3.20 \(\int \frac{(a+b \tanh ^{-1}(c x))^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=953 \[ \text{result too large to display} \]

[Out]

(3*b*c*(a + b*ArcTanh[c*x])^2)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^3/(2*e*(d + e*x)^2) - (3*b
^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[
2/(1 - c*x)])/(4*e*(c*d + e)^2) - (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2
) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) - (3*b*c^2*(a + b*ArcTanh[c*x]
)^2*Log[2/(1 + c*x)])/(4*(c*d - e)^2*e) + (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/((c*d - e)^2*(c*
d + e)^2) + (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d +
e)^2) - (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2
) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d - e)*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLo
g[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (3*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^
2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*(c*d + e)) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyL
og[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/((c
*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c
*d + e)^2) + (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d -
e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(8*e*(c*d + e)^2) + (3*b^3*c^2*PolyLog[3, 1 - 2/(1
+ c*x)])/(8*(c*d - e)^2*e) - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^2) + (3*b^3*c
^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c*d + e)^2)

________________________________________________________________________________________

Rubi [A]  time = 1.02713, antiderivative size = 953, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5928, 5918, 5948, 6058, 6610, 6056, 2402, 2315, 5920, 2447, 5922} \[ -\frac{3 c^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) b^3}{4 (c d-e) (c d+e)^2}-\frac{3 c^2 \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) b^3}{4 (c d-e)^2 (c d+e)}+\frac{3 c^2 e \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 e \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right ) b^3}{8 e (c d+e)^2}+\frac{3 c^2 \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right ) b^3}{8 (c d-e)^2 e}-\frac{3 c^3 d \text{PolyLog}\left (3,1-\frac{2}{c x+1}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}+\frac{3 c^3 d \text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^3}{2 (c d-e)^2 (c d+e)^2}-\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right ) b^2}{2 (c d-e) (c d+e)^2}+\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{c x+1}\right ) b^2}{2 (c d-e)^2 (c d+e)}-\frac{3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) b^2}{4 e (c d+e)^2}+\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) b^2}{4 (c d-e)^2 e}-\frac{3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c^3 d \left (a+b \tanh ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac{3 c \left (a+b \tanh ^{-1}(c x)\right )^2 b}{2 \left (c^2 d^2-e^2\right ) (d+e x)}+\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-c x}\right ) b}{4 e (c d+e)^2}-\frac{3 c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{c x+1}\right ) b}{4 (c d-e)^2 e}+\frac{3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac{3 c^3 d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])^3/(d + e*x)^3,x]

[Out]

(3*b*c*(a + b*ArcTanh[c*x])^2)/(2*(c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcTanh[c*x])^3/(2*e*(d + e*x)^2) - (3*b
^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + (3*b*c^2*(a + b*ArcTanh[c*x])^2*Log[
2/(1 - c*x)])/(4*e*(c*d + e)^2) - (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2
) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) - (3*b*c^2*(a + b*ArcTanh[c*x]
)^2*Log[2/(1 + c*x)])/(4*(c*d - e)^2*e) + (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/((c*d - e)^2*(c*
d + e)^2) + (3*b^2*c^2*e*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d +
e)^2) - (3*b*c^3*d*(a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2
) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 - c*x)])/(4*(c*d - e)*(c*d + e)^2) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyLo
g[2, 1 - 2/(1 - c*x)])/(4*e*(c*d + e)^2) + (3*b^3*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^
2) - (3*b^3*c^2*PolyLog[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*(c*d + e)) + (3*b^2*c^2*(a + b*ArcTanh[c*x])*PolyL
og[2, 1 - 2/(1 + c*x)])/(4*(c*d - e)^2*e) - (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/((c
*d - e)^2*(c*d + e)^2) - (3*b^3*c^2*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c
*d + e)^2) + (3*b^2*c^3*d*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d -
e)^2*(c*d + e)^2) - (3*b^3*c^2*PolyLog[3, 1 - 2/(1 - c*x)])/(8*e*(c*d + e)^2) + (3*b^3*c^2*PolyLog[3, 1 - 2/(1
+ c*x)])/(8*(c*d - e)^2*e) - (3*b^3*c^3*d*PolyLog[3, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)^2) + (3*b^3*c
^3*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*(c*d - e)^2*(c*d + e)^2)

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 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 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 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]

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 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 5922

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTanh[c*x])^2*Log[
2/(1 + c*x)])/e, x] + (Simp[((a + b*ArcTanh[c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(
b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/e, x] - Simp[(b*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c*(
d + e*x))/((c*d + e)*(1 + c*x))])/e, x] + Simp[(b^2*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e), x] - Simp[(b^2*PolyLog
[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2,
0]

Rubi steps

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

Mathematica [F]  time = 87.2254, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^3}{(d+e x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^3,x]

[Out]

Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x)^3, x]

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Maple [C]  time = 3.486, size = 53538, normalized size = 56.2 \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))^3/(e*x+d)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

-3/4*((4*c^2*d*log(e*x + d)/(c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - c*log(c*x + 1)/(c^2*d^2*e - 2*c*d*e^2 + e^3) + c
*log(c*x - 1)/(c^2*d^2*e + 2*c*d*e^2 + e^3) - 2/(c^2*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x))*c + 2*arctanh(c*x)/(e
^3*x^2 + 2*d*e^2*x + d^2*e))*a^2*b - 1/2*a^3/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/16*(((c^4*d^2*e^2 - 2*c^3*d*e^3
+ c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e - 2*c^3*d^2*e^2 + c^2*d*e^3)*b^3*x - (2*c^3*d^3*e - 3*c^2*d^2*e^2 + e^4)*b^
3)*log(-c*x + 1)^3 - 3*(2*(c^3*d^2*e^2 - c*e^4)*b^3*x - 2*(c^4*d^4 - 2*c^2*d^2*e^2 + e^4)*a*b^2 + 2*(c^3*d^3*e
- c*d*e^3)*b^3 + ((c^4*d^2*e^2 + 2*c^3*d*e^3 + c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e + 2*c^3*d^2*e^2 + c^2*d*e^3)*b
^3*x + (2*c^3*d^3*e + 3*c^2*d^2*e^2 - e^4)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/(c^4*d^6*e - 2*c^2*d^4*e^3 + d^
2*e^5 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)*x^2 + 2*(c^4*d^5*e^2 - 2*c^2*d^3*e^4 + d*e^6)*x) - integrate(1/8*(
((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*b^3)*log(c*x
+ 1)^3 + 6*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*a*b^2*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*
a*b^2)*log(c*x + 1)^2 - 3*(2*(c^3*d*e^3 - c^2*e^4)*b^3*x^2 - 2*(c^4*d^4 - c^3*d^3*e - c^2*d^2*e^2 + c*d*e^3)*a
*b^2 + 2*(c^3*d^3*e - c^2*d^2*e^2)*b^3 + ((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c
^2*d^2*e^2 - c*d*e^3 + e^4)*b^3)*log(c*x + 1)^2 - 2*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*a*b^2 - 2*(
c^3*d^2*e^2 - c^2*d*e^3)*b^3)*x + ((c^4*d*e^3 + c^3*e^4)*b^3*x^3 + 3*(c^4*d^2*e^2 + c^3*d*e^3)*b^3*x^2 - 4*(c^
3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*a*b^2 + (2*c^3*d^3*e + c^2*d^2*e^2 - c*d*e^3)*b^3 + (4*(c^4*d^3*e - c^3
*d^2*e^2 - c^2*d*e^3 + c*e^4)*a*b^2 + (2*c^4*d^3*e + 4*c^3*d^2*e^2 + c^2*d*e^3 - c*e^4)*b^3)*x)*log(c*x + 1))*
log(-c*x + 1))/(c^3*d^6*e - c^2*d^5*e^2 - c*d^4*e^3 + d^3*e^4 - (c^4*d^3*e^4 - c^3*d^2*e^5 - c^2*d*e^6 + c*e^7
)*x^4 - (3*c^4*d^4*e^3 - 4*c^3*d^3*e^4 - 2*c^2*d^2*e^5 + 4*c*d*e^6 - e^7)*x^3 - 3*(c^4*d^5*e^2 - 2*c^3*d^4*e^3
+ 2*c*d^2*e^5 - d*e^6)*x^2 - (c^4*d^6*e - 4*c^3*d^5*e^2 + 2*c^2*d^4*e^3 + 4*c*d^3*e^4 - 3*d^2*e^5)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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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 )}^{3}}{{\left (e x + d\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x))^3/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x) + a)^3/(e*x + d)^3, x)