3.215 \(\int \frac{(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^8} \, dx\)

Optimal. Leaf size=183 \[ -\frac{8}{105} a^7 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{17 a^4}{630 x^3}-\frac{a^2}{105 x^5}-\frac{8 a^5 \tanh ^{-1}(a x)}{105 x^2}-\frac{a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{9 a^3 \tanh ^{-1}(a x)}{70 x^4}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^6}{210 x}+\frac{8}{105} a^7 \tanh ^{-1}(a x)^2-\frac{1}{210} a^7 \tanh ^{-1}(a x)+\frac{16}{105} a^7 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{21 x^6}-\frac{\tanh ^{-1}(a x)^2}{7 x^7} \]

[Out]

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

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Rubi [A]  time = 0.81617, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6012, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ -\frac{8}{105} a^7 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{17 a^4}{630 x^3}-\frac{a^2}{105 x^5}-\frac{8 a^5 \tanh ^{-1}(a x)}{105 x^2}-\frac{a^4 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{9 a^3 \tanh ^{-1}(a x)}{70 x^4}+\frac{2 a^2 \tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^6}{210 x}+\frac{8}{105} a^7 \tanh ^{-1}(a x)^2-\frac{1}{210} a^7 \tanh ^{-1}(a x)+\frac{16}{105} a^7 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{21 x^6}-\frac{\tanh ^{-1}(a x)^2}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^8,x]

[Out]

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

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 5982

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 5988

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

Rule 5932

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTanh[c*
x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [A]  time = 1.38135, size = 140, normalized size = 0.77 \[ \frac{-48 a^7 x^7 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+a^2 x^2 \left (3 a^4 x^4+17 a^2 x^2-6\right )+6 \left (8 a^7 x^7-35 a^4 x^4+42 a^2 x^2-15\right ) \tanh ^{-1}(a x)^2+3 a x \tanh ^{-1}(a x) \left (-a^6 x^6-16 a^4 x^4+27 a^2 x^2+32 a^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-10\right )}{630 x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^8,x]

[Out]

(a^2*x^2*(-6 + 17*a^2*x^2 + 3*a^4*x^4) + 6*(-15 + 42*a^2*x^2 - 35*a^4*x^4 + 8*a^7*x^7)*ArcTanh[a*x]^2 + 3*a*x*
ArcTanh[a*x]*(-10 + 27*a^2*x^2 - 16*a^4*x^4 - a^6*x^6 + 32*a^6*x^6*Log[1 - E^(-2*ArcTanh[a*x])]) - 48*a^7*x^7*
PolyLog[2, E^(-2*ArcTanh[a*x])])/(630*x^7)

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Maple [A]  time = 0.064, size = 292, normalized size = 1.6 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{7\,{x}^{7}}}+{\frac{2\,{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5\,{x}^{5}}}-{\frac{{a}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{8\,{a}^{7}{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{105}}-{\frac{a{\it Artanh} \left ( ax \right ) }{21\,{x}^{6}}}+{\frac{9\,{a}^{3}{\it Artanh} \left ( ax \right ) }{70\,{x}^{4}}}-{\frac{8\,{a}^{5}{\it Artanh} \left ( ax \right ) }{105\,{x}^{2}}}+{\frac{16\,{a}^{7}{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) }{105}}-{\frac{8\,{a}^{7}{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{105}}-{\frac{8\,{a}^{7}{\it dilog} \left ( ax \right ) }{105}}-{\frac{8\,{a}^{7}{\it dilog} \left ( ax+1 \right ) }{105}}-{\frac{8\,{a}^{7}\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{105}}-{\frac{2\,{a}^{7} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{105}}+{\frac{8\,{a}^{7}}{105}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,{a}^{7}\ln \left ( ax-1 \right ) }{105}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{2\,{a}^{7} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{105}}+{\frac{4\,{a}^{7}}{105}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,{a}^{7}\ln \left ( ax+1 \right ) }{105}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{{a}^{7}\ln \left ( ax-1 \right ) }{420}}+{\frac{{a}^{6}}{210\,x}}-{\frac{{a}^{2}}{105\,{x}^{5}}}+{\frac{17\,{a}^{4}}{630\,{x}^{3}}}-{\frac{{a}^{7}\ln \left ( ax+1 \right ) }{420}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x)

[Out]

-1/7*arctanh(a*x)^2/x^7+2/5*a^2*arctanh(a*x)^2/x^5-1/3*a^4*arctanh(a*x)^2/x^3-8/105*a^7*arctanh(a*x)*ln(a*x-1)
-1/21*a*arctanh(a*x)/x^6+9/70*a^3*arctanh(a*x)/x^4-8/105*a^5*arctanh(a*x)/x^2+16/105*a^7*arctanh(a*x)*ln(a*x)-
8/105*a^7*arctanh(a*x)*ln(a*x+1)-8/105*a^7*dilog(a*x)-8/105*a^7*dilog(a*x+1)-8/105*a^7*ln(a*x)*ln(a*x+1)-2/105
*a^7*ln(a*x-1)^2+8/105*a^7*dilog(1/2+1/2*a*x)+4/105*a^7*ln(a*x-1)*ln(1/2+1/2*a*x)+2/105*a^7*ln(a*x+1)^2+4/105*
a^7*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)-4/105*a^7*ln(-1/2*a*x+1/2)*ln(a*x+1)+1/420*a^7*ln(a*x-1)+1/210*a^6/x-1/10
5*a^2/x^5+17/630*a^4/x^3-1/420*a^7*ln(a*x+1)

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Maxima [A]  time = 0.994082, size = 343, normalized size = 1.87 \begin{align*} \frac{1}{1260} \,{\left (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} - 96 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )} a^{5} + 96 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )} a^{5} - 3 \, a^{5} \log \left (a x + 1\right ) + 3 \, a^{5} \log \left (a x - 1\right ) + \frac{2 \,{\left (12 \, a^{5} x^{5} \log \left (a x + 1\right )^{2} - 24 \, a^{5} x^{5} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 12 \, a^{5} x^{5} \log \left (a x - 1\right )^{2} + 3 \, a^{4} x^{4} + 17 \, a^{2} x^{2} - 6\right )}}{x^{5}}\right )} a^{2} - \frac{1}{210} \,{\left (16 \, a^{6} \log \left (a^{2} x^{2} - 1\right ) - 16 \, a^{6} \log \left (x^{2}\right ) + \frac{16 \, a^{4} x^{4} - 27 \, a^{2} x^{2} + 10}{x^{6}}\right )} a \operatorname{artanh}\left (a x\right ) - \frac{{\left (35 \, a^{4} x^{4} - 42 \, a^{2} x^{2} + 15\right )} \operatorname{artanh}\left (a x\right )^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="maxima")

[Out]

1/1260*(96*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a^5 - 96*(log(a*x + 1)*log(x) + dilog(-a*
x))*a^5 + 96*(log(-a*x + 1)*log(x) + dilog(a*x))*a^5 - 3*a^5*log(a*x + 1) + 3*a^5*log(a*x - 1) + 2*(12*a^5*x^5
*log(a*x + 1)^2 - 24*a^5*x^5*log(a*x + 1)*log(a*x - 1) - 12*a^5*x^5*log(a*x - 1)^2 + 3*a^4*x^4 + 17*a^2*x^2 -
6)/x^5)*a^2 - 1/210*(16*a^6*log(a^2*x^2 - 1) - 16*a^6*log(x^2) + (16*a^4*x^4 - 27*a^2*x^2 + 10)/x^6)*a*arctanh
(a*x) - 1/105*(35*a^4*x^4 - 42*a^2*x^2 + 15)*arctanh(a*x)^2/x^7

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="fricas")

[Out]

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^8, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{x^{8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**8,x)

[Out]

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**8, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^8,x, algorithm="giac")

[Out]

integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2/x^8, x)