3.1272 \(\int \frac{(a+b \tan ^{-1}(c x))^2}{x (d+e x^2)^2} \, dx\)

Optimal. Leaf size=1087 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.86097, antiderivative size = 1087, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 16, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.696, Rules used = {4980, 4850, 4988, 4884, 4994, 6610, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4920, 4854, 4858} \[ \frac{i c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}-\frac{i c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\frac{\text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right ) b^2}{2 d^2}-\frac{\text{PolyLog}\left (3,1-\frac{2}{i c x+1}\right ) b^2}{2 d^2}+\frac{\text{PolyLog}\left (3,\frac{2}{i c x+1}-1\right ) b^2}{2 d^2}-\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}{4 d^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}{4 d^2}-\frac{c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\frac{c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}-\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) b}{d^2}-\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{i c x+1}\right ) b}{d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{2}{i c x+1}-1\right ) b}{d^2}+\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}{2 d^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}{2 d^2}-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{i c x+1}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{d^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 d^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 )}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 4978

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Dist[1/(4*d^2*Rt[-(e
/d), 2]), Int[(a + b*ArcTan[c*x])^p/(1 - Rt[-(e/d), 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-(e/d), 2]), Int[(a + b
*ArcTan[c*x])^p/(1 + Rt[-(e/d), 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 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 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 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 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 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]

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (d+e x^2\right )^2} \, dx &=\int \left (\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )^2}{d \left (d+e x^2\right )^2}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^2}-\frac{e \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{d^2}-\frac{e \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{d}\\ &=\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{(4 b c) \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{\sqrt{e} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 (-d)^{5/2}}-\frac{\sqrt{e} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 (-d)^{5/2}}-\frac{e \int \left (-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{d^2}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{(b c) \int \left (\frac{\sqrt{-d} 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 d \left (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^2}-\frac{(b c) \int \left (\frac{\sqrt{-d} 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 (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^2}+\frac{(2 b c) \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{(2 b c) \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{\sqrt{e} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 d^2}-\frac{\sqrt{e} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 d^2}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{d^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 d^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 d^2}-\frac{i b \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 \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 \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 \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 d^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 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 d^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 d^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 d^2}+\frac{\left (i b^2 c\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 (i b^2 c\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 (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 (-d)^{3/2} \left (c^2 d-e\right )}-\frac{\left (b c^3\right ) \int \frac{\left (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}-\frac{(b c e) \int \frac{a+b \tan ^{-1}(c x)}{-\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{3/2} \left (c^2 d-e\right )}+\frac{(b c e) \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 (-d)^{3/2} \left (c^2 d-e\right )}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{d^2}-\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}+\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}-\frac{i b \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 \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 \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 \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 d^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 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^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 d^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 d^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 (-d)^{3/2} \left (c^2 d-e\right )}-\frac{\left (b c^3\right ) \int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac{\sqrt{-d} \sqrt{e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 d^2 \left (c^2 d-e\right )}+\frac{\left (b^2 c^2 \sqrt{e}\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 (-d)^{3/2} \left (c^2 d-e\right )}-\frac{\left (b^2 c^2 \sqrt{e}\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 (-d)^{3/2} \left (c^2 d-e\right )}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{d^2}-\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}+\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}-\frac{i b \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 \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 \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 \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\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 d^2}-\frac{i b^2 c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\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 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^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 d^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 d^2}-2 \frac{\left (b c^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 d \left (c^2 d-e\right )}\\ &=-\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (c^2 d-e\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 d^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{d^2}-\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}+\frac{b c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\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 d^2}-\frac{i b \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 \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 \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 \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\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 d^2}-\frac{i b^2 c \sqrt{e} \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 (-d)^{3/2} \left (c^2 d-e\right )}+\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 d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 d^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^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 d^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 d^2}\\ \end{align*}

Mathematica [F]  time = 15.9039, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (d+e x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 10.853, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}}{x \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{1}{d e x^{2} + d^{2}} - \frac{\log \left (e x^{2} + d\right )}{d^{2}} + \frac{2 \, \log \left (x\right )}{d^{2}}\right )} + \int \frac{b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right )}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*(1/(d*e*x^2 + d^2) - log(e*x^2 + d)/d^2 + 2*log(x)/d^2) + integrate((b^2*arctan(c*x)^2 + 2*a*b*arctan(
c*x))/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arctan(c*x)^2 + 2*a*b*arctan(c*x) + a^2)/(e^2*x^5 + 2*d*e*x^3 + d^2*x), x)

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Sympy [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*atan(c*x))**2/x/(e*x**2+d)**2,x)

[Out]

Timed out

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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 (e x^{2} + d\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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