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

Optimal. Leaf size=371 \[ \frac{2 b c \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 b c \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{2 b c \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{b^2 c \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{d^2}-\frac{b^2 c \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{d^2}+\frac{b^2 c \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )}{d^2}+\frac{b^2 c \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{d^2}-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}+\frac{3 c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}-\frac{4 c \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac{2 b c \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 c \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (c x+1)}+\frac{b^2 c \tanh ^{-1}(c x)}{2 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.797292, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {5940, 5916, 5988, 5932, 2447, 5914, 6052, 5948, 6058, 6610, 5928, 5926, 627, 44, 207, 5918, 6056} \[ \frac{2 b c \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 b c \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}+\frac{2 b c \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{b^2 c \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{d^2}-\frac{b^2 c \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )}{d^2}+\frac{b^2 c \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )}{d^2}+\frac{b^2 c \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right )}{d^2}-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d^2 (c x+1)}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 x}-\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2 (c x+1)}+\frac{3 c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d^2}-\frac{4 c \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}+\frac{2 b c \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d^2}-\frac{2 c \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (c x+1)}+\frac{b^2 c \tanh ^{-1}(c x)}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5940

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[E
xpandIntegrand[(a + b*ArcTanh[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[
p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 5916

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

Rule 5988

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

Rule 5932

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

Rule 2447

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

Rule 5914

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

Rule 6052

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

Rule 5948

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

Rule 6058

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[((a + b*ArcT
anh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p)/2, Int[((a + b*ArcTanh[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[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 -
2/(1 - 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 5928

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

Rule 5926

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

Rule 627

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 6056

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p)/2, Int[((a + b*ArcTanh[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[c^2*d + e, 0] && EqQ[(1 - u)^2 - (1 - 2
/(1 + c*x))^2, 0]

Rubi steps

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

Mathematica [C]  time = 1.71242, size = 347, normalized size = 0.94 \[ \frac{6 a b c \left (4 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+4 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+\sinh \left (2 \tanh ^{-1}(c x)\right )-\cosh \left (2 \tanh ^{-1}(c x)\right )+\tanh ^{-1}(c x) \left (-\frac{4}{c x}-8 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+2 \sinh \left (2 \tanh ^{-1}(c x)\right )-2 \cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+b^2 c \left (-24 \tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )-12 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )+16 \tanh ^{-1}(c x)^3-\frac{12 \tanh ^{-1}(c x)^2}{c x}+12 \tanh ^{-1}(c x)^2-24 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+24 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+6 \tanh ^{-1}(c x)^2 \sinh \left (2 \tanh ^{-1}(c x)\right )+6 \tanh ^{-1}(c x) \sinh \left (2 \tanh ^{-1}(c x)\right )+3 \sinh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x)^2 \cosh \left (2 \tanh ^{-1}(c x)\right )-6 \tanh ^{-1}(c x) \cosh \left (2 \tanh ^{-1}(c x)\right )-3 \cosh \left (2 \tanh ^{-1}(c x)\right )-i \pi ^3\right )-\frac{12 a^2 c}{c x+1}-24 a^2 c \log (x)+24 a^2 c \log (c x+1)-\frac{12 a^2}{x}}{12 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-12*a^2)/x - (12*a^2*c)/(1 + c*x) - 24*a^2*c*Log[x] + 24*a^2*c*Log[1 + c*x] + b^2*c*((-I)*Pi^3 + 12*ArcTanh[
c*x]^2 - (12*ArcTanh[c*x]^2)/(c*x) + 16*ArcTanh[c*x]^3 - 3*Cosh[2*ArcTanh[c*x]] - 6*ArcTanh[c*x]*Cosh[2*ArcTan
h[c*x]] - 6*ArcTanh[c*x]^2*Cosh[2*ArcTanh[c*x]] + 24*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*
x]^2*Log[1 - E^(2*ArcTanh[c*x])] - 12*PolyLog[2, E^(-2*ArcTanh[c*x])] - 24*ArcTanh[c*x]*PolyLog[2, E^(2*ArcTan
h[c*x])] + 12*PolyLog[3, E^(2*ArcTanh[c*x])] + 3*Sinh[2*ArcTanh[c*x]] + 6*ArcTanh[c*x]*Sinh[2*ArcTanh[c*x]] +
6*ArcTanh[c*x]^2*Sinh[2*ArcTanh[c*x]]) + 6*a*b*c*(-Cosh[2*ArcTanh[c*x]] + 4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*P
olyLog[2, E^(-2*ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + ArcTanh[c*x]*(-4/(c*x) - 2*Cosh[2*ArcTanh[c*x]] - 8*Lo
g[1 - E^(-2*ArcTanh[c*x])] + 2*Sinh[2*ArcTanh[c*x]])))/(12*d^2)

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Maple [C]  time = 0.682, size = 7397, normalized size = 19.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -a^{2}{\left (\frac{2 \, c x + 1}{c d^{2} x^{2} + d^{2} x} - \frac{2 \, c \log \left (c x + 1\right )}{d^{2}} + \frac{2 \, c \log \left (x\right )}{d^{2}}\right )} - \frac{{\left (2 \, b^{2} c x + b^{2} - 2 \,{\left (b^{2} c^{2} x^{2} + b^{2} c x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{4 \,{\left (c d^{2} x^{2} + d^{2} x\right )}} - \int -\frac{{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c x - a b\right )} \log \left (c x + 1\right ) + 2 \,{\left (2 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + 2 \, a b -{\left (2 \, a b c - b^{2} c\right )} x -{\left (2 \, b^{2} c^{4} x^{4} + 4 \, b^{2} c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} + b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{3} d^{2} x^{5} + c^{2} d^{2} x^{4} - c d^{2} x^{3} - d^{2} x^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*((2*c*x + 1)/(c*d^2*x^2 + d^2*x) - 2*c*log(c*x + 1)/d^2 + 2*c*log(x)/d^2) - 1/4*(2*b^2*c*x + b^2 - 2*(b^2
*c^2*x^2 + b^2*c*x)*log(c*x + 1))*log(-c*x + 1)^2/(c*d^2*x^2 + d^2*x) - integrate(-1/4*((b^2*c*x - b^2)*log(c*
x + 1)^2 + 4*(a*b*c*x - a*b)*log(c*x + 1) + 2*(2*b^2*c^3*x^3 + 3*b^2*c^2*x^2 + 2*a*b - (2*a*b*c - b^2*c)*x - (
2*b^2*c^4*x^4 + 4*b^2*c^3*x^3 + 2*b^2*c^2*x^2 + b^2*c*x - b^2)*log(c*x + 1))*log(-c*x + 1))/(c^3*d^2*x^5 + c^2
*d^2*x^4 - c*d^2*x^3 - d^2*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x\right ) + a^{2}}{c^{2} d^{2} x^{4} + 2 \, c d^{2} x^{3} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arctanh(c*x)^2 + 2*a*b*arctanh(c*x) + a^2)/(c^2*d^2*x^4 + 2*c*d^2*x^3 + d^2*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**2/(c**2*x**4 + 2*c*x**3 + x**2), x) + Integral(b**2*atanh(c*x)**2/(c**2*x**4 + 2*c*x**3 + x**2),
x) + Integral(2*a*b*atanh(c*x)/(c**2*x**4 + 2*c*x**3 + x**2), x))/d**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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