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

Optimal. Leaf size=143 \[ \frac{2}{15} a^5 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{30 x^3}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{a^4}{30 x}-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{1}{30} a^5 \tanh ^{-1}(a x)-\frac{4}{15} a^5 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}-\frac{\tanh ^{-1}(a x)^2}{5 x^5} \]

[Out]

-a^2/(30*x^3) + a^4/(30*x) - (a^5*ArcTanh[a*x])/30 - (a*ArcTanh[a*x])/(10*x^4) + (2*a^3*ArcTanh[a*x])/(15*x^2)
- (2*a^5*ArcTanh[a*x]^2)/15 - ArcTanh[a*x]^2/(5*x^5) + (a^2*ArcTanh[a*x]^2)/(3*x^3) - (4*a^5*ArcTanh[a*x]*Log
[2 - 2/(1 + a*x)])/15 + (2*a^5*PolyLog[2, -1 + 2/(1 + a*x)])/15

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Rubi [A]  time = 0.445357, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6014, 5916, 5982, 325, 206, 5988, 5932, 2447} \[ \frac{2}{15} a^5 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{a^2}{30 x^3}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{a^4}{30 x}-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{1}{30} a^5 \tanh ^{-1}(a x)-\frac{4}{15} a^5 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}-\frac{\tanh ^{-1}(a x)^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(30*x^3) + a^4/(30*x) - (a^5*ArcTanh[a*x])/30 - (a*ArcTanh[a*x])/(10*x^4) + (2*a^3*ArcTanh[a*x])/(15*x^2)
- (2*a^5*ArcTanh[a*x]^2)/15 - ArcTanh[a*x]^2/(5*x^5) + (a^2*ArcTanh[a*x]^2)/(3*x^3) - (4*a^5*ArcTanh[a*x]*Log
[2 - 2/(1 + a*x)])/15 + (2*a^5*PolyLog[2, -1 + 2/(1 + a*x)])/15

Rule 6014

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

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 ) \tanh ^{-1}(a x)^2}{x^6} \, dx &=-\left (a^2 \int \frac{\tanh ^{-1}(a x)^2}{x^4} \, dx\right )+\int \frac{\tanh ^{-1}(a x)^2}{x^6} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{1}{5} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^5 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{1}{5} (2 a) \int \frac{\tanh ^{-1}(a x)}{x^5} \, dx+\frac{1}{5} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3} \, dx-\frac{1}{3} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{a^3 \tanh ^{-1}(a x)}{3 x^2}-\frac{1}{3} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}+\frac{1}{10} a^2 \int \frac{1}{x^4 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x)}{x^3} \, dx-\frac{1}{3} a^4 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac{1}{3} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{a^4}{3 x}-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{2}{3} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{1}{10} a^4 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} a^4 \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac{1}{5} \left (2 a^5\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac{1}{3} a^6 \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{3} \left (2 a^6\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{a^4}{30 x}-\frac{1}{3} a^5 \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{4}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{1}{3} a^5 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+\frac{1}{10} a^6 \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{5} a^6 \int \frac{1}{1-a^2 x^2} \, dx-\frac{1}{5} \left (2 a^6\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a^2}{30 x^3}+\frac{a^4}{30 x}-\frac{1}{30} a^5 \tanh ^{-1}(a x)-\frac{a \tanh ^{-1}(a x)}{10 x^4}+\frac{2 a^3 \tanh ^{-1}(a x)}{15 x^2}-\frac{2}{15} a^5 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{5 x^5}+\frac{a^2 \tanh ^{-1}(a x)^2}{3 x^3}-\frac{4}{15} a^5 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{2}{15} a^5 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.430777, size = 114, normalized size = 0.8 \[ \frac{4 a^5 x^5 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+a^2 x^2 \left (a^2 x^2-1\right )-2 \left (2 a^5 x^5-5 a^2 x^2+3\right ) \tanh ^{-1}(a x)^2-a x \tanh ^{-1}(a x) \left (a^4 x^4-4 a^2 x^2+8 a^4 x^4 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+3\right )}{30 x^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*x^2*(-1 + a^2*x^2) - 2*(3 - 5*a^2*x^2 + 2*a^5*x^5)*ArcTanh[a*x]^2 - a*x*ArcTanh[a*x]*(3 - 4*a^2*x^2 + a^4
*x^4 + 8*a^4*x^4*Log[1 - E^(-2*ArcTanh[a*x])]) + 4*a^5*x^5*PolyLog[2, E^(-2*ArcTanh[a*x])])/(30*x^5)

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Maple [B]  time = 0.062, size = 258, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{5\,{x}^{5}}}+{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}}+{\frac{2\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{15}}-{\frac{a{\it Artanh} \left ( ax \right ) }{10\,{x}^{4}}}+{\frac{2\,{a}^{3}{\it Artanh} \left ( ax \right ) }{15\,{x}^{2}}}-{\frac{4\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) }{15}}+{\frac{2\,{a}^{5}{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{15}}+{\frac{{a}^{5}\ln \left ( ax-1 \right ) }{60}}-{\frac{{a}^{2}}{30\,{x}^{3}}}+{\frac{{a}^{4}}{30\,x}}-{\frac{{a}^{5}\ln \left ( ax+1 \right ) }{60}}+{\frac{2\,{a}^{5}{\it dilog} \left ( ax \right ) }{15}}+{\frac{2\,{a}^{5}{\it dilog} \left ( ax+1 \right ) }{15}}+{\frac{2\,{a}^{5}\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{15}}+{\frac{{a}^{5} \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{30}}-{\frac{2\,{a}^{5}}{15}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{{a}^{5}\ln \left ( ax-1 \right ) }{15}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{{a}^{5}\ln \left ( ax+1 \right ) }{15}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{{a}^{5}}{15}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{{a}^{5} \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*arctanh(a*x)^2/x^5+1/3*a^2*arctanh(a*x)^2/x^3+2/15*a^5*arctanh(a*x)*ln(a*x-1)-1/10*a*arctanh(a*x)/x^4+2/1
5*a^3*arctanh(a*x)/x^2-4/15*a^5*arctanh(a*x)*ln(a*x)+2/15*a^5*arctanh(a*x)*ln(a*x+1)+1/60*a^5*ln(a*x-1)-1/30*a
^2/x^3+1/30*a^4/x-1/60*a^5*ln(a*x+1)+2/15*a^5*dilog(a*x)+2/15*a^5*dilog(a*x+1)+2/15*a^5*ln(a*x)*ln(a*x+1)+1/30
*a^5*ln(a*x-1)^2-2/15*a^5*dilog(1/2+1/2*a*x)-1/15*a^5*ln(a*x-1)*ln(1/2+1/2*a*x)+1/15*a^5*ln(-1/2*a*x+1/2)*ln(a
*x+1)-1/15*a^5*ln(-1/2*a*x+1/2)*ln(1/2+1/2*a*x)-1/30*a^5*ln(a*x+1)^2

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Maxima [A]  time = 0.99753, size = 308, normalized size = 2.15 \begin{align*} -\frac{1}{60} \,{\left (8 \,{\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} - 8 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )} a^{3} + 8 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )} a^{3} + a^{3} \log \left (a x + 1\right ) - a^{3} \log \left (a x - 1\right ) + \frac{2 \,{\left (a^{3} x^{3} \log \left (a x + 1\right )^{2} - 2 \, a^{3} x^{3} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a^{3} x^{3} \log \left (a x - 1\right )^{2} - a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a^{2} + \frac{1}{30} \,{\left (4 \, a^{4} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{4} \log \left (x^{2}\right ) + \frac{4 \, a^{2} x^{2} - 3}{x^{4}}\right )} a \operatorname{artanh}\left (a x\right ) + \frac{{\left (5 \, a^{2} x^{2} - 3\right )} \operatorname{artanh}\left (a x\right )^{2}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(8*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a^3 - 8*(log(a*x + 1)*log(x) + dilog(-a*x))
*a^3 + 8*(log(-a*x + 1)*log(x) + dilog(a*x))*a^3 + a^3*log(a*x + 1) - a^3*log(a*x - 1) + 2*(a^3*x^3*log(a*x +
1)^2 - 2*a^3*x^3*log(a*x + 1)*log(a*x - 1) - a^3*x^3*log(a*x - 1)^2 - a^2*x^2 + 1)/x^3)*a^2 + 1/30*(4*a^4*log(
a^2*x^2 - 1) - 4*a^4*log(x^2) + (4*a^2*x^2 - 3)/x^4)*a*arctanh(a*x) + 1/15*(5*a^2*x^2 - 3)*arctanh(a*x)^2/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-atanh(a*x)**2/x**6, x) - Integral(a**2*atanh(a*x)**2/x**4, 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 )} \operatorname{artanh}\left (a x\right )^{2}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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