3.78 \(\int x (a+b \tanh ^{-1}(c x^2))^3 \, dx\)

Optimal. Leaf size=134 \[ -\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{2 c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x^2}\right )}{4 c}+\frac{1}{2} x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3+\frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{2 c}-\frac{3 b \log \left (\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{2 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 2.32313, antiderivative size = 390, normalized size of antiderivative = 2.91, number of steps used = 82, number of rules used = 23, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.643, Rules used = {6099, 2454, 2389, 2296, 2295, 6715, 2430, 2416, 2396, 2433, 2374, 6589, 2411, 2346, 2301, 6742, 43, 2394, 2393, 2391, 2375, 2317, 2425} \[ -\frac{3 b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^2\right )\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (3,\frac{1}{2} \left (1-c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (3,\frac{1}{2} \left (c x^2+1\right )\right )}{4 c}+\frac{3 b^3 \log \left (c x^2+1\right ) \text{PolyLog}\left (2,\frac{1}{2} \left (c x^2+1\right )\right )}{4 c}+\frac{3}{16} b^2 x^2 \log ^2\left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{3 b^2 \log ^2\left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{16 c}+\frac{3 b \log \left (\frac{1}{2} \left (c x^2+1\right )\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{8 c}+\frac{3}{16} b x^2 \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac{3 b \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{b^3 \left (c x^2+1\right ) \log ^3\left (c x^2+1\right )}{16 c}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (c x^2+1\right )}{8 c} \]

Warning: Unable to verify antiderivative.

[In]

Int[x*(a + b*ArcTanh[c*x^2])^3,x]

[Out]

-((1 - c*x^2)*(2*a - b*Log[1 - c*x^2])^3)/(16*c) + (3*b*(2*a - b*Log[1 - c*x^2])^2*Log[(1 + c*x^2)/2])/(8*c) -
(3*b*(2*a - b*Log[1 - c*x^2])^2*Log[1 + c*x^2])/(16*c) + (3*b*x^2*(2*a - b*Log[1 - c*x^2])^2*Log[1 + c*x^2])/
16 + (3*b^3*Log[(1 - c*x^2)/2]*Log[1 + c*x^2]^2)/(8*c) + (3*b^2*(2*a - b*Log[1 - c*x^2])*Log[1 + c*x^2]^2)/(16
*c) + (3*b^2*x^2*(2*a - b*Log[1 - c*x^2])*Log[1 + c*x^2]^2)/16 + (b^3*(1 + c*x^2)*Log[1 + c*x^2]^3)/(16*c) - (
3*b^2*(2*a - b*Log[1 - c*x^2])*PolyLog[2, (1 - c*x^2)/2])/(4*c) + (3*b^3*Log[1 + c*x^2]*PolyLog[2, (1 + c*x^2)
/2])/(4*c) - (3*b^3*PolyLog[3, (1 - c*x^2)/2])/(4*c) - (3*b^3*PolyLog[3, (1 + c*x^2)/2])/(4*c)

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 2430

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[(x
*(a + b*Log[c*(d + e*x)^n])^p)/(i + j*x), x], x] - Dist[b*e*n*p, Int[(x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*(f
+ g*Log[h*(i + j*x)^m]))/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 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 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

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

Rule 6589

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2346

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 6742

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

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 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]

Rule 2375

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 2317

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

Rule 2425

Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)))/(x_), x_Symbol] :> Simp[(Log[
f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m), x] - Dist[(b*e*n)/(2*m), Int[Log[f*x^m]^2/(d + e*x), x], x] /; Fre
eQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin{align*} \int x \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x \left (2 a-b \log \left (1-c x^2\right )\right )^3+\frac{3}{8} b x \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )-\frac{3}{8} b^2 x \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{1}{8} b^3 x \log ^3\left (1+c x^2\right )\right ) \, dx\\ &=\frac{1}{8} \int x \left (2 a-b \log \left (1-c x^2\right )\right )^3 \, dx+\frac{1}{8} (3 b) \int x \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right ) \, dx-\frac{1}{8} \left (3 b^2\right ) \int x \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right ) \, dx+\frac{1}{8} b^3 \int x \log ^3\left (1+c x^2\right ) \, dx\\ &=\frac{1}{16} \operatorname{Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^2\right )+\frac{1}{16} b^3 \operatorname{Subst}\left (\int \log ^3(1+c x) \, dx,x,x^2\right )\\ &=\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^2\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+c x^2\right )}{16 c}-\frac{1}{16} (3 b c) \operatorname{Subst}\left (\int \frac{x (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{x \log ^2(1+c x)}{1-c x} \, dx,x,x^2\right )\\ &=-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{(3 b) \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{16 c}-\frac{1}{16} (3 b c) \operatorname{Subst}\left (\int \left (\frac{(-2 a+b \log (1-c x))^2}{c}-\frac{(-2 a+b \log (1-c x))^2}{c (1+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c (1+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c}-\frac{\log ^2(1+c x)}{c (-1+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )-\frac{3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{1}{16} (3 b) \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,x^2\right )+\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{-1+c x} \, dx,x,x^2\right )-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^2\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{8 c}\\ &=\frac{3}{4} a b^2 x^2-\frac{3 b^3 x^2}{8}-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c}+\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )-\frac{3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}+\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x)) \log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right ) \log (1+c x)}{1+c x} \, dx,x,x^2\right )+\frac{(3 b) \operatorname{Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{16 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^2\right )}{8 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{8 c}\\ &=\frac{3}{4} a b^2 x^2+\frac{3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{8 c}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c}+\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^2\right )}{16 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) (-2 a+b \log (x))}{x} \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+c x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{8 c}\\ &=\frac{3 b^3 x^2}{8}+\frac{3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{8 c}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{8 c}+\frac{3 b^3 \log \left (1+c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{8 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{8 c}\\ &=-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac{3 b^3 \log \left (1+c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1-c x^2\right )\right )}{8 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1+c x^2\right )\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{8 c}\\ &=-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{16 c}+\frac{3}{16} b x^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{16 c}+\frac{3}{16} b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{4 c}+\frac{3 b^3 \log \left (1+c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1-c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1+c x^2\right )\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.222373, size = 213, normalized size = 1.59 \[ \frac{3 a b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+\tanh ^{-1}\left (c x^2\right ) \left (c x^2 \tanh ^{-1}\left (c x^2\right )-\tanh ^{-1}\left (c x^2\right )-2 \log \left (e^{-2 \tanh ^{-1}\left (c x^2\right )}+1\right )\right )\right )}{2 c}+\frac{b^3 \left (3 \tanh ^{-1}\left (c x^2\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+\tanh ^{-1}\left (c x^2\right )^2 \left (c x^2 \tanh ^{-1}\left (c x^2\right )-\tanh ^{-1}\left (c x^2\right )-3 \log \left (e^{-2 \tanh ^{-1}\left (c x^2\right )}+1\right )\right )\right )}{2 c}+\frac{3 a^2 b \log \left (1-c^2 x^4\right )}{4 c}+\frac{3}{2} a^2 b x^2 \tanh ^{-1}\left (c x^2\right )+\frac{a^3 x^2}{2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcTanh[c*x^2])^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^2*ArcTanh[c*x^2])/2 + (3*a^2*b*Log[1 - c^2*x^4])/(4*c) + (3*a*b^2*(ArcTanh[c*x^2]*(-A
rcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 2*Log[1 + E^(-2*ArcTanh[c*x^2])]) + PolyLog[2, -E^(-2*ArcTanh[c*x^2])])
)/(2*c) + (b^3*(ArcTanh[c*x^2]^2*(-ArcTanh[c*x^2] + c*x^2*ArcTanh[c*x^2] - 3*Log[1 + E^(-2*ArcTanh[c*x^2])]) +
3*ArcTanh[c*x^2]*PolyLog[2, -E^(-2*ArcTanh[c*x^2])] + (3*PolyLog[3, -E^(-2*ArcTanh[c*x^2])])/2))/(2*c)

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Maple [B]  time = 0.007, size = 298, normalized size = 2.2 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{{b}^{3}{x}^{2} \left ({\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{3}}{2}}+{\frac{{b}^{3} \left ({\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{3}}{2\,c}}-{\frac{3\,{b}^{3} \left ({\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{2\,c}\ln \left ({\frac{ \left ( c{x}^{2}+1 \right ) ^{2}}{-{c}^{2}{x}^{4}+1}}+1 \right ) }-{\frac{3\,{b}^{3}{\it Artanh} \left ( c{x}^{2} \right ) }{2\,c}{\it polylog} \left ( 2,-{\frac{ \left ( c{x}^{2}+1 \right ) ^{2}}{-{c}^{2}{x}^{4}+1}} \right ) }+{\frac{3\,{b}^{3}}{4\,c}{\it polylog} \left ( 3,-{\frac{ \left ( c{x}^{2}+1 \right ) ^{2}}{-{c}^{2}{x}^{4}+1}} \right ) }+{\frac{3\, \left ({\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}{x}^{2}a{b}^{2}}{2}}-3\,{\frac{{\it Artanh} \left ( c{x}^{2} \right ) a{b}^{2}}{c}\ln \left ({\frac{ \left ( c{x}^{2}+1 \right ) ^{2}}{-{c}^{2}{x}^{4}+1}}+1 \right ) }+{\frac{3\,a{b}^{2} \left ({\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{2\,c}}-{\frac{3\,a{b}^{2}}{2\,c}{\it polylog} \left ( 2,-{\frac{ \left ( c{x}^{2}+1 \right ) ^{2}}{-{c}^{2}{x}^{4}+1}} \right ) }+{\frac{3\,{x}^{2}{a}^{2}b{\it Artanh} \left ( c{x}^{2} \right ) }{2}}+{\frac{3\,{a}^{2}b\ln \left ( -{c}^{2}{x}^{4}+1 \right ) }{4\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arctanh(c*x^2))^3,x)

[Out]

1/2*x^2*a^3+1/2*b^3*x^2*arctanh(c*x^2)^3+1/2/c*b^3*arctanh(c*x^2)^3-3/2/c*b^3*arctanh(c*x^2)^2*ln((c*x^2+1)^2/
(-c^2*x^4+1)+1)-3/2/c*b^3*arctanh(c*x^2)*polylog(2,-(c*x^2+1)^2/(-c^2*x^4+1))+3/4/c*b^3*polylog(3,-(c*x^2+1)^2
/(-c^2*x^4+1))+3/2*arctanh(c*x^2)^2*x^2*a*b^2-3/c*ln((c*x^2+1)^2/(-c^2*x^4+1)+1)*arctanh(c*x^2)*a*b^2+3/2/c*a*
b^2*arctanh(c*x^2)^2-3/2/c*polylog(2,-(c*x^2+1)^2/(-c^2*x^4+1))*a*b^2+3/2*x^2*a^2*b*arctanh(c*x^2)+3/4/c*a^2*b
*ln(-c^2*x^4+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{3} x^{2} + \frac{3 \,{\left (2 \, c x^{2} \operatorname{artanh}\left (c x^{2}\right ) + \log \left (-c^{2} x^{4} + 1\right )\right )} a^{2} b}{4 \, c} - \frac{{\left (b^{3} c x^{2} - b^{3}\right )} \log \left (-c x^{2} + 1\right )^{3} - 3 \,{\left (2 \, a b^{2} c x^{2} +{\left (b^{3} c x^{2} + b^{3}\right )} \log \left (c x^{2} + 1\right )\right )} \log \left (-c x^{2} + 1\right )^{2}}{16 \, c} - \int -\frac{{\left (b^{3} c x^{3} - b^{3} x\right )} \log \left (c x^{2} + 1\right )^{3} + 6 \,{\left (a b^{2} c x^{3} - a b^{2} x\right )} \log \left (c x^{2} + 1\right )^{2} - 3 \,{\left (4 \, a b^{2} c x^{3} +{\left (b^{3} c x^{3} - b^{3} x\right )} \log \left (c x^{2} + 1\right )^{2} + 2 \,{\left ({\left (2 \, a b^{2} c + b^{3} c\right )} x^{3} -{\left (2 \, a b^{2} - b^{3}\right )} x\right )} \log \left (c x^{2} + 1\right )\right )} \log \left (-c x^{2} + 1\right )}{8 \,{\left (c x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x^2))^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x^2))^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*atanh(c*x**2))**3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctanh(c*x^2))^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^2) + a)^3*x, x)