3.272 \(\int \frac {\tanh ^{-1}(a x)^2}{x^3 (1-a^2 x^2)^2} \, dx\)
Optimal. Leaf size=205 \[ -a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )-2 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \log (x)+\frac {2}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2+2 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \]
[Out]
1/4*a^2/(-a^2*x^2+1)-a*arctanh(a*x)/x-1/2*a^3*x*arctanh(a*x)/(-a^2*x^2+1)+1/4*a^2*arctanh(a*x)^2-1/2*arctanh(a
*x)^2/x^2+1/2*a^2*arctanh(a*x)^2/(-a^2*x^2+1)+2/3*a^2*arctanh(a*x)^3+a^2*ln(x)-1/2*a^2*ln(-a^2*x^2+1)+2*a^2*ar
ctanh(a*x)^2*ln(2-2/(a*x+1))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(a*x+1))-a^2*polylog(3,-1+2/(a*x+1))
________________________________________________________________________________________
Rubi [A] time = 0.70, antiderivative size = 205, normalized size of antiderivative = 1.00,
number of steps used = 22, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used
= {6030, 5982, 5916, 266, 36, 29, 31, 5948, 5988, 5932, 6056, 6610, 5994, 5956, 261}
\[ -a^2 \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-2 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \log (x)+\frac {2}{3} a^2 \tanh ^{-1}(a x)^3+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2+2 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}-\frac {a \tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
[In]
Int[ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^2),x]
[Out]
a^2/(4*(1 - a^2*x^2)) - (a*ArcTanh[a*x])/x - (a^3*x*ArcTanh[a*x])/(2*(1 - a^2*x^2)) + (a^2*ArcTanh[a*x]^2)/4 -
ArcTanh[a*x]^2/(2*x^2) + (a^2*ArcTanh[a*x]^2)/(2*(1 - a^2*x^2)) + (2*a^2*ArcTanh[a*x]^3)/3 + a^2*Log[x] - (a^
2*Log[1 - a^2*x^2])/2 + 2*a^2*ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - 2*a^2*ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 +
a*x)] - a^2*PolyLog[3, -1 + 2/(1 + a*x)]
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 261
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -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 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 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 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 5956
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[(x*(a + b*ArcTanh[c*x
])^p)/(2*d*(d + e*x^2)), x] + (-Dist[(b*c*p)/2, Int[(x*(a + b*ArcTanh[c*x])^(p - 1))/(d + e*x^2)^2, x], x] + S
imp[(a + b*ArcTanh[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] &&
GtQ[p, 0]
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 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 5994
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)
^(q + 1)*(a + b*ArcTanh[c*x])^p)/(2*e*(q + 1)), x] + Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan
h[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0] && NeQ[q, -1]
Rule 6030
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int
[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTanh
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[
m, 0] && NeQ[p, -1]
Rule 6056
Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[((a + b*ArcTa
nh[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 {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=2 \left (a^2 \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx\right )+a^4 \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a \int \frac {\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\right )-a^3 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\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+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\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\right )+\frac {1}{2} a^4 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \int \frac {1}{x \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+a^3 \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\right )\\ &=\frac {a^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\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 )\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^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\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 )\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^2}{4 \left (1-a^2 x^2\right )}-\frac {a \tanh ^{-1}(a x)}{x}-\frac {a^3 x \tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1-a^2 x^2\right )+2 \left (\frac {1}{3} a^2 \tanh ^{-1}(a x)^3+a^2 \tanh ^{-1}(a x)^2 \log \left (2-\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 )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.02, size = 146, normalized size = 0.71 \[ a^2 \left (2 \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{24} \left (24 \log \left (\frac {a x}{\sqrt {1-a^2 x^2}}\right )+6 \tanh ^{-1}(a x)^2 \left (-\frac {2}{a^2 x^2}+8 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+\cosh \left (2 \tanh ^{-1}(a x)\right )+2\right )-24 \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )-16 \tanh ^{-1}(a x)^3-\frac {6 \tanh ^{-1}(a x) \left (a x \sinh \left (2 \tanh ^{-1}(a x)\right )+4\right )}{a x}+3 \cosh \left (2 \tanh ^{-1}(a x)\right )+2 i \pi ^3\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcTanh[a*x]^2/(x^3*(1 - a^2*x^2)^2),x]
[Out]
a^2*(2*ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((2*I)*Pi^3 - 16*ArcTanh[a*x]^3 + 3*Cosh[2*ArcTanh[a*x]]
+ 6*ArcTanh[a*x]^2*(2 - 2/(a^2*x^2) + Cosh[2*ArcTanh[a*x]] + 8*Log[1 - E^(2*ArcTanh[a*x])]) + 24*Log[(a*x)/Sqr
t[1 - a^2*x^2]] - 24*PolyLog[3, E^(2*ArcTanh[a*x])] - (6*ArcTanh[a*x]*(4 + a*x*Sinh[2*ArcTanh[a*x]]))/(a*x))/2
4)
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x, algorithm="fricas")
[Out]
integral(arctanh(a*x)^2/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x, algorithm="giac")
[Out]
integrate(arctanh(a*x)^2/((a^2*x^2 - 1)^2*x^3), x)
________________________________________________________________________________________
maple [C] time = 2.26, size = 3040, normalized size = 14.83 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x)
[Out]
I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi
-I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*csgn
(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi+1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/
2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi+1/2*I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+
1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi-I*a^2/(a*x-1)/(
a*x+1)*arctanh(a*x)^2*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)))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi+1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^
2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi-I*a^4*x^2/(a*x-1)/(a*x+1)*arcta
nh(a*x)^2*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*P
i-1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*csgn
(I*(a*x+1)^2/(a^2*x^2-1))*Pi+4*a^2*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+4*a^2*arctanh(a*x)*poly
log(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+a^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)-1/16*a^4/(a*x-1)/(a*x+1)*x^2+2/3*a^2/(a
*x-1)/(a*x+1)*arctanh(a*x)^3-1/4*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2+a^2/(a*x-1)/(a*x+1)*arctanh(a*x)-1/2*arcta
nh(a*x)^2/x^2-4*a^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-4*a^2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+I*a^4*x
^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi-1/2*I*a^2
/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(
-a^2*x^2+1)))^2*Pi+1/2*I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^
2+1)))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi-1/2*I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)
^(1/2))^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))*Pi-I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1
/2))*csgn(I*(a*x+1)^2/(a^2*x^2-1))^2*Pi+1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2
-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi+1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)
)^3*Pi-I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi+I*a^2/(a*x-1)/(a*x+1)*
arctanh(a*x)^2*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*Pi+I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*cs
gn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi+I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1))
)^3*Pi+I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*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)))*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1))*Pi-1/2*I*a^4*x^2/(a*x-1)/(a*x+1)*arctanh
(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))*csgn(I*(a*
x+1)^2/(a^2*x^2-1))*Pi-2/3*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^3+1/4*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2-a
^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)-1/2*a^3*x/(a*x-1)/(a*x+1)*arctanh(a*x)+a/x/(a*x-1)/(a*x+1)*arctanh(a*x)-2*
a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*ln(2)-1/2*I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1)
/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*Pi-I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*((a*x+1)^2/(-a^2*x^2+1)-1)/(1+(a
*x+1)^2/(-a^2*x^2+1)))^3*Pi-1/2*I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3*Pi+a^2*ln
(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^2*Pi+I*
a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*Pi-I*a^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*csgn(I/(1+(a*x+1)^2/(-a^2*x^2+1
)))^3*Pi-3/16*a^2/(a*x-1)/(a*x+1)+2*a^4*x^2/(a*x-1)/(a*x+1)*arctanh(a*x)^2*ln(2)-I*a^2/(a*x-1)/(a*x+1)*arctanh
(a*x)^2*Pi+2*a^2*arctanh(a*x)^2*ln(a*x)-1/4*a^2*arctanh(a*x)^2/(a*x-1)-a^2*arctanh(a*x)^2*ln(a*x-1)-2*a^2*arct
anh(a*x)^2*ln((a*x+1)^2/(-a^2*x^2+1)-1)+1/4*a^2*arctanh(a*x)^2/(a*x+1)-a^2*arctanh(a*x)^2*ln(a*x+1)+2*a^2*arct
anh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+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)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a^{6} \int \frac {x^{6} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} + \frac {1}{2} \, a^{5} \int \frac {x^{5} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {1}{16} \, {\left (a {\left (\frac {2}{a^{4} x - a^{3}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} + \frac {4 \, \log \left (-a x + 1\right )}{a^{4} x^{2} - a^{2}}\right )} a^{4} - \frac {1}{2} \, a^{4} \int \frac {x^{4} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {1}{2} \, a^{3} \int \frac {x^{3} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} + \frac {1}{2} \, a^{3} \int \frac {x^{3} \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {1}{4} \, a^{2} \int \frac {x^{2} \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {1}{4} \, a \int \frac {x \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {2 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{3} + 3 \, {\left (2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) - 1\right )} \log \left (-a x + 1\right )^{2}}{24 \, {\left (a^{2} x^{4} - x^{2}\right )}} + \frac {1}{4} \, \int \frac {\log \left (a x + 1\right )^{2}}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} - \frac {1}{2} \, \int \frac {\log \left (a x + 1\right ) \log \left (-a x + 1\right )}{a^{4} x^{7} - 2 \, a^{2} x^{5} + x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arctanh(a*x)^2/x^3/(-a^2*x^2+1)^2,x, algorithm="maxima")
[Out]
1/2*a^6*integrate(x^6*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) + 1/2*a^5*integrate(x^5*log(a
*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/16*(a*(2/(a^4*x - a^3) - log(a*x + 1)/a^3 + log(a*x
- 1)/a^3) + 4*log(-a*x + 1)/(a^4*x^2 - a^2))*a^4 - 1/2*a^4*integrate(x^4*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 -
2*a^2*x^5 + x^3), x) - 1/2*a^3*integrate(x^3*log(a*x + 1)*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) + 1/2
*a^3*integrate(x^3*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/4*a^2*integrate(x^2*log(-a*x + 1)/(a^4*x^
7 - 2*a^2*x^5 + x^3), x) - 1/4*a*integrate(x*log(-a*x + 1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/24*(2*(a^4*x^4
- a^2*x^2)*log(-a*x + 1)^3 + 3*(2*a^2*x^2 + 2*(a^4*x^4 - a^2*x^2)*log(a*x + 1) - 1)*log(-a*x + 1)^2)/(a^2*x^4
- x^2) + 1/4*integrate(log(a*x + 1)^2/(a^4*x^7 - 2*a^2*x^5 + x^3), x) - 1/2*integrate(log(a*x + 1)*log(-a*x +
1)/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(atanh(a*x)^2/(x^3*(a^2*x^2 - 1)^2),x)
[Out]
int(atanh(a*x)^2/(x^3*(a^2*x^2 - 1)^2), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{3} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(atanh(a*x)**2/x**3/(-a**2*x**2+1)**2,x)
[Out]
Integral(atanh(a*x)**2/(x**3*(a*x - 1)**2*(a*x + 1)**2), x)
________________________________________________________________________________________