3.3.77 \(\int \frac {\tanh ^{-1}(a x)^3}{x (1-a^2 x^2)^2} \, dx\) [277]

Optimal. Leaf size=193 \[ -\frac {3 a x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} \tanh ^{-1}(a x)+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \]

[Out]

-3/8*a*x/(-a^2*x^2+1)-3/8*arctanh(a*x)+3/4*arctanh(a*x)/(-a^2*x^2+1)-3/4*a*x*arctanh(a*x)^2/(-a^2*x^2+1)-1/4*a
rctanh(a*x)^3+1/2*arctanh(a*x)^3/(-a^2*x^2+1)+1/4*arctanh(a*x)^4+arctanh(a*x)^3*ln(2-2/(a*x+1))-3/2*arctanh(a*
x)^2*polylog(2,-1+2/(a*x+1))-3/2*arctanh(a*x)*polylog(3,-1+2/(a*x+1))-3/4*polylog(4,-1+2/(a*x+1))

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6177, 6135, 6079, 6095, 6203, 6207, 6745, 6141, 6103, 205, 212} \begin {gather*} -\frac {3 a x}{8 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3}{4} \text {Li}_4\left (\frac {2}{a x+1}-1\right )-\frac {3}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2-\frac {3}{2} \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{4} \tanh ^{-1}(a x)^3-\frac {3}{8} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*x)/(8*(1 - a^2*x^2)) - (3*ArcTanh[a*x])/8 + (3*ArcTanh[a*x])/(4*(1 - a^2*x^2)) - (3*a*x*ArcTanh[a*x]^2)/
(4*(1 - a^2*x^2)) - ArcTanh[a*x]^3/4 + ArcTanh[a*x]^3/(2*(1 - a^2*x^2)) + ArcTanh[a*x]^4/4 + ArcTanh[a*x]^3*Lo
g[2 - 2/(1 + a*x)] - (3*ArcTanh[a*x]^2*PolyLog[2, -1 + 2/(1 + a*x)])/2 - (3*ArcTanh[a*x]*PolyLog[3, -1 + 2/(1
+ a*x)])/2 - (3*PolyLog[4, -1 + 2/(1 + a*x)])/4

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6079

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 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 6103

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 6135

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 6141

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 6177

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 6203

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan
h[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 6207

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(a
+ b*ArcTanh[c*x])^p)*(PolyLog[k + 1, u]/(2*c*d)), x] + Dist[b*(p/2), Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog
[k + 1, u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[c^2*d + e, 0] && EqQ[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 {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx\\ &=\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{2} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\\ &=-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-(3 a) \int \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{4} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+(3 a) \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a x}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {1}{8} (3 a) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{2} (3 a) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} \tanh ^{-1}(a x)+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {3}{4} \text {Li}_4\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 135, normalized size = 0.70 \begin {gather*} \frac {1}{64} \left (\pi ^4-16 \tanh ^{-1}(a x)^4+24 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+16 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+96 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-96 \tanh ^{-1}(a x) \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+48 \text {PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )-12 \sinh \left (2 \tanh ^{-1}(a x)\right )-24 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Pi^4 - 16*ArcTanh[a*x]^4 + 24*ArcTanh[a*x]*Cosh[2*ArcTanh[a*x]] + 16*ArcTanh[a*x]^3*Cosh[2*ArcTanh[a*x]] + 64
*ArcTanh[a*x]^3*Log[1 - E^(2*ArcTanh[a*x])] + 96*ArcTanh[a*x]^2*PolyLog[2, E^(2*ArcTanh[a*x])] - 96*ArcTanh[a*
x]*PolyLog[3, E^(2*ArcTanh[a*x])] + 48*PolyLog[4, E^(2*ArcTanh[a*x])] - 12*Sinh[2*ArcTanh[a*x]] - 24*ArcTanh[a
*x]^2*Sinh[2*ArcTanh[a*x]])/64

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 276.65, size = 1387, normalized size = 7.19

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1387\)
default	\(\text {Expression too large to display}\)	\(1387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*arctanh(a*x)*(a*x-1)/(a*x+1)-3/16*arctanh(a*x)*(a*x+1)/(a*x-1)-3/16*arctanh(a*x)^2*(a*x-1)/(a*x+1)+1/2*I
*Pi*arctanh(a*x)^3*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))+1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))^3+1/4*I*Pi*arct
anh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3+1/2*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^
2/(-a^2*x^2+1)+1))^3-1/2*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))^2+1/2*I*Pi*arctanh(a*x)^3*csgn
(I*((a*x+1)^2/(-a^2*x^2+1)-1)/((a*x+1)^2/(-a^2*x^2+1)+1))^3+6*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*polylog
(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/4*I*Pi*arctanh(a*x)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^
2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)^2/(-a^2*x^2+1)+1))+3/16*(a*x+1)*arctanh(a*x)^2/(a*x-1)+1/2*I*P
i*arctanh(a*x)^3+3*arctanh(a*x)^2*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*arctanh(a*x)^2*polylog(2,(a*x+1)/(-
a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*arctanh(a*x)*polylog(3,(a*x+1)/(-a^2
*x^2+1)^(1/2))-1/4*arctanh(a*x)^4-1/4*arctanh(a*x)^3+arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a
*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)/(-a^2*x^2+1)^(1/2))^2*csgn(I*(a*
x+1)^2/(a^2*x^2-1))-1/2*I*Pi*arctanh(a*x)^3*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+1/2*I*Pi*arctanh(a*x)^3*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)^3*csgn(I/((a*x+1)^2/(-a^2*x^2+1)+1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x+1)
^2/(-a^2*x^2+1)+1))^2-1/4*I*Pi*arctanh(a*x)^3*csgn(I*(a*x+1)^2/(a^2*x^2-1))*csgn(I*(a*x+1)^2/(a^2*x^2-1)/((a*x
+1)^2/(-a^2*x^2+1)+1))^2-1/2*I*Pi*arctanh(a*x)^3*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+1/4*arctanh(a*x)^3/(a*x+1)-1/2*arctanh(a*x)^3*ln(a*x+1)-1/4*arctanh(a*x)
^3/(a*x-1)-1/2*arctanh(a*x)^3*ln(a*x-1)+arctanh(a*x)^3*ln((a*x+1)/(-a^2*x^2+1)^(1/2))-3/32*(a*x-1)/(a*x+1)+3/3
2*(a*x+1)/(a*x-1)+arctanh(a*x)^3*ln(a*x)-arctanh(a*x)^3*ln((a*x+1)^2/(-a^2*x^2+1)-1)+ln(2)*arctanh(a*x)^3

________________________________________________________________________________________

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

[Out]

1/64*((a^2*x^2 - 1)*log(-a*x + 1)^4 + 4*((a^2*x^2 - 1)*log(a*x + 1) + 1)*log(-a*x + 1)^3)/(a^2*x^2 - 1) - 1/8*
integrate(-1/2*(2*log(a*x + 1)^3 - 6*log(a*x + 1)^2*log(-a*x + 1) - 3*(a^2*x^2 + a*x + (a^4*x^4 + a^3*x^3 - a^
2*x^2 - a*x - 2)*log(a*x + 1))*log(-a*x + 1)^2)/(a^4*x^5 - 2*a^2*x^3 + x), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________