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

Optimal. Leaf size=178 \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{105 a^3}+\frac{a^2 x^5}{105}+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac{x}{210 a^2}+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{\tanh ^{-1}(a x)}{210 a^3}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{105 a^3}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{17 x^3}{630} \]

[Out]

-x/(210*a^2) - (17*x^3)/630 + (a^2*x^5)/105 + ArcTanh[a*x]/(210*a^3) + (8*x^2*ArcTanh[a*x])/(105*a) - (9*a*x^4
*ArcTanh[a*x])/70 + (a^3*x^6*ArcTanh[a*x])/21 + (8*ArcTanh[a*x]^2)/(105*a^3) + (x^3*ArcTanh[a*x]^2)/3 - (2*a^2
*x^5*ArcTanh[a*x]^2)/5 + (a^4*x^7*ArcTanh[a*x]^2)/7 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(105*a^3) - (8*PolyLo
g[2, 1 - 2/(1 - a*x)])/(105*a^3)

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Rubi [A]  time = 0.779231, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6012, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 302} \[ -\frac{8 \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{105 a^3}+\frac{a^2 x^5}{105}+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac{x}{210 a^2}+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{\tanh ^{-1}(a x)}{210 a^3}-\frac{16 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{105 a^3}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{17 x^3}{630} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(210*a^2) - (17*x^3)/630 + (a^2*x^5)/105 + ArcTanh[a*x]/(210*a^3) + (8*x^2*ArcTanh[a*x])/(105*a) - (9*a*x^4
*ArcTanh[a*x])/70 + (a^3*x^6*ArcTanh[a*x])/21 + (8*ArcTanh[a*x]^2)/(105*a^3) + (x^3*ArcTanh[a*x]^2)/3 - (2*a^2
*x^5*ArcTanh[a*x]^2)/5 + (a^4*x^7*ArcTanh[a*x]^2)/7 - (16*ArcTanh[a*x]*Log[2/(1 - a*x)])/(105*a^3) - (8*PolyLo
g[2, 1 - 2/(1 - a*x)])/(105*a^3)

Rule 6012

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[E
xpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[
c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]

Rule 5916

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

Rule 5980

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int x^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^2 \tanh ^{-1}(a x)^2-2 a^2 x^4 \tanh ^{-1}(a x)^2+a^4 x^6 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^4 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^6 \tanh ^{-1}(a x)^2 \, dx+\int x^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{1}{3} (2 a) \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{5} \left (4 a^3\right ) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac{1}{7} \left (2 a^5\right ) \int \frac{x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{3 a}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}-\frac{1}{5} (4 a) \int x^3 \tanh ^{-1}(a x) \, dx+\frac{1}{5} (4 a) \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{7} \left (2 a^3\right ) \int x^5 \tanh ^{-1}(a x) \, dx-\frac{1}{7} \left (2 a^3\right ) \int \frac{x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{x^2 \tanh ^{-1}(a x)}{3 a}-\frac{1}{5} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{\tanh ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{1}{3} \int \frac{x^2}{1-a^2 x^2} \, dx-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{3 a^2}-\frac{4 \int x \tanh ^{-1}(a x) \, dx}{5 a}+\frac{4 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}+\frac{1}{7} (2 a) \int x^3 \tanh ^{-1}(a x) \, dx-\frac{1}{7} (2 a) \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac{1}{5} a^2 \int \frac{x^4}{1-a^2 x^2} \, dx-\frac{1}{21} a^4 \int \frac{x^6}{1-a^2 x^2} \, dx\\ &=\frac{x}{3 a^2}-\frac{x^2 \tanh ^{-1}(a x)}{15 a}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{2 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{3 a^3}+\frac{2}{5} \int \frac{x^2}{1-a^2 x^2} \, dx-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{3 a^2}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^2}+\frac{4 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^2}+\frac{2 \int x \tanh ^{-1}(a x) \, dx}{7 a}-\frac{2 \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}-\frac{1}{14} a^2 \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{1}{5} a^2 \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac{1}{21} a^4 \int \left (-\frac{1}{a^6}-\frac{x^2}{a^4}-\frac{x^4}{a^2}+\frac{1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=-\frac{23 x}{105 a^2}-\frac{16 x^3}{315}+\frac{a^2 x^5}{105}-\frac{\tanh ^{-1}(a x)}{3 a^3}+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac{2 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{15 a^3}-\frac{1}{7} \int \frac{x^2}{1-a^2 x^2} \, dx-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{3 a^3}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{21 a^2}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{5 a^2}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^2}+\frac{2 \int \frac{1}{1-a^2 x^2} \, dx}{5 a^2}-\frac{4 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^2}-\frac{1}{14} a^2 \int \left (-\frac{1}{a^4}-\frac{x^2}{a^2}+\frac{1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=-\frac{x}{210 a^2}-\frac{17 x^3}{630}+\frac{a^2 x^5}{105}+\frac{23 \tanh ^{-1}(a x)}{105 a^3}+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{105 a^3}-\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{3 a^3}+\frac{4 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{5 a^3}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{14 a^2}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{7 a^2}+\frac{2 \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^2}\\ &=-\frac{x}{210 a^2}-\frac{17 x^3}{630}+\frac{a^2 x^5}{105}+\frac{\tanh ^{-1}(a x)}{210 a^3}+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{105 a^3}+\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{15 a^3}-\frac{2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{7 a^3}\\ &=-\frac{x}{210 a^2}-\frac{17 x^3}{630}+\frac{a^2 x^5}{105}+\frac{\tanh ^{-1}(a x)}{210 a^3}+\frac{8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac{9}{70} a x^4 \tanh ^{-1}(a x)+\frac{1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac{8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} x^3 \tanh ^{-1}(a x)^2-\frac{2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac{1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac{16 \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{105 a^3}-\frac{8 \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{105 a^3}\\ \end{align*}

Mathematica [A]  time = 1.09864, size = 121, normalized size = 0.68 \[ \frac{48 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+a x \left (6 a^4 x^4-17 a^2 x^2-3\right )+6 \left (15 a^7 x^7-42 a^5 x^5+35 a^3 x^3-8\right ) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (30 a^6 x^6-81 a^4 x^4+48 a^2 x^2-96 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+3\right )}{630 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x*(-3 - 17*a^2*x^2 + 6*a^4*x^4) + 6*(-8 + 35*a^3*x^3 - 42*a^5*x^5 + 15*a^7*x^7)*ArcTanh[a*x]^2 + ArcTanh[a*
x]*(3 + 48*a^2*x^2 - 81*a^4*x^4 + 30*a^6*x^6 - 96*Log[1 + E^(-2*ArcTanh[a*x])]) + 48*PolyLog[2, -E^(-2*ArcTanh
[a*x])])/(630*a^3)

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Maple [A]  time = 0.05, size = 239, normalized size = 1.3 \begin{align*}{\frac{{a}^{4}{x}^{7} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{7}}-{\frac{2\,{a}^{2}{x}^{5} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{x}^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{{a}^{3}{x}^{6}{\it Artanh} \left ( ax \right ) }{21}}-{\frac{9\,a{x}^{4}{\it Artanh} \left ( ax \right ) }{70}}+{\frac{8\,{x}^{2}{\it Artanh} \left ( ax \right ) }{105\,a}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{105\,{a}^{3}}}+{\frac{8\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{105\,{a}^{3}}}+{\frac{{x}^{5}{a}^{2}}{105}}-{\frac{17\,{x}^{3}}{630}}-{\frac{x}{210\,{a}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{420\,{a}^{3}}}+{\frac{\ln \left ( ax+1 \right ) }{420\,{a}^{3}}}+{\frac{2\, \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{105\,{a}^{3}}}-{\frac{8}{105\,{a}^{3}}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,\ln \left ( ax-1 \right ) }{105\,{a}^{3}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4}{105\,{a}^{3}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,\ln \left ( ax+1 \right ) }{105\,{a}^{3}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{2\, \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{105\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^4*x^7*arctanh(a*x)^2-2/5*a^2*x^5*arctanh(a*x)^2+1/3*x^3*arctanh(a*x)^2+1/21*a^3*x^6*arctanh(a*x)-9/70*a*
x^4*arctanh(a*x)+8/105*x^2*arctanh(a*x)/a+8/105/a^3*arctanh(a*x)*ln(a*x-1)+8/105/a^3*arctanh(a*x)*ln(a*x+1)+1/
105*x^5*a^2-17/630*x^3-1/210*x/a^2-1/420/a^3*ln(a*x-1)+1/420/a^3*ln(a*x+1)+2/105/a^3*ln(a*x-1)^2-8/105/a^3*dil
og(1/2+1/2*a*x)-4/105/a^3*ln(a*x-1)*ln(1/2+1/2*a*x)-4/105/a^3*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)+4/105/a^3*ln(-1
/2*a*x+1/2)*ln(a*x+1)-2/105/a^3*ln(a*x+1)^2

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Maxima [A]  time = 0.983677, size = 267, normalized size = 1.5 \begin{align*} \frac{1}{1260} \, a^{2}{\left (\frac{12 \, a^{5} x^{5} - 34 \, a^{3} x^{3} - 6 \, a x - 24 \, \log \left (a x + 1\right )^{2} + 48 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 24 \, \log \left (a x - 1\right )^{2} - 3 \, \log \left (a x - 1\right )}{a^{5}} - \frac{96 \,{\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^{5}} + \frac{3 \, \log \left (a x + 1\right )}{a^{5}}\right )} + \frac{1}{210} \, a{\left (\frac{10 \, a^{4} x^{6} - 27 \, a^{2} x^{4} + 16 \, x^{2}}{a^{2}} + \frac{16 \, \log \left (a x + 1\right )}{a^{4}} + \frac{16 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname{artanh}\left (a x\right ) + \frac{1}{105} \,{\left (15 \, a^{4} x^{7} - 42 \, a^{2} x^{5} + 35 \, x^{3}\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*a^2*((12*a^5*x^5 - 34*a^3*x^3 - 6*a*x - 24*log(a*x + 1)^2 + 48*log(a*x + 1)*log(a*x - 1) + 24*log(a*x -
1)^2 - 3*log(a*x - 1))/a^5 - 96*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a^5 + 3*log(a*x + 1
)/a^5) + 1/210*a*((10*a^4*x^6 - 27*a^2*x^4 + 16*x^2)/a^2 + 16*log(a*x + 1)/a^4 + 16*log(a*x - 1)/a^4)*arctanh(
a*x) + 1/105*(15*a^4*x^7 - 42*a^2*x^5 + 35*x^3)*arctanh(a*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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