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

Optimal. Leaf size=338 \[ \frac{24 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{35 a}-\frac{48 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{16 \tanh ^{-1}(a x)^3}{35 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{48 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{35 a} \]

[Out]

(-13*(1 - a^2*x^2))/(210*a) - (1 - a^2*x^2)^2/(140*a) - (14*x*ArcTanh[a*x])/15 - (13*x*(1 - a^2*x^2)*ArcTanh[a
*x])/105 - (x*(1 - a^2*x^2)^2*ArcTanh[a*x])/35 + (12*(1 - a^2*x^2)*ArcTanh[a*x]^2)/(35*a) + (9*(1 - a^2*x^2)^2
*ArcTanh[a*x]^2)/(70*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(14*a) + (16*ArcTanh[a*x]^3)/(35*a) + (16*x*ArcTanh
[a*x]^3)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3)/35 + (x*(1 - a^2*x^
2)^3*ArcTanh[a*x]^3)/7 - (48*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(35*a) - (7*Log[1 - a^2*x^2])/(15*a) - (48*ArcTa
nh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a) + (24*PolyLog[3, 1 - 2/(1 - a*x)])/(35*a)

________________________________________________________________________________________

Rubi [A]  time = 0.333408, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {5944, 5910, 5984, 5918, 5948, 6058, 6610, 260, 5942} \[ \frac{24 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{35 a}-\frac{48 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{16 \tanh ^{-1}(a x)^3}{35 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{48 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)^3*ArcTanh[a*x]^3,x]

[Out]

(-13*(1 - a^2*x^2))/(210*a) - (1 - a^2*x^2)^2/(140*a) - (14*x*ArcTanh[a*x])/15 - (13*x*(1 - a^2*x^2)*ArcTanh[a
*x])/105 - (x*(1 - a^2*x^2)^2*ArcTanh[a*x])/35 + (12*(1 - a^2*x^2)*ArcTanh[a*x]^2)/(35*a) + (9*(1 - a^2*x^2)^2
*ArcTanh[a*x]^2)/(70*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x]^2)/(14*a) + (16*ArcTanh[a*x]^3)/(35*a) + (16*x*ArcTanh
[a*x]^3)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^3)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^3)/35 + (x*(1 - a^2*x^
2)^3*ArcTanh[a*x]^3)/7 - (48*ArcTanh[a*x]^2*Log[2/(1 - a*x)])/(35*a) - (7*Log[1 - a^2*x^2])/(15*a) - (48*ArcTa
nh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a) + (24*PolyLog[3, 1 - 2/(1 - a*x)])/(35*a)

Rule 5944

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

Rule 5910

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

Rule 5984

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

Rule 5918

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

Rule 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 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 5942

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

Rubi steps

\begin{align*} \int \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3 \, dx &=\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{1}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx+\frac{6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{4}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx-\frac{9}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx+\frac{24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{8}{105} \int \tanh ^{-1}(a x) \, dx-\frac{6}{35} \int \tanh ^{-1}(a x) \, dx+\frac{16}{35} \int \tanh ^{-1}(a x)^3 \, dx-\frac{24}{35} \int \tanh ^{-1}(a x) \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3+\frac{1}{105} (8 a) \int \frac{x}{1-a^2 x^2} \, dx+\frac{1}{35} (6 a) \int \frac{x}{1-a^2 x^2} \, dx+\frac{1}{35} (24 a) \int \frac{x}{1-a^2 x^2} \, dx-\frac{1}{35} (48 a) \int \frac{x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{16 \tanh ^{-1}(a x)^3}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}-\frac{48}{35} \int \frac{\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{16 \tanh ^{-1}(a x)^3}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{48 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{35 a}-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}+\frac{96}{35} \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{16 \tanh ^{-1}(a x)^3}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{48 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{35 a}-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}-\frac{48 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a}+\frac{48}{35} \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{13 \left (1-a^2 x^2\right )}{210 a}-\frac{\left (1-a^2 x^2\right )^2}{140 a}-\frac{14}{15} x \tanh ^{-1}(a x)-\frac{13}{105} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)-\frac{1}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)+\frac{12 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{35 a}+\frac{9 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{70 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2}{14 a}+\frac{16 \tanh ^{-1}(a x)^3}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^3+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3-\frac{48 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{35 a}-\frac{7 \log \left (1-a^2 x^2\right )}{15 a}-\frac{48 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a}+\frac{24 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{35 a}\\ \end{align*}

Mathematica [A]  time = 1.09682, size = 231, normalized size = 0.68 \[ -\frac{-576 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-288 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+3 a^4 x^4-32 a^2 x^2+196 \log \left (1-a^2 x^2\right )+60 a^7 x^7 \tanh ^{-1}(a x)^3+30 a^6 x^6 \tanh ^{-1}(a x)^2-252 a^5 x^5 \tanh ^{-1}(a x)^3+12 a^5 x^5 \tanh ^{-1}(a x)-144 a^4 x^4 \tanh ^{-1}(a x)^2+420 a^3 x^3 \tanh ^{-1}(a x)^3-76 a^3 x^3 \tanh ^{-1}(a x)+342 a^2 x^2 \tanh ^{-1}(a x)^2-420 a x \tanh ^{-1}(a x)^3+456 a x \tanh ^{-1}(a x)+192 \tanh ^{-1}(a x)^3-228 \tanh ^{-1}(a x)^2+576 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+29}{420 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(29 - 32*a^2*x^2 + 3*a^4*x^4 + 456*a*x*ArcTanh[a*x] - 76*a^3*x^3*ArcTanh[a*x] + 12*a^5*x^5*ArcTanh[a*x] - 228
*ArcTanh[a*x]^2 + 342*a^2*x^2*ArcTanh[a*x]^2 - 144*a^4*x^4*ArcTanh[a*x]^2 + 30*a^6*x^6*ArcTanh[a*x]^2 + 192*Ar
cTanh[a*x]^3 - 420*a*x*ArcTanh[a*x]^3 + 420*a^3*x^3*ArcTanh[a*x]^3 - 252*a^5*x^5*ArcTanh[a*x]^3 + 60*a^7*x^7*A
rcTanh[a*x]^3 + 576*ArcTanh[a*x]^2*Log[1 + E^(-2*ArcTanh[a*x])] + 196*Log[1 - a^2*x^2] - 576*ArcTanh[a*x]*Poly
Log[2, -E^(-2*ArcTanh[a*x])] - 288*PolyLog[3, -E^(-2*ArcTanh[a*x])])/(420*a)

________________________________________________________________________________________

Maple [C]  time = 2.487, size = 932, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-29/420/a+19/35*arctanh(a*x)^2/a+16/35*arctanh(a*x)^3/a+x*arctanh(a*x)^3-38/35*x*arctanh(a*x)-1/7*a^6*arctanh(
a*x)^3*x^7-1/14*a^5*arctanh(a*x)^2*x^6-1/35*a^4*arctanh(a*x)*x^5+19/105*a^2*arctanh(a*x)*x^3+8/105*a*x^2-a^2*a
rctanh(a*x)^3*x^3-57/70*a*arctanh(a*x)^2*x^2+24/35/a*arctanh(a*x)^2*ln(a*x-1)+24/35/a*arctanh(a*x)^2*ln(a*x+1)
-48/35/a*arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+12/35*a^3*arctanh(a*x)^2*x^4+3/5*a^4*arctanh(a*x)^3*x^5
-48/35/a*arctanh(a*x)^2*ln(2)-48/35/a*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-1/140*x^4*a^3-14/15*arct
anh(a*x)/a+14/15/a*ln((a*x+1)^2/(-a^2*x^2+1)+1)+24/35/a*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))-24/35*I/a*Pi*csgn(I
*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*arctanh(a*x)^2-12/35*I/a*Pi*csgn(I*(a*x+1)/(-a^2*
x^2+1)^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*arctanh(a*x)^2-12/35*I/a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*c
sgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2+12/35*I/a*Pi*csgn(I*(a*x+1)^2/(a^2*x^
2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2-24/35*I/a*Pi*arctanh(a*x)^2+12
/35*I/a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x
+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^2-12/35*I/a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*
arctanh(a*x)^2-12/35*I/a*Pi*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*arctanh(a*x)^2-24/35*I/a*Pi*csgn(I/((a*x+1)^2/(-a^
2*x^2+1)+1))^3*arctanh(a*x)^2+24/35*I/a*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arctanh(a*x)^2

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

[Out]

1/19600*(150*a^7*x^7 - 175*a^6*x^6 - 672*a^5*x^5 + 840*a^4*x^4 + 1330*a^3*x^3 - 1995*a^2*x^2 - 3360*a*x - 210*
(5*a^7*x^7 - 21*a^5*x^5 + 35*a^3*x^3 - 35*a*x - 16)*log(a*x + 1))*log(-a*x + 1)^2/a - 1/8*(log(-a*x + 1)^3 - 3
*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1)/a + 1/691488000*(36000*(343*log(-a*x + 1)^3 - 147*log(-a*x +
1)^2 + 42*log(-a*x + 1) - 6)*(a*x - 1)^7 + 2401000*(36*log(-a*x + 1)^3 - 18*log(-a*x + 1)^2 + 6*log(-a*x + 1)
- 1)*(a*x - 1)^6 + 2074464*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x + 1) - 6)*(a*x - 1)^5 + 13
505625*(32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*log(-a*x + 1) - 3)*(a*x - 1)^4 + 48020000*(9*log(-a*x + 1
)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 64827000*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 +
6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 86436000*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x
- 1))/a - 1/480000*(288*(125*log(-a*x + 1)^3 - 75*log(-a*x + 1)^2 + 30*log(-a*x + 1) - 6)*(a*x - 1)^5 + 5625*(
32*log(-a*x + 1)^3 - 24*log(-a*x + 1)^2 + 12*log(-a*x + 1) - 3)*(a*x - 1)^4 + 40000*(9*log(-a*x + 1)^3 - 9*log
(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 90000*(4*log(-a*x + 1)^3 - 6*log(-a*x + 1)^2 + 6*log(-a*x +
1) - 3)*(a*x - 1)^2 + 180000*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6)*(a*x - 1))/a + 1/288*
(4*(9*log(-a*x + 1)^3 - 9*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 2)*(a*x - 1)^3 + 27*(4*log(-a*x + 1)^3 - 6*log(-
a*x + 1)^2 + 6*log(-a*x + 1) - 3)*(a*x - 1)^2 + 108*(log(-a*x + 1)^3 - 3*log(-a*x + 1)^2 + 6*log(-a*x + 1) - 6
)*(a*x - 1))/a + 1/8*integrate(-1/1225*(1225*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^
2 - a*x + 1)*log(a*x + 1)^3 + (150*a^7*x^7 - 175*a^6*x^6 - 672*a^5*x^5 + 840*a^4*x^4 + 1330*a^3*x^3 - 1995*a^2
*x^2 - 3675*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*log(a*x + 1)^2 - 336
0*a*x - 210*(5*a^7*x^7 - 21*a^5*x^5 + 35*a^3*x^3 - 35*a*x - 16)*log(a*x + 1))*log(-a*x + 1))/(a*x - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*arctanh(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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