3.2.79 \(\int \frac {(1-a^2 x^2) \tanh ^{-1}(a x)^2}{x^3} \, dx\) [179]

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

[Out]

-a*arctanh(a*x)/x+1/2*a^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2/x^2+2*a^2*arctanh(a*x)^2*arctanh(-1+2/(-a*x+1))+a^
2*ln(x)-1/2*a^2*ln(-a^2*x^2+1)+a^2*arctanh(a*x)*polylog(2,1-2/(-a*x+1))-a^2*arctanh(a*x)*polylog(2,-1+2/(-a*x+
1))-1/2*a^2*polylog(3,1-2/(-a*x+1))+1/2*a^2*polylog(3,-1+2/(-a*x+1))

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Rubi [A]
time = 0.25, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6161, 6037, 6129, 272, 36, 29, 31, 6095, 6033, 6199, 6205, 6745} \begin {gather*} -\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 (\frac {2}{1-a x}-1\right )+a^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)-a^2 \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)-\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} \end {gather*}

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 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 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 272

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 6033

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 6037

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

Rule 6095

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 6129

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 6161

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 6199

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 6205

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 6745

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 \text {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 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} a^4 \text {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*}

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Mathematica [A]
time = 0.05, size = 174, normalized size = 1.01 \begin {gather*} -\frac {a \tanh ^{-1}(a x)}{x}+\frac {\left (-1+a^2 x^2\right ) \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 {PolyLog}\left (2,\frac {-1-a x}{-1+a x}\right )+a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {1+a x}{-1+a x}\right )+\frac {1}{2} a^2 \text {PolyLog}\left (3,\frac {-1-a x}{-1+a x}\right )-\frac {1}{2} a^2 \text {PolyLog}\left (3,\frac {1+a x}{-1+a x}\right ) \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 19.02, size = 736, normalized size = 4.28

method	result	size
derivativedivides	\(a^{2} \left (-\arctanh \left (a x \right )^{2} \ln \left (a x \right )-\frac {\arctanh \left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}}{2}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\frac {\arctanh \left (a x \right )^{2}}{2}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )}{2}-\frac {\left (-\sqrt {-a^{2} x^{2}+1}+a x +1\right ) \arctanh \left (a x \right )}{2 a x}-\frac {\arctanh \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}+a x +1\right )}{2 a x}+\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}\right )\)	\(736\)
default	\(a^{2} \left (-\arctanh \left (a x \right )^{2} \ln \left (a x \right )-\frac {\arctanh \left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {i \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{2} \arctanh \left (a x \right )^{2}}{2}-\frac {i \arctanh \left (a x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )^{3}}{2}+\ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\frac {\arctanh \left (a x \right )^{2}}{2}+\ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {i \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}+1}\right )}{2}-\frac {\left (-\sqrt {-a^{2} x^{2}+1}+a x +1\right ) \arctanh \left (a x \right )}{2 a x}-\frac {\arctanh \left (a x \right ) \left (\sqrt {-a^{2} x^{2}+1}+a x +1\right )}{2 a x}+\arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )-\arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )-\frac {\polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}\right )\)	\(736\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)*arctanh(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(-arctanh(a*x)^2*ln(a*x)-1/2*arctanh(a*x)^2/a^2/x^2-1/2*I*arctanh(a*x)^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))+1/2*I*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+1/2*I*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+ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+1/2*arctanh(a*x)^2+ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*I*a
rctanh(a*x)^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3-1/2*(-(-a^2*x^2+1)^(1/2)+a*x+
1)/a/x*arctanh(a*x)-1/2*arctanh(a*x)*((-a^2*x^2+1)^(1/2)+a*x+1)/a/x+arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1
)-arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*poly
log(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*arctanh(a*x)*polylog(2,(a
*x+1)/(-a^2*x^2+1)^(1/2))+2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+
1))-1/2*polylog(3,-(a*x+1)^2/(-a^2*x^2+1)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{3}}\right )\, dx - \int \frac {a^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(atanh(a*x)^2*(a^2*x^2 - 1))/x^3,x)

[Out]

-int((atanh(a*x)^2*(a^2*x^2 - 1))/x^3, x)

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