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

Optimal. Leaf size=196 \[ \frac {11}{32 \left (1-a^2 x^2\right )}+\frac {1}{32 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {11 a x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {1}{2} \text {Li}_3\left (\frac {2}{a x+1}-1\right )-\text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {1}{3} \tanh ^{-1}(a x)^3-\frac {11}{32} \tanh ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

[Out]

1/32/(-a^2*x^2+1)^2+11/32/(-a^2*x^2+1)-1/8*a*x*arctanh(a*x)/(-a^2*x^2+1)^2-11/16*a*x*arctanh(a*x)/(-a^2*x^2+1)
-11/32*arctanh(a*x)^2+1/4*arctanh(a*x)^2/(-a^2*x^2+1)^2+1/2*arctanh(a*x)^2/(-a^2*x^2+1)+1/3*arctanh(a*x)^3+arc
tanh(a*x)^2*ln(2-2/(a*x+1))-arctanh(a*x)*polylog(2,-1+2/(a*x+1))-1/2*polylog(3,-1+2/(a*x+1))

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Rubi [A]  time = 0.45, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6030, 5988, 5932, 5948, 6056, 6610, 5994, 5956, 261, 5960} \[ -\frac {1}{2} \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-\tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {11}{32 \left (1-a^2 x^2\right )}+\frac {1}{32 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {11 a x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac {1}{3} \tanh ^{-1}(a x)^3-\frac {11}{32} \tanh ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(32*(1 - a^2*x^2)^2) + 11/(32*(1 - a^2*x^2)) - (a*x*ArcTanh[a*x])/(8*(1 - a^2*x^2)^2) - (11*a*x*ArcTanh[a*x]
)/(16*(1 - a^2*x^2)) - (11*ArcTanh[a*x]^2)/32 + ArcTanh[a*x]^2/(4*(1 - a^2*x^2)^2) + ArcTanh[a*x]^2/(2*(1 - a^
2*x^2)) + ArcTanh[a*x]^3/3 + ArcTanh[a*x]^2*Log[2 - 2/(1 + a*x)] - ArcTanh[a*x]*PolyLog[2, -1 + 2/(1 + a*x)] -
PolyLog[3, -1 + 2/(1 + a*x)]/2

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 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 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

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 \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{2} a \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+a^2 \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 \left (1-a^2 x^2\right )} \, dx\\ &=\frac {1}{32 \left (1-a^2 x^2\right )^2}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \tanh ^{-1}(a x)^3-\frac {1}{8} (3 a) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-a \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx\\ &=\frac {1}{32 \left (1-a^2 x^2\right )^2}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {11 a x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^2+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-(2 a) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{16} \left (3 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx+\frac {1}{2} a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {1}{32 \left (1-a^2 x^2\right )^2}+\frac {11}{32 \left (1-a^2 x^2\right )}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {11 a x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^2+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+a \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {1}{32 \left (1-a^2 x^2\right )^2}+\frac {11}{32 \left (1-a^2 x^2\right )}-\frac {a x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )^2}-\frac {11 a x \tanh ^{-1}(a x)}{16 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^2+\frac {\tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^2}{2 \left (1-a^2 x^2\right )}+\frac {1}{3} \tanh ^{-1}(a x)^3+\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{2} \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.41, size = 129, normalized size = 0.66 \[ \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {1}{768} \left (-384 \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )-256 \tanh ^{-1}(a x)^3-12 \tanh ^{-1}(a x) \left (24 \sinh \left (2 \tanh ^{-1}(a x)\right )+\sinh \left (4 \tanh ^{-1}(a x)\right )\right )+144 \cosh \left (2 \tanh ^{-1}(a x)\right )+3 \cosh \left (4 \tanh ^{-1}(a x)\right )+24 \tanh ^{-1}(a x)^2 \left (32 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+12 \cosh \left (2 \tanh ^{-1}(a x)\right )+\cosh \left (4 \tanh ^{-1}(a x)\right )\right )+32 i \pi ^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ArcTanh[a*x]*PolyLog[2, E^(2*ArcTanh[a*x])] + ((32*I)*Pi^3 - 256*ArcTanh[a*x]^3 + 144*Cosh[2*ArcTanh[a*x]] + 3
*Cosh[4*ArcTanh[a*x]] + 24*ArcTanh[a*x]^2*(12*Cosh[2*ArcTanh[a*x]] + Cosh[4*ArcTanh[a*x]] + 32*Log[1 - E^(2*Ar
cTanh[a*x])]) - 384*PolyLog[3, E^(2*ArcTanh[a*x])] - 12*ArcTanh[a*x]*(24*Sinh[2*ArcTanh[a*x]] + Sinh[4*ArcTanh
[a*x]]))/768

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{2}}{a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/x/(-a^2*x^2+1)^3,x, algorithm="fricas")

[Out]

integral(-arctanh(a*x)^2/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/x/(-a^2*x^2+1)^3,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.84, size = 1392, normalized size = 7.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(a*x+1)/(a*x-1)-2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-2*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/3*arc
tanh(a*x)^3-11/32*arctanh(a*x)^2+1/512*(a*x+1)^2/(a*x-1)^2-1/2*arctanh(a*x)^2*ln(a*x-1)-1/2*arctanh(a*x)^2*ln(
a*x+1)+arctanh(a*x)^2*ln((a*x+1)/(-a^2*x^2+1)^(1/2))+1/512*(a*x-1)^2/(a*x+1)^2+arctanh(a*x)^2*ln(2)-3/32*(a*x-
1)/(a*x+1)+1/16*arctanh(a*x)^2/(a*x-1)^2-5/16*arctanh(a*x)^2/(a*x-1)+1/16*arctanh(a*x)^2/(a*x+1)^2+5/16*arctan
h(a*x)^2/(a*x+1)+arctanh(a*x)^2*ln(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))+arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+1/2*I*Pi*arctanh(a*x)^2-3/16*arctanh(a*x
)*(a*x-1)/(a*x+1)+1/128*arctanh(a*x)*(a*x-1)^2/(a*x+1)^2-1/128*arctanh(a*x)*(a*x+1)^2/(a*x-1)^2+3/16*(a*x+1)*a
rctanh(a*x)/(a*x-1)+2*arctanh(a*x)*polylog(2,-(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))+1/2*I*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)))-1/4*I*Pi*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))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/(1+(a*x+1)^2/(-a^2*x^2+1)))+1/2*I*Pi*cs
gn(I/(1+(a*x+1)^2/(-a^2*x^2+1)))^3*arctanh(a*x)^2+1/4*I*Pi*arctanh(a*x)^2*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+1/4*
I*Pi*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)))^3-1/2*I*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*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+1/4*I
*Pi*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))+1/4*I*Pi*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+1/2*I*Pi*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))^2-1/4*I*Pi*arctanh(a*x)^2*csgn(I*(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)))^2-1/2*I*Pi*csgn(I/(1+(a*x+1)^2/(-a
^2*x^2+1)))^2*arctanh(a*x)^2+1/2*I*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

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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 )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} + \frac {1}{2} \, a^{5} \int \frac {x^{5} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} - \frac {1}{256} \, {\left (a {\left (\frac {2 \, {\left (5 \, a^{2} x^{2} + 3 \, a x - 6\right )}}{a^{8} x^{3} - a^{7} x^{2} - a^{6} x + a^{5}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} + \frac {16 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}}\right )} a^{4} - a^{4} \int \frac {x^{4} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} - a^{3} \int \frac {x^{3} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} + \frac {1}{2} \, a^{3} \int \frac {x^{3} \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} - \frac {3}{512} \, {\left (a {\left (\frac {2 \, {\left (3 \, a^{2} x^{2} - 3 \, a x - 2\right )}}{a^{6} x^{3} - a^{5} x^{2} - a^{4} x + a^{3}} - \frac {3 \, \log \left (a x + 1\right )}{a^{3}} + \frac {3 \, \log \left (a x - 1\right )}{a^{3}}\right )} - \frac {16 \, \log \left (-a x + 1\right )}{a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}}\right )} a^{2} + \frac {1}{2} \, a^{2} \int \frac {x^{2} \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} + \frac {1}{2} \, a \int \frac {x \log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} - \frac {3}{4} \, a \int \frac {x \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} - \frac {2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )^{3} + 3 \, {\left (2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - 3\right )} \log \left (-a x + 1\right )^{2}}{48 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )}} - \frac {1}{2} \, \int \frac {\log \left (a x + 1\right )^{2}}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} + \int \frac {\log \left (a x + 1\right ) \log \left (-a x + 1\right )}{2 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/x/(-a^2*x^2+1)^3,x, algorithm="maxima")

[Out]

1/2*a^6*integrate(1/2*x^6*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) + 1/2*a^5*integ
rate(1/2*x^5*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 1/256*(a*(2*(5*a^2*x^2 + 3
*a*x - 6)/(a^8*x^3 - a^7*x^2 - a^6*x + a^5) - 5*log(a*x + 1)/a^5 + 5*log(a*x - 1)/a^5) + 16*(2*a^2*x^2 - 1)*lo
g(-a*x + 1)/(a^8*x^4 - 2*a^6*x^2 + a^4))*a^4 - a^4*integrate(1/2*x^4*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a
^4*x^5 + 3*a^2*x^3 - x), x) - a^3*integrate(1/2*x^3*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^
3 - x), x) + 1/2*a^3*integrate(1/2*x^3*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 3/512*(a*(2*(
3*a^2*x^2 - 3*a*x - 2)/(a^6*x^3 - a^5*x^2 - a^4*x + a^3) - 3*log(a*x + 1)/a^3 + 3*log(a*x - 1)/a^3) - 16*log(-
a*x + 1)/(a^6*x^4 - 2*a^4*x^2 + a^2))*a^2 + 1/2*a^2*integrate(1/2*x^2*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*
a^4*x^5 + 3*a^2*x^3 - x), x) + 1/2*a*integrate(1/2*x*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x
^3 - x), x) - 3/4*a*integrate(1/2*x*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) - 1/48*(2*(a^4*x^4
- 2*a^2*x^2 + 1)*log(-a*x + 1)^3 + 3*(2*a^2*x^2 + 2*(a^4*x^4 - 2*a^2*x^2 + 1)*log(a*x + 1) - 3)*log(-a*x + 1)
^2)/(a^4*x^4 - 2*a^2*x^2 + 1) - 1/2*integrate(1/2*log(a*x + 1)^2/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x) + i
ntegrate(1/2*log(a*x + 1)*log(-a*x + 1)/(a^6*x^7 - 3*a^4*x^5 + 3*a^2*x^3 - x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{a^{6} x^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**2/x/(-a**2*x**2+1)**3,x)

[Out]

-Integral(atanh(a*x)**2/(a**6*x**7 - 3*a**4*x**5 + 3*a**2*x**3 - x), x)

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