3.29 \(\int (d+e x) (a+b \tanh ^{-1}(c x^2))^2 \, dx\)

Optimal. Leaf size=1085 \[ \text{result too large to display} \]

[Out]

a^2*d*x + (2*a*b*d*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*d*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (2*a*b*d*ArcTanh[Sqrt[
c]*x])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] + (e*(a + b*ArcTanh[c*x^2])^2)/(2*c) + (e*x^2*(a + b*Arc
Tanh[c*x^2])^2)/2 + (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTan[Sqrt[c]*x]*L
og[2/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/
Sqrt[c] + (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1
+ Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1
+ Sqrt[c]*x))])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1
+ Sqrt[c]*x))])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] -
(b*e*(a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c - a*b*d*x*Log[1 - c*x^2] - (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 -
c*x^2])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*d*x*Log[1 - c*x^2]^2)/4 + a*b*d*x*
Log[1 + c*x^2] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/
Sqrt[c] - (b^2*d*x*Log[1 - c*x^2]*Log[1 + c*x^2])/2 + (b^2*d*x*Log[1 + c*x^2]^2)/4 + (b^2*d*PolyLog[2, 1 - 2/(
1 - Sqrt[c]*x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] - ((I/2)*b^2*d*PolyLog[2, 1 -
((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/Sqrt[c]
+ (b^2*d*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] - (b^2*d*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sq
rt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*d*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-
c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - ((I/2)*b^2*d*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sq
rt[c]*x)])/Sqrt[c] - (b^2*e*PolyLog[2, 1 - 2/(1 - c*x^2)])/(2*c)

________________________________________________________________________________________

Rubi [A]  time = 2.43804, antiderivative size = 1216, normalized size of antiderivative = 1.12, number of steps used = 104, number of rules used = 39, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2.167, Rules used = {6742, 6091, 298, 203, 206, 6097, 260, 6093, 2450, 2476, 2448, 321, 2470, 12, 5984, 5918, 2402, 2315, 2556, 5992, 5920, 2447, 4928, 4856, 4920, 4854, 6099, 2454, 2389, 2296, 2295, 30, 2557, 2475, 43, 2416, 2394, 2393, 2391} \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(d + e*x)^2)/(2*e) + (2*a*b*d*ArcTan[Sqrt[c]*x])/Sqrt[c] + (I*b^2*d*ArcTan[Sqrt[c]*x]^2)/Sqrt[c] - (2*a*b
*d*ArcTanh[Sqrt[c]*x])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]^2)/Sqrt[c] + 2*a*b*d*x*ArcTanh[c*x^2] + a*b*e*x^2*A
rcTanh[c*x^2] + (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2
/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt
[c] + (2*b^2*d*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/Sqrt[c] - (2*b^2*d*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sq
rt[c]*x)])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sq
rt[c]*x))])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sq
rt[c]*x))])/Sqrt[c] + (b^2*d*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] - (b^
2*d*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 - c*x^2])/Sqrt[c] + (b^2*d*x*L
og[1 - c*x^2]^2)/4 - (b^2*e*(1 - c*x^2)*Log[1 - c*x^2]^2)/(8*c) - (b^2*e*Log[1 - c*x^2]*Log[(1 + c*x^2)/2])/(4
*c) + (b^2*d*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] - (b^2*d*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/Sqrt[c] + (
b^2*e*Log[(1 - c*x^2)/2]*Log[1 + c*x^2])/(4*c) - (b^2*d*x*Log[1 - c*x^2]*Log[1 + c*x^2])/2 - (b^2*e*x^2*Log[1
- c*x^2]*Log[1 + c*x^2])/4 + (b^2*d*x*Log[1 + c*x^2]^2)/4 + (b^2*e*(1 + c*x^2)*Log[1 + c*x^2]^2)/(8*c) + (a*b*
e*Log[1 - c^2*x^4])/(2*c) - (b^2*e*PolyLog[2, (1 - c*x^2)/2])/(4*c) + (b^2*e*PolyLog[2, (1 + c*x^2)/2])/(4*c)
+ (b^2*d*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*Sqrt[c]*x)])/Sqrt[c] -
((I/2)*b^2*d*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c] + (I*b^2*d*PolyLog[2, 1 - 2/
(1 + I*Sqrt[c]*x)])/Sqrt[c] + (b^2*d*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])/Sqrt[c] - (b^2*d*PolyLog[2, 1 + (2*Sqr
t[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - (b^2*d*PolyLog[2, 1 - (2*Sqrt[c]
*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(2*Sqrt[c]) - ((I/2)*b^2*d*PolyLog[2, 1 - ((1 - I)
*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/Sqrt[c]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 6091

Int[ArcTanh[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTanh[c*x^n], x] - Dist[c*n, Int[x^n/(1 - c^2*x^(2*n)), x]
, x] /; FreeQ[{c, n}, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 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 206

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

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 6093

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + (b*Log[1 + c*x^n])/2
- (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && IntegerQ[n]

Rule 2450

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[(x^n*(a + b*Log[c*(d + e*x^n)^p])^(q - 1))/(d + e*x^n), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 321

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 5984

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

Rule 5918

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^p*
Log[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTanh[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 2556

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[(x*Log[w]*D[v, x])/v, x],
x] - Int[SimplifyIntegrand[(x*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]

Rule 5992

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 5920

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])*Log[2/(1
+ c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d +
e*x))/((c*d + e)*(1 + c*x))]/(1 - c^2*x^2), x], x] + Simp[((a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)
*(1 + c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[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 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
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 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 6099

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^
m*(a + (b*Log[1 + c*x^n])/2 - (b*Log[1 - c*x^n])/2)^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] &&
IntegerQ[m] && IntegerQ[n]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 3.18411, size = 684, normalized size = 0.63 \[ \frac{-b^2 d \sqrt{c x^2} \left (-\text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{c x^2}\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )+\text{PolyLog}\left (2,\left (-\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-1\right )\right )+\text{PolyLog}\left (2,\frac{1}{2} \left (\sqrt{c x^2}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )-\text{PolyLog}\left (2,\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}+1\right )\right )+\frac{1}{2} i \text{PolyLog}\left (2,-e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )-\frac{1}{2} \log ^2\left (1-\sqrt{c x^2}\right )+\frac{1}{2} \log ^2\left (\sqrt{c x^2}+1\right )+\log (2) \log \left (1-\sqrt{c x^2}\right )+\log \left (1-\sqrt{c x^2}\right ) \log \left (\left (\frac{1}{2}+\frac{i}{2}\right ) \left (\sqrt{c x^2}-i\right )\right )-\log \left (\frac{1}{2} \left ((1+i)-(1-i) \sqrt{c x^2}\right )\right ) \log \left (\sqrt{c x^2}+1\right )-\log \left (\left (-\frac{1}{2}-\frac{i}{2}\right ) \left (\sqrt{c x^2}+i\right )\right ) \log \left (\sqrt{c x^2}+1\right )-\log (2) \log \left (\sqrt{c x^2}+1\right )+\log \left (1-\sqrt{c x^2}\right ) \log \left (\frac{1}{2} \left ((1-i) \sqrt{c x^2}+(1+i)\right )\right )+2 i \tan ^{-1}\left (\sqrt{c x^2}\right )^2-2 \sqrt{c x^2} \tanh ^{-1}\left (c x^2\right )^2-2 \tan ^{-1}\left (\sqrt{c x^2}\right ) \log \left (1+e^{4 i \tan ^{-1}\left (\sqrt{c x^2}\right )}\right )-2 \log \left (1-\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )+2 \log \left (\sqrt{c x^2}+1\right ) \tanh ^{-1}\left (c x^2\right )-4 \tan ^{-1}\left (\sqrt{c x^2}\right ) \tanh ^{-1}\left (c x^2\right )\right )+b^2 e x \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+2 a^2 c d x^2+a^2 c e x^3+a b e x \left (\log \left (1-c^2 x^4\right )+2 c x^2 \tanh ^{-1}\left (c x^2\right )\right )+4 a b c d x^2 \tanh ^{-1}\left (c x^2\right )+4 a b d \sqrt{c x^2} \left (\tan ^{-1}\left (\sqrt{c x^2}\right )-\tanh ^{-1}\left (\sqrt{c x^2}\right )\right )+b^2 e x \tanh ^{-1}\left (c x^2\right ) \left (\left (c x^2-1\right ) \tanh ^{-1}\left (c x^2\right )-2 \log \left (e^{-2 \tanh ^{-1}\left (c x^2\right )}+1\right )\right )}{2 c x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a^2*c*d*x^2 + a^2*c*e*x^3 + 4*a*b*c*d*x^2*ArcTanh[c*x^2] + 4*a*b*d*Sqrt[c*x^2]*(ArcTan[Sqrt[c*x^2]] - ArcTa
nh[Sqrt[c*x^2]]) + b^2*e*x*ArcTanh[c*x^2]*((-1 + c*x^2)*ArcTanh[c*x^2] - 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + a
*b*e*x*(2*c*x^2*ArcTanh[c*x^2] + Log[1 - c^2*x^4]) + b^2*e*x*PolyLog[2, -E^(-2*ArcTanh[c*x^2])] - b^2*d*Sqrt[c
*x^2]*((2*I)*ArcTan[Sqrt[c*x^2]]^2 - 4*ArcTan[Sqrt[c*x^2]]*ArcTanh[c*x^2] - 2*Sqrt[c*x^2]*ArcTanh[c*x^2]^2 - 2
*ArcTan[Sqrt[c*x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c*x^2]])] - 2*ArcTanh[c*x^2]*Log[1 - Sqrt[c*x^2]] + Log[2]*L
og[1 - Sqrt[c*x^2]] - Log[1 - Sqrt[c*x^2]]^2/2 + Log[1 - Sqrt[c*x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c*x^2])] + 2*
ArcTanh[c*x^2]*Log[1 + Sqrt[c*x^2]] - Log[2]*Log[1 + Sqrt[c*x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c*x^2])/2]*Log
[1 + Sqrt[c*x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c*x^2])]*Log[1 + Sqrt[c*x^2]] + Log[1 + Sqrt[c*x^2]]^2/2 + Log[
1 - Sqrt[c*x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c*x^2])/2] + (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c*x^2]])] - P
olyLog[2, (1 - Sqrt[c*x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c*x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + S
qrt[c*x^2])] + PolyLog[2, (1 + Sqrt[c*x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c*x^2])] - PolyLog[2, (1/2 +
I/2)*(1 + Sqrt[c*x^2])]))/(2*c*x)

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Maple [F]  time = 0.269, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(a^2*e*x + a^2*d + (b^2*e*x + b^2*d)*arctanh(c*x^2)^2 + 2*(a*b*e*x + a*b*d)*arctanh(c*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x**2))**2*(d + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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