)^(1/2)-x*e^(1/2))-1/4*(a+b*arctan(c*x))^2/e^(3/2)/((-d)^(1/2)+x*e^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 1.95, antiderivative size = 1033, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4980, 4914, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858} \[ -\frac {i c \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b^2}{2 \left (c^2 d-e\right ) e}-\frac {i c \text {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b^2}{2 \left (c^2 d-e\right ) e}+\frac {i c \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e}+\frac {i c \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e}+\frac {\text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 \sqrt {-d} e^{3/2}}-\frac {\text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 \sqrt {-d} e^{3/2}}+\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right ) b}{\left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{i c x+1}\right ) b}{\left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e}-\frac {c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e}-\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 \sqrt {-d} e^{3/2}}+\frac {i \left (a+b \tan ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 \sqrt {-d} e^{3/2}}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*c*(a + b*ArcTan[c*x])^2)/((c^2*d - e)*e) + (a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) -
(a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/((c^2*
d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/((c^2*d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(
Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) + ((a + b*ArcTan[c*x])^2*Log
[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) - (b*c*(a + b*ArcT
an[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) - ((a + b
*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2))
- ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/((c^2*d - e)*e) - ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(
(c^2*d - e)*e) + ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x
))])/((c^2*d - e)*e) - ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] -
I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*S
qrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e) + ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-
d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*e^(3/2)) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[
-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2)) - (b^2*PolyLog[3, 1 - (2*c*(Sq
rt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2))

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 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 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 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 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4858

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^2*Log[2/
(1 - I*c*x)])/e, x] + (Simp[((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] + Sim
p[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e, x] - Simp[(I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 -
(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e, x] - Simp[(b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e), x] + Simp
[(b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && Ne
Q[c^2*d^2 + e^2, 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 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 4914

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && 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 4980

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

Rule 4984

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{e}-\frac {d \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac {\int \left (\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e}-\frac {d \int \left (-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{4 d \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac {1}{4} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx+\frac {1}{4} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx+\frac {1}{2} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-d e-e^2 x^2} \, dx-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d} e}-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d} e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {1}{2} \int \left (-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )^2}{2 d e \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx+\frac {(b c) \int \left (\frac {\sqrt {e} \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {c^2 \left (-\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}-\frac {(b c) \int \left (-\frac {\sqrt {e} \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {e} \left (-c^2 d+e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (b c^3\right ) \int \frac {\left (-\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (b c^3\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 \sqrt {-d} e}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 \sqrt {-d} e}-\frac {(b c) \int \frac {a+b \tan ^{-1}(c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) \sqrt {e}}-\frac {(b c) \int \frac {a+b \tan ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) \sqrt {e}}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}+\frac {\left (b c^3\right ) \int \left (-\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac {\left (b c^3\right ) \int \left (\frac {\sqrt {-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac {\sqrt {e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^{3/2}}-2 \frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}+\frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2 c \left (-\sqrt {-d}+\sqrt {e} x\right )}{\left (-c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}+\frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-2 \frac {\left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 \left (c^2 d-e\right ) e}+2 \frac {\left (b c^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d-e\right ) e}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}+2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 \left (c^2 d-e\right ) e}-\frac {\left (b c^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 \left (c^2 d-e\right ) e}\right )\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d-e\right ) e}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}+2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 \left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\right )\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d-e\right ) e}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}+2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 \left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 \left (c^2 d-e\right ) e}\right )\\ &=\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 \left (c^2 d-e\right ) e}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 \left (c^2 d-e\right ) e}+2 \left (-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^2}{4 \left (c^2 d-e\right ) e}-\frac {b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{2 \left (c^2 d-e\right ) e}-\frac {i b^2 c \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{4 \left (c^2 d-e\right ) e}\right )+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b^2 c \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 \sqrt {-d} e^{3/2}}\\ \end {align*}

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Mathematica [F]  time = 45.35, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \arctan \left (c x\right )^{2} + 2 \, a b x^{2} \arctan \left (c x\right ) + a^{2} x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^2*arctan(c*x)^2 + 2*a*b*x^2*arctan(c*x) + a^2*x^2)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [C]  time = 3.10, size = 6575, normalized size = 6.36 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a^{2} {\left (\frac {x}{e^{2} x^{2} + d e} - \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e}\right )} - \frac {4 \, b^{2} x \arctan \left (c x\right )^{2} - b^{2} x \log \left (c^{2} x^{2} + 1\right )^{2} - 2 \, {\left (e^{2} x^{2} + d e\right )} \int \frac {12 \, {\left (b^{2} c^{2} e x^{4} + b^{2} e x^{2}\right )} \arctan \left (c x\right )^{2} + {\left (b^{2} c^{2} e x^{4} + b^{2} e x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + 4 \, {\left (8 \, a b c^{2} e x^{4} + b^{2} c e x^{3} + b^{2} c d x + 8 \, a b e x^{2}\right )} \arctan \left (c x\right ) - 2 \, {\left (b^{2} c^{2} e x^{4} + b^{2} c^{2} d x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2} e^{3} x^{6} + {\left (2 \, c^{2} d e^{2} + e^{3}\right )} x^{4} + d^{2} e + {\left (c^{2} d^{2} e + 2 \, d e^{2}\right )} x^{2}}\,{d x}}{32 \, {\left (e^{2} x^{2} + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(x/(e^2*x^2 + d*e) - arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e)) - 1/32*(4*b^2*x*arctan(c*x)^2 - b^2*x*log(c
^2*x^2 + 1)^2 - 32*(e^2*x^2 + d*e)*integrate(1/16*(12*(b^2*c^2*e*x^4 + b^2*e*x^2)*arctan(c*x)^2 + (b^2*c^2*e*x
^4 + b^2*e*x^2)*log(c^2*x^2 + 1)^2 + 4*(8*a*b*c^2*e*x^4 + b^2*c*e*x^3 + b^2*c*d*x + 8*a*b*e*x^2)*arctan(c*x) -
2*(b^2*c^2*e*x^4 + b^2*c^2*d*x^2)*log(c^2*x^2 + 1))/(c^2*e^3*x^6 + (2*c^2*d*e^2 + e^3)*x^4 + d^2*e + (c^2*d^2
*e + 2*d*e^2)*x^2), x))/(e^2*x^2 + d*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*atan(c*x))^2)/(d + e*x^2)^2,x)

[Out]

int((x^2*(a + b*atan(c*x))^2)/(d + e*x^2)^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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