3.212 \(\int \frac {(1-a^2 x^2)^2 \tanh ^{-1}(a x)^2}{x^5} \, dx\)

Optimal. Leaf size=214 \[ \frac {1}{2} a^4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {Li}_3\left (\frac {2}{1-a x}-1\right )-a^4 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+a^4 \text {Li}_2\left (\frac {2}{1-a x}-1\right ) \tanh ^{-1}(a x)-\frac {4}{3} a^4 \log (x)-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {a^2}{12 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+\frac {2}{3} a^4 \log \left (1-a^2 x^2\right )-\frac {\tanh ^{-1}(a x)^2}{4 x^4}-\frac {a \tanh ^{-1}(a x)}{6 x^3} \]

[Out]

-1/12*a^2/x^2-1/6*a*arctanh(a*x)/x^3+3/2*a^3*arctanh(a*x)/x-3/4*a^4*arctanh(a*x)^2-1/4*arctanh(a*x)^2/x^4+a^2*
arctanh(a*x)^2/x^2-2*a^4*arctanh(a*x)^2*arctanh(-1+2/(-a*x+1))-4/3*a^4*ln(x)+2/3*a^4*ln(-a^2*x^2+1)-a^4*arctan
h(a*x)*polylog(2,1-2/(-a*x+1))+a^4*arctanh(a*x)*polylog(2,-1+2/(-a*x+1))+1/2*a^4*polylog(3,1-2/(-a*x+1))-1/2*a
^4*polylog(3,-1+2/(-a*x+1))

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Rubi [A]  time = 0.55, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6012, 5916, 5982, 266, 44, 36, 29, 31, 5948, 5914, 6052, 6058, 6610} \[ \frac {1}{2} a^4 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {PolyLog}\left (3,\frac {2}{1-a x}-1\right )-a^4 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{1-a x}-1\right )-\frac {a^2}{12 x^2}+\frac {2}{3} a^4 \log \left (1-a^2 x^2\right )+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}-\frac {4}{3} a^4 \log (x)-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {a \tanh ^{-1}(a x)}{6 x^3}-\frac {\tanh ^{-1}(a x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^5,x]

[Out]

-a^2/(12*x^2) - (a*ArcTanh[a*x])/(6*x^3) + (3*a^3*ArcTanh[a*x])/(2*x) - (3*a^4*ArcTanh[a*x]^2)/4 - ArcTanh[a*x
]^2/(4*x^4) + (a^2*ArcTanh[a*x]^2)/x^2 + 2*a^4*ArcTanh[a*x]^2*ArcTanh[1 - 2/(1 - a*x)] - (4*a^4*Log[x])/3 + (2
*a^4*Log[1 - a^2*x^2])/3 - a^4*ArcTanh[a*x]*PolyLog[2, 1 - 2/(1 - a*x)] + a^4*ArcTanh[a*x]*PolyLog[2, -1 + 2/(
1 - a*x)] + (a^4*PolyLog[3, 1 - 2/(1 - a*x)])/2 - (a^4*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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 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 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 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 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 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 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 )^2 \tanh ^{-1}(a x)^2}{x^5} \, dx &=\int \left (\frac {\tanh ^{-1}(a x)^2}{x^5}-\frac {2 a^2 \tanh ^{-1}(a x)^2}{x^3}+\frac {a^4 \tanh ^{-1}(a x)^2}{x}\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x^3} \, dx\right )+a^4 \int \frac {\tanh ^{-1}(a x)^2}{x} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^5} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{x^4 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx-\left (4 a^5\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}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{x^4} \, dx+\frac {1}{2} a^3 \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx-\left (2 a^3\right ) \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx-\left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\left (2 a^5\right ) \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-\left (2 a^5\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)}{6 x^3}+\frac {2 a^3 \tanh ^{-1}(a x)}{x}-a^4 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{6} a^2 \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^3 \int \frac {\tanh ^{-1}(a x)}{x^2} \, dx-\left (2 a^4\right ) \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^5 \int \frac {\tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+a^5 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-a^5 \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)}{6 x^3}+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a^4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )+\frac {1}{12} a^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )+\frac {1}{2} a^4 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx-a^4 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-a^4 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a^4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )+\frac {1}{12} a^2 \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{x \left (1-a^2 x\right )} \, dx,x,x^2\right )-a^4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-a^6 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {11}{6} a^4 \log (x)+\frac {11}{12} a^4 \log \left (1-a^2 x^2\right )-a^4 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a^4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )+\frac {1}{4} a^4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{4} a^6 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a^2}{12 x^2}-\frac {a \tanh ^{-1}(a x)}{6 x^3}+\frac {3 a^3 \tanh ^{-1}(a x)}{2 x}-\frac {3}{4} a^4 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{4 x^4}+\frac {a^2 \tanh ^{-1}(a x)^2}{x^2}+2 a^4 \tanh ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1-a x}\right )-\frac {4}{3} a^4 \log (x)+\frac {2}{3} a^4 \log \left (1-a^2 x^2\right )-a^4 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )+a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-a x}\right )+\frac {1}{2} a^4 \text {Li}_3\left (1-\frac {2}{1-a x}\right )-\frac {1}{2} a^4 \text {Li}_3\left (-1+\frac {2}{1-a x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 238, normalized size = 1.11 \[ \frac {1}{24} \left (24 a^4 \tanh ^{-1}(a x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+24 a^4 \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+12 a^4 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )-12 a^4 \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )-16 a^4 \tanh ^{-1}(a x)^3-18 a^4 \tanh ^{-1}(a x)^2-24 a^4 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+24 a^4 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+i \pi ^3 a^4+2 a^4+\frac {36 a^3 \tanh ^{-1}(a x)}{x}-\frac {2 a^2}{x^2}+\frac {24 a^2 \tanh ^{-1}(a x)^2}{x^2}-32 a^4 \log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )-\frac {6 \tanh ^{-1}(a x)^2}{x^4}-\frac {4 a \tanh ^{-1}(a x)}{x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - a^2*x^2)^2*ArcTanh[a*x]^2)/x^5,x]

[Out]

(2*a^4 + I*a^4*Pi^3 - (2*a^2)/x^2 - (4*a*ArcTanh[a*x])/x^3 + (36*a^3*ArcTanh[a*x])/x - 18*a^4*ArcTanh[a*x]^2 -
(6*ArcTanh[a*x]^2)/x^4 + (24*a^2*ArcTanh[a*x]^2)/x^2 - 16*a^4*ArcTanh[a*x]^3 - 24*a^4*ArcTanh[a*x]^2*Log[1 +
E^(-2*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]^2*Log[1 - E^(2*ArcTanh[a*x])] - 32*a^4*Log[(a*x)/Sqrt[1 - a^2*x^2]]
+ 24*a^4*ArcTanh[a*x]*PolyLog[2, -E^(-2*ArcTanh[a*x])] + 24*a^4*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] +
12*a^4*PolyLog[3, -E^(-2*ArcTanh[a*x])] - 12*a^4*PolyLog[3, E^(2*ArcTanh[a*x])])/24

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="fricas")

[Out]

integral((a^4*x^4 - 2*a^2*x^2 + 1)*arctanh(a*x)^2/x^5, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="giac")

[Out]

integrate((a^2*x^2 - 1)^2*arctanh(a*x)^2/x^5, x)

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maple [C]  time = 2.76, size = 927, normalized size = 4.33 \[ \frac {3 a^{3} \arctanh \left (a x \right )}{2 x}+\frac {a^{2} \arctanh \left (a x \right )^{2}}{x^{2}}-\frac {a \arctanh \left (a x \right )}{6 x^{3}}+a^{4} \arctanh \left (a x \right )^{2} \ln \left (a x \right )-a^{4} \arctanh \left (a x \right )^{2} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )+a^{4} \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a^{4} \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{4} \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a^{4} \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-a^{4} \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )+\frac {i a^{4} \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 )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{3}}{2}+\frac {a^{4}}{24 \sqrt {-a^{2} x^{2}+1}-24}-\frac {i a^{4} \arctanh \left (a x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2}}{2}-\frac {i a^{4} \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 \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )^{2}}{2}-\frac {3 a^{4} \arctanh \left (a x \right )^{2}}{4}-\frac {\arctanh \left (a x \right )^{2}}{4 x^{4}}-2 a^{4} \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {4 a^{4} \arctanh \left (a x \right )}{3}-2 a^{4} \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{4} \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2}+\frac {a^{4} \sqrt {-a^{2} x^{2}+1}}{12 \sqrt {-a^{2} x^{2}+1}+12 a x +12}+\frac {a^{5} x}{24 \sqrt {-a^{2} x^{2}+1}+24}-\frac {a^{4} \sqrt {-a^{2} x^{2}+1}}{12 \left (a x +1-\sqrt {-a^{2} x^{2}+1}\right )}-\frac {a^{5} x}{24 \left (\sqrt {-a^{2} x^{2}+1}-1\right )}+\frac {i a^{4} \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}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}}\right )}{2}-\frac {4 a^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {4 a^{4} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{3}-\frac {a^{4}}{24 \left (\sqrt {-a^{2} x^{2}+1}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x)

[Out]

-3/4*a^4*arctanh(a*x)^2-1/4*arctanh(a*x)^2/x^4+a^2*arctanh(a*x)^2/x^2-2*a^4*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/
2))-1/6*a*arctanh(a*x)/x^3+3/2*a^3*arctanh(a*x)/x-1/2*I*a^4*arctanh(a*x)^2*Pi*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)
))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2-1/2*I*a^4*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)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2+4/3*a^4*arctanh(a*x)+1/2*I
*a^4*arctanh(a*x)^2*Pi*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3+a^4*arctanh(a*x)^2*ln(a
*x)-a^4*arctanh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+a^4*arctanh(a*x)^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+2*a^4*
arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+a^4*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+2*a^4*a
rctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))-a^4*arctanh(a*x)*polylog(2,-(a*x+1)^2/(-a^2*x^2+1))-1/12*a^
4/(a*x+1-(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2)+1/12*a^4/((-a^2*x^2+1)^(1/2)+a*x+1)*(-a^2*x^2+1)^(1/2)-1/24*a^
5/((-a^2*x^2+1)^(1/2)-1)*x+1/24*a^5/((-a^2*x^2+1)^(1/2)+1)*x+1/2*I*a^4*arctanh(a*x)^2*Pi*csgn(I*((a*x+1)^2/(-a
^2*x^2+1)-1))*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))
-4/3*a^4*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-2*a^4*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*a^4*polylog(3,-(a*x
+1)^2/(-a^2*x^2+1))-4/3*a^4*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+1/24*a^4/((-a^2*x^2+1)^(1/2)-1)-1/24*a^4/((-a^2*x
^2+1)^(1/2)+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (4 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )^{2}}{16 \, x^{4}} - \frac {1}{4} \, \int -\frac {2 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )^{2} - {\left (4 \, a^{3} x^{3} - a x + 4 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )}{2 \, {\left (a x^{6} - x^{5}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*x^2+1)^2*arctanh(a*x)^2/x^5,x, algorithm="maxima")

[Out]

1/16*(4*a^2*x^2 - 1)*log(-a*x + 1)^2/x^4 - 1/4*integrate(-1/2*(2*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 +
a*x - 1)*log(a*x + 1)^2 - (4*a^3*x^3 - a*x + 4*(a^5*x^5 - a^4*x^4 - 2*a^3*x^3 + 2*a^2*x^2 + a*x - 1)*log(a*x +
1))*log(-a*x + 1))/(a*x^6 - x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^5,x)

[Out]

int((atanh(a*x)^2*(a^2*x^2 - 1)^2)/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*x**2+1)**2*atanh(a*x)**2/x**5,x)

[Out]

Integral((a*x - 1)**2*(a*x + 1)**2*atanh(a*x)**2/x**5, x)

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