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

Optimal. Leaf size=172 \[ -\frac{1}{2} a^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )+a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )-\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x} \]

[Out]

-((a*ArcTanh[a*x])/x) + (a^2*ArcTanh[a*x]^2)/2 - ArcTanh[a*x]^2/(2*x^2) - 2*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(
1 - a*x)] + a^2*Log[x] - (a^2*Log[1 - a^2*x^2])/2 + a^2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] - a^2*ArcTanh
[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] - (a^2*PolyLog[3, 1 - 2/(1 - a*x)])/2 + (a^2*PolyLog[3, -1 + 2/(1 - a*x)])/
2

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Rubi [A]  time = 0.334493, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6014, 5916, 5982, 266, 36, 29, 31, 5948, 5914, 6052, 6058, 6610} \[ -\frac{1}{2} a^2 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \text{PolyLog}\left (3,\frac{2}{1-a x}-1\right )+a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{1-a x}-1\right )-\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \log (x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-\frac{a \tanh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTanh[a*x])/x) + (a^2*ArcTanh[a*x]^2)/2 - ArcTanh[a*x]^2/(2*x^2) - 2*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(
1 - a*x)] + a^2*Log[x] - (a^2*Log[1 - a^2*x^2])/2 + a^2*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] - a^2*ArcTanh
[a*x]*PolyLog[2, -1 + 2/(1 - a*x)] - (a^2*PolyLog[3, 1 - 2/(1 - a*x)])/2 + (a^2*PolyLog[3, -1 + 2/(1 - a*x)])/
2

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 5914

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

Rule 6052

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

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]

Rubi steps

\begin{align*} \int \frac{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x^3} \, dx &=-\left (a^2 \int \frac{\tanh ^{-1}(a x)^2}{x} \, dx\right )+\int \frac{\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+\left (4 a^3\right ) \int \frac{\tanh ^{-1}(a x) \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a \int \frac{\tanh ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac{\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+\left (2 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )+a^2 \int \frac{1}{x \left (1-a^2 x^2\right )} \, dx-a^3 \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx+a^3 \int \frac{\text{Li}_2\left (-1+\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-\frac{1}{2} a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-\frac{1}{2} a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a \tanh ^{-1}(a x)}{x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{2 x^2}-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )+a^2 \log (x)-\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-a x}\right )-\frac{1}{2} a^2 \text{Li}_3\left (1-\frac{2}{1-a x}\right )+\frac{1}{2} a^2 \text{Li}_3\left (-1+\frac{2}{1-a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0735436, size = 174, normalized size = 1.01 \[ \frac{1}{2} a^2 \text{PolyLog}\left (3,\frac{-a x-1}{a x-1}\right )-\frac{1}{2} a^2 \text{PolyLog}\left (3,\frac{a x+1}{a x-1}\right )-a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{-a x-1}{a x-1}\right )+a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{a x+1}{a x-1}\right )-\frac{1}{2} a^2 \log \left (1-a^2 x^2\right )+\frac{\left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-2 a^2 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1-a x}\right )-\frac{a \tanh ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTanh[a*x])/x) + ((-1 + a^2*x^2)*ArcTanh[a*x]^2)/(2*x^2) - 2*a^2*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)
] + a^2*Log[x] - (a^2*Log[1 - a^2*x^2])/2 - a^2*ArcTanh[a*x]*PolyLog[2, (-1 - a*x)/(-1 + a*x)] + a^2*ArcTanh[a
*x]*PolyLog[2, (1 + a*x)/(-1 + a*x)] + (a^2*PolyLog[3, (-1 - a*x)/(-1 + a*x)])/2 - (a^2*PolyLog[3, (1 + a*x)/(
-1 + a*x)])/2

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Maple [C]  time = 1.208, size = 741, normalized size = 4.3 \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)*arctanh(a*x)^2/x^3,x)

[Out]

-a^2*arctanh(a*x)^2*ln(a*x)-1/2*arctanh(a*x)^2/x^2+1/2*a^2*arctanh(a*x)^2+a^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
-1/2*I*a^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^
2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))*arctanh(a*x)^2+a^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)-a^2*arctanh(a*x)-a*arc
tanh(a*x)/x-1/2*I*a^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3*arctanh(a*x)^2+1/2*I*
a^2*Pi*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^2*arct
anh(a*x)^2+1/2*I*a^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*
x^2+1)+1))^2*arctanh(a*x)^2+a^2*arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)-a^2*arctanh(a*x)^2*ln(1-(a*x+1)/(-
a^2*x^2+1)^(1/2))-2*a^2*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+2*a^2*polylog(3,(a*x+1)/(-a^2*x^2+1
)^(1/2))-a^2*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-2*a^2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1
)^(1/2))+2*a^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+a^2*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-1/2*
a^2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (-a x + 1\right )^{2}}{8 \, x^{2}} + \frac{1}{4} \, \int -\frac{{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right )^{2} -{\left (a x + 2 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{a x^{4} - x^{3}}\,{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^3,x, algorithm="maxima")

[Out]

-1/8*log(-a*x + 1)^2/x^2 + 1/4*integrate(-((a^3*x^3 - a^2*x^2 - a*x + 1)*log(a*x + 1)^2 - (a*x + 2*(a^3*x^3 -
a^2*x^2 - a*x + 1)*log(a*x + 1))*log(-a*x + 1))/(a*x^4 - x^3), x)

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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^{3}}, 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^3,x, algorithm="fricas")

[Out]

integral(-(a^2*x^2 - 1)*arctanh(a*x)^2/x^3, 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^{3}}\, dx - \int \frac{a^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{x}\, 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**3,x)

[Out]

-Integral(-atanh(a*x)**2/x**3, x) - Integral(a**2*atanh(a*x)**2/x, 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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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