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

Optimal. Leaf size=94 \[ \frac{1}{2} b^2 c \text{PolyLog}\left (2,\frac{2}{\frac{c}{x^2}+1}-1\right )+\frac{1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right )^2-\frac{1}{2} c \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right )^2-b c \log \left (2-\frac{2}{\frac{c}{x^2}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{x^2}{c}\right )\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 0.696246, antiderivative size = 404, normalized size of antiderivative = 4.3, number of steps used = 34, number of rules used = 19, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.357, Rules used = {6099, 2454, 2397, 2392, 2391, 2455, 263, 260, 6715, 2448, 31, 6742, 2556, 12, 2462, 2416, 2394, 2315, 2393} \[ \frac{1}{4} b^2 c \text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\frac{1}{4} b^2 c \text{PolyLog}\left (2,\frac{c}{x^2}\right )-\frac{1}{4} b^2 c \text{PolyLog}\left (2,\frac{c-x^2}{2 c}\right )+\frac{1}{4} b^2 c \text{PolyLog}\left (2,\frac{c+x^2}{2 c}\right )-\frac{1}{4} b^2 c \text{PolyLog}\left (2,\frac{c+x^2}{c}\right )+\frac{1}{4} b^2 c \text{PolyLog}\left (2,1-\frac{x^2}{c}\right )+\frac{1}{2} a b x^2 \log \left (\frac{c}{x^2}+1\right )+\frac{1}{2} a b c \log \left (c+x^2\right )+\frac{1}{8} x^2 \left (1-\frac{c}{x^2}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2+a b c \log (x)+\frac{1}{8} b^2 x^2 \left (\frac{c}{x^2}+1\right ) \log ^2\left (\frac{c}{x^2}+1\right )-\frac{1}{4} b^2 x^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (\frac{c}{x^2}+1\right )-\frac{1}{4} b^2 c \log \left (1-\frac{c}{x^2}\right ) \log \left (-c-x^2\right )-\frac{1}{4} b^2 c \log \left (-\frac{x^2}{c}\right ) \log \left (-c-x^2\right )+\frac{1}{4} b^2 c \log \left (-c-x^2\right ) \log \left (\frac{c-x^2}{2 c}\right )+\frac{1}{4} b^2 c \log \left (\frac{c}{x^2}+1\right ) \log \left (x^2-c\right )+\frac{1}{4} b^2 c \log \left (\frac{x^2}{c}\right ) \log \left (x^2-c\right )-\frac{1}{4} b^2 c \log \left (x^2-c\right ) \log \left (\frac{c+x^2}{2 c}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x*(a + b*ArcTanh[c/x^2])^2,x]

[Out]

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

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 2397

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

Rule 2392

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + (e*x)/d]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 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 2455

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 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 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 31

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

Rule 6742

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

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 12

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

Rule 2462

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

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.13381, size = 107, normalized size = 1.14 \[ \frac{1}{2} \left (b^2 c \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x^2}\right )}\right )+a \left (a x^2+b c \log \left (1-\frac{c^2}{x^4}\right )-2 b c \log \left (\frac{c}{x^2}\right )\right )+2 b \tanh ^{-1}\left (\frac{c}{x^2}\right ) \left (a x^2-b c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x^2}\right )}\right )\right )+b^2 \left (x^2-c\right ) \tanh ^{-1}\left (\frac{c}{x^2}\right )^2\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b^2*x*arctanh(c/x^2)^2 + 2*a*b*x*arctanh(c/x^2) + a^2*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 (\frac{c}{x^{2}} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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