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

Optimal. Leaf size=227 \[ -\frac{16 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a}-\frac{1}{105} a^4 x^5+\frac{19 a^2 x^3}{315}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{16 \tanh ^{-1}(a x)^2}{35 a}-\frac{32 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}-\frac{38 x}{105} \]

[Out]

(-38*x)/105 + (19*a^2*x^3)/315 - (a^4*x^5)/105 + (8*(1 - a^2*x^2)*ArcTanh[a*x])/(35*a) + (3*(1 - a^2*x^2)^2*Ar
cTanh[a*x])/(35*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21*a) + (16*ArcTanh[a*x]^2)/(35*a) + (16*x*ArcTanh[a*x]^2
)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^2)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/35 + (x*(1 - a^2*x^2)^3*Ar
cTanh[a*x]^2)/7 - (32*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a) - (16*PolyLog[2, 1 - 2/(1 - a*x)])/(35*a)

________________________________________________________________________________________

Rubi [A]  time = 0.171278, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {5944, 5910, 5984, 5918, 2402, 2315, 8, 194} \[ -\frac{16 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{35 a}-\frac{1}{105} a^4 x^5+\frac{19 a^2 x^3}{315}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{16 \tanh ^{-1}(a x)^2}{35 a}-\frac{32 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a}-\frac{38 x}{105} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2*x^2)^3*ArcTanh[a*x]^2,x]

[Out]

(-38*x)/105 + (19*a^2*x^3)/315 - (a^4*x^5)/105 + (8*(1 - a^2*x^2)*ArcTanh[a*x])/(35*a) + (3*(1 - a^2*x^2)^2*Ar
cTanh[a*x])/(35*a) + ((1 - a^2*x^2)^3*ArcTanh[a*x])/(21*a) + (16*ArcTanh[a*x]^2)/(35*a) + (16*x*ArcTanh[a*x]^2
)/35 + (8*x*(1 - a^2*x^2)*ArcTanh[a*x]^2)/35 + (6*x*(1 - a^2*x^2)^2*ArcTanh[a*x]^2)/35 + (x*(1 - a^2*x^2)^3*Ar
cTanh[a*x]^2)/7 - (32*ArcTanh[a*x]*Log[2/(1 - a*x)])/(35*a) - (16*PolyLog[2, 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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2 \, dx &=\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{1}{21} \int \left (1-a^2 x^2\right )^2 \, dx+\frac{6}{7} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{1}{21} \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac{3}{35} \int \left (1-a^2 x^2\right ) \, dx+\frac{24}{35} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac{2 x}{15}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{8 \int 1 \, dx}{35}+\frac{16}{35} \int \tanh ^{-1}(a x)^2 \, dx\\ &=-\frac{38 x}{105}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{1}{35} (32 a) \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac{38 x}{105}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{16 \tanh ^{-1}(a x)^2}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{32}{35} \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac{38 x}{105}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{16 \tanh ^{-1}(a x)^2}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{32 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a}+\frac{32}{35} \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{38 x}{105}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{16 \tanh ^{-1}(a x)^2}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{32 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a}-\frac{32 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{35 a}\\ &=-\frac{38 x}{105}+\frac{19 a^2 x^3}{315}-\frac{a^4 x^5}{105}+\frac{8 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{35 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}{35 a}+\frac{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)}{21 a}+\frac{16 \tanh ^{-1}(a x)^2}{35 a}+\frac{16}{35} x \tanh ^{-1}(a x)^2+\frac{8}{35} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2+\frac{6}{35} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+\frac{1}{7} x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2-\frac{32 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{35 a}-\frac{16 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{35 a}\\ \end{align*}

Mathematica [A]  time = 1.18337, size = 124, normalized size = 0.55 \[ -\frac{-144 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+3 a^5 x^5-19 a^3 x^3+9 (a x-1)^4 \left (5 a^3 x^3+20 a^2 x^2+29 a x+16\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (5 a^6 x^6-24 a^4 x^4+57 a^2 x^2+96 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-38\right )+114 a x}{315 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(114*a*x - 19*a^3*x^3 + 3*a^5*x^5 + 9*(-1 + a*x)^4*(16 + 29*a*x + 20*a^2*x^2 + 5*a^3*x^3)*ArcTanh[a*x]^2 + 3*
ArcTanh[a*x]*(-38 + 57*a^2*x^2 - 24*a^4*x^4 + 5*a^6*x^6 + 96*Log[1 + E^(-2*ArcTanh[a*x])]) - 144*PolyLog[2, -E
^(-2*ArcTanh[a*x])])/(315*a)

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 250, normalized size = 1.1 \begin{align*} -{\frac{{a}^{6} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{7}}{7}}+{\frac{3\,{a}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{5}}{5}}-{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}+x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-{\frac{{a}^{5}{\it Artanh} \left ( ax \right ){x}^{6}}{21}}+{\frac{8\,{a}^{3}{\it Artanh} \left ( ax \right ){x}^{4}}{35}}-{\frac{19\,a{\it Artanh} \left ( ax \right ){x}^{2}}{35}}+{\frac{16\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{35\,a}}+{\frac{16\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{35\,a}}+{\frac{4\, \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{35\,a}}-{\frac{16}{35\,a}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{8\,\ln \left ( ax-1 \right ) }{35\,a}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\, \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{35\,a}}-{\frac{8}{35\,a}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{8\,\ln \left ( ax+1 \right ) }{35\,a}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{{a}^{4}{x}^{5}}{105}}+{\frac{19\,{x}^{3}{a}^{2}}{315}}-{\frac{38\,x}{105}}-{\frac{19\,\ln \left ( ax-1 \right ) }{105\,a}}+{\frac{19\,\ln \left ( ax+1 \right ) }{105\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^6*arctanh(a*x)^2*x^7+3/5*a^4*arctanh(a*x)^2*x^5-a^2*arctanh(a*x)^2*x^3+x*arctanh(a*x)^2-1/21*a^5*arctan
h(a*x)*x^6+8/35*a^3*arctanh(a*x)*x^4-19/35*a*arctanh(a*x)*x^2+16/35/a*arctanh(a*x)*ln(a*x-1)+16/35/a*arctanh(a
*x)*ln(a*x+1)+4/35/a*ln(a*x-1)^2-16/35/a*dilog(1/2+1/2*a*x)-8/35/a*ln(a*x-1)*ln(1/2+1/2*a*x)-4/35/a*ln(a*x+1)^
2-8/35/a*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)+8/35/a*ln(-1/2*a*x+1/2)*ln(a*x+1)-1/105*a^4*x^5+19/315*x^3*a^2-38/10
5*x-19/105/a*ln(a*x-1)+19/105/a*ln(a*x+1)

________________________________________________________________________________________

Maxima [A]  time = 0.976557, size = 269, normalized size = 1.19 \begin{align*} -\frac{1}{315} \, a^{2}{\left (\frac{3 \, a^{5} x^{5} - 19 \, a^{3} x^{3} + 114 \, a x + 36 \, \log \left (a x + 1\right )^{2} - 72 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 36 \, \log \left (a x - 1\right )^{2} + 57 \, \log \left (a x - 1\right )}{a^{3}} + \frac{144 \,{\left (\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )\right )}}{a^{3}} - \frac{57 \, \log \left (a x + 1\right )}{a^{3}}\right )} - \frac{1}{105} \,{\left (5 \, a^{4} x^{6} - 24 \, a^{2} x^{4} + 57 \, x^{2} - \frac{48 \, \log \left (a x + 1\right )}{a^{2}} - \frac{48 \, \log \left (a x - 1\right )}{a^{2}}\right )} a \operatorname{artanh}\left (a x\right ) - \frac{1}{35} \,{\left (5 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 35 \, a^{2} x^{3} - 35 \, x\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*a^2*((3*a^5*x^5 - 19*a^3*x^3 + 114*a*x + 36*log(a*x + 1)^2 - 72*log(a*x + 1)*log(a*x - 1) - 36*log(a*x
- 1)^2 + 57*log(a*x - 1))/a^3 + 144*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^3 - 57*log(a*x
+ 1)/a^3) - 1/105*(5*a^4*x^6 - 24*a^2*x^4 + 57*x^2 - 48*log(a*x + 1)/a^2 - 48*log(a*x - 1)/a^2)*a*arctanh(a*x
) - 1/35*(5*a^6*x^7 - 21*a^4*x^5 + 35*a^2*x^3 - 35*x)*arctanh(a*x)^2

________________________________________________________________________________________

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 )^{2}, 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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

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