3.344 \(\int \frac{1}{(1-a^2 x^2)^3 \tanh ^{-1}(a x)^6} \, dx\)

Optimal. Leaf size=257 \[ \frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac{a^2 x^2+1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{32}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{15 a}+\frac{16 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{15 a} \]

[Out]

-1/(5*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^5) - x/(5*(1 - a^2*x^2)^2*ArcTanh[a*x]^4) - 4/(15*a*(1 - a^2*x^2)^2*ArcTa
nh[a*x]^3) + 1/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (8*x)/(15*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + x/(5*(1 - a^2*
x^2)*ArcTanh[a*x]^2) - 32/(15*a*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 8/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (1 + a^2*
x^2)/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(15*a) + (16*SinhIntegral[4*ArcTanh[a
*x]])/(15*a)

________________________________________________________________________________________

Rubi [A]  time = 1.34905, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 49, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {5966, 6032, 6028, 5996, 6034, 5448, 12, 3298} \[ \frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac{a^2 x^2+1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{32}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{15 a}+\frac{16 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{15 a} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6),x]

[Out]

-1/(5*a*(1 - a^2*x^2)^2*ArcTanh[a*x]^5) - x/(5*(1 - a^2*x^2)^2*ArcTanh[a*x]^4) - 4/(15*a*(1 - a^2*x^2)^2*ArcTa
nh[a*x]^3) + 1/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]^3) - (8*x)/(15*(1 - a^2*x^2)^2*ArcTanh[a*x]^2) + x/(5*(1 - a^2*
x^2)*ArcTanh[a*x]^2) - 32/(15*a*(1 - a^2*x^2)^2*ArcTanh[a*x]) + 8/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (1 + a^2*
x^2)/(5*a*(1 - a^2*x^2)*ArcTanh[a*x]) + (2*SinhIntegral[2*ArcTanh[a*x]])/(15*a) + (16*SinhIntegral[4*ArcTanh[a
*x]])/(15*a)

Rule 5966

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

Rule 6032

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

Rule 6028

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

Rule 5996

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTa
nh[c*x])^(p + 1))/(b*c*d*(p + 1)*(d + e*x^2)), x] + (Dist[4/(b^2*(p + 1)*(p + 2)), Int[(x*(a + b*ArcTanh[c*x])
^(p + 2))/(d + e*x^2)^2, x], x] + Simp[((1 + c^2*x^2)*(a + b*ArcTanh[c*x])^(p + 2))/(b^2*e*(p + 1)*(p + 2)*(d
+ e*x^2)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[p, -1] && NeQ[p, -2]

Rule 6034

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c^(
m + 1), Subst[Int[((a + b*x)^p*Sinh[x]^m)/Cosh[x]^(m + 2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c,
d, e, p}, x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 12

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^6} \, dx &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}+\frac{1}{5} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^5} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac{1}{5} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx+\frac{1}{5} \left (3 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{1}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{3}{5} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx-\frac{3}{5} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4} \, dx+\frac{1}{15} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{2 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{2}{15} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-\frac{1}{5} (2 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx+\frac{1}{5} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+\frac{1}{5} \left (2 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \left (\frac{2}{5} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\right )-\frac{2}{5} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\frac{1}{15} (8 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx-\frac{1}{5} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\frac{1}{5} \left (6 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{6}{5} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-\frac{6}{5} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{15 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}-\frac{1}{5} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{5} (8 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\right )\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{4}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{15 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\right )-\frac{1}{5} (12 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\frac{1}{5} (24 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{4}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{15 a}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{15 a}-\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\right )-\frac{12 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+\frac{24 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{4}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{4 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{15 a}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{15 a}-\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\right )-\frac{12 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}+\frac{24 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{4}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{15 a}+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{5 a}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{5 a}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{5 a}\\ &=-\frac{1}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^5}-\frac{x}{5 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{4}{15 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{8 x}{15 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{5 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{4}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{8}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{1+a^2 x^2}{5 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{2 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}+2 \left (-\frac{2}{5 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{5 a}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{5 a}\right )\\ \end{align*}

Mathematica [A]  time = 0.300222, size = 166, normalized size = 0.65 \[ -\frac{-2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^5 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )-16 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^5 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )+3 a^4 x^4 \tanh ^{-1}(a x)^4+24 a^2 x^2 \tanh ^{-1}(a x)^4+3 a^3 x^3 \tanh ^{-1}(a x)^3+3 a^2 x^2 \tanh ^{-1}(a x)^2+5 \tanh ^{-1}(a x)^4+5 a x \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x)+3}{15 a \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - a^2*x^2)^3*ArcTanh[a*x]^6),x]

[Out]

-(3 + 3*a*x*ArcTanh[a*x] + ArcTanh[a*x]^2 + 3*a^2*x^2*ArcTanh[a*x]^2 + 5*a*x*ArcTanh[a*x]^3 + 3*a^3*x^3*ArcTan
h[a*x]^3 + 5*ArcTanh[a*x]^4 + 24*a^2*x^2*ArcTanh[a*x]^4 + 3*a^4*x^4*ArcTanh[a*x]^4 - 2*(-1 + a^2*x^2)^2*ArcTan
h[a*x]^5*SinhIntegral[2*ArcTanh[a*x]] - 16*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5*SinhIntegral[4*ArcTanh[a*x]])/(15*a
*(-1 + a^2*x^2)^2*ArcTanh[a*x]^5)

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Maple [A]  time = 0.07, size = 182, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ( -{\frac{3}{40\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{5}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{10\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{5}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{20\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{30\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{30\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{15\,{\it Artanh} \left ( ax \right ) }}+{\frac{2\,{\it Shi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{15}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{40\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{5}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{40\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{30\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{15\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{4\,\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{15\,{\it Artanh} \left ( ax \right ) }}+{\frac{16\,{\it Shi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x)

[Out]

1/a*(-3/40/arctanh(a*x)^5-1/10/arctanh(a*x)^5*cosh(2*arctanh(a*x))-1/20/arctanh(a*x)^4*sinh(2*arctanh(a*x))-1/
30/arctanh(a*x)^3*cosh(2*arctanh(a*x))-1/30/arctanh(a*x)^2*sinh(2*arctanh(a*x))-1/15/arctanh(a*x)*cosh(2*arcta
nh(a*x))+2/15*Shi(2*arctanh(a*x))-1/40/arctanh(a*x)^5*cosh(4*arctanh(a*x))-1/40/arctanh(a*x)^4*sinh(4*arctanh(
a*x))-1/30/arctanh(a*x)^3*cosh(4*arctanh(a*x))-1/15/arctanh(a*x)^2*sinh(4*arctanh(a*x))-4/15/arctanh(a*x)*cosh
(4*arctanh(a*x))+16/15*Shi(4*arctanh(a*x)))

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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(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="maxima")

[Out]

-2/15*((3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1)^4 + (3*a^4*x^4 + 24*a^2*x^2 + 5)*log(-a*x + 1)^4 + 2*(3*a^3*x
^3 + 5*a*x)*log(a*x + 1)^3 - 2*(3*a^3*x^3 + 5*a*x + 2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log(a*x + 1))*log(-a*x + 1)
^3 + 24*a*x*log(a*x + 1) + 4*(3*a^2*x^2 + 1)*log(a*x + 1)^2 + 2*(6*a^2*x^2 + 3*(3*a^4*x^4 + 24*a^2*x^2 + 5)*lo
g(a*x + 1)^2 + 3*(3*a^3*x^3 + 5*a*x)*log(a*x + 1) + 2)*log(-a*x + 1)^2 - 2*(2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log
(a*x + 1)^3 + 3*(3*a^3*x^3 + 5*a*x)*log(a*x + 1)^2 + 12*a*x + 4*(3*a^2*x^2 + 1)*log(a*x + 1))*log(-a*x + 1) +
48)/((a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^5 - 5*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^4*log(-a*x + 1) + 10*
(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^3*log(-a*x + 1)^2 - 10*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)^2*log(-a*
x + 1)^3 + 5*(a^5*x^4 - 2*a^3*x^2 + a)*log(a*x + 1)*log(-a*x + 1)^4 - (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)^
5) + integrate(-8/15*(15*a^3*x^3 + 17*a*x)/((a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(a*x + 1) - (a^6*x^6 - 3*
a^4*x^4 + 3*a^2*x^2 - 1)*log(-a*x + 1)), x)

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Fricas [A]  time = 1.99863, size = 803, normalized size = 3.12 \begin{align*} \frac{{\left (8 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 8 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) +{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) -{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (-\frac{a x - 1}{a x + 1}\right )\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{5} - 2 \,{\left (3 \, a^{4} x^{4} + 24 \, a^{2} x^{2} + 5\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} - 4 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} - 48 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 8 \,{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 96}{15 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="fricas")

[Out]

1/15*((8*(a^4*x^4 - 2*a^2*x^2 + 1)*log_integral((a^2*x^2 + 2*a*x + 1)/(a^2*x^2 - 2*a*x + 1)) - 8*(a^4*x^4 - 2*
a^2*x^2 + 1)*log_integral((a^2*x^2 - 2*a*x + 1)/(a^2*x^2 + 2*a*x + 1)) + (a^4*x^4 - 2*a^2*x^2 + 1)*log_integra
l(-(a*x + 1)/(a*x - 1)) - (a^4*x^4 - 2*a^2*x^2 + 1)*log_integral(-(a*x - 1)/(a*x + 1)))*log(-(a*x + 1)/(a*x -
1))^5 - 2*(3*a^4*x^4 + 24*a^2*x^2 + 5)*log(-(a*x + 1)/(a*x - 1))^4 - 4*(3*a^3*x^3 + 5*a*x)*log(-(a*x + 1)/(a*x
- 1))^3 - 48*a*x*log(-(a*x + 1)/(a*x - 1)) - 8*(3*a^2*x^2 + 1)*log(-(a*x + 1)/(a*x - 1))^2 - 96)/((a^5*x^4 -
2*a^3*x^2 + a)*log(-(a*x + 1)/(a*x - 1))^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a^{6} x^{6} \operatorname{atanh}^{6}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{6}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{6}{\left (a x \right )} - \operatorname{atanh}^{6}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*x**2+1)**3/atanh(a*x)**6,x)

[Out]

-Integral(1/(a**6*x**6*atanh(a*x)**6 - 3*a**4*x**4*atanh(a*x)**6 + 3*a**2*x**2*atanh(a*x)**6 - atanh(a*x)**6),
x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^3/arctanh(a*x)^6,x, algorithm="giac")

[Out]

integrate(-1/((a^2*x^2 - 1)^3*arctanh(a*x)^6), x)