\(\int \frac {\text {arctanh}(a x)^3}{x^2 (1-a^2 x^2)^{3/2}} \, dx\) [409]

Optimal result

Integrand size = 24, antiderivative size = 187 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 a}{\sqrt {1-a^2 x^2}}+\frac {6 a^2 x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-6 a \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2+\frac {a^2 x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}-6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \] Output:

-6*a/(-a^2*x^2+1)^(1/2)+6*a^2*x*arctanh(a*x)/(-a^2*x^2+1)^(1/2)-3*a*arctan 
h(a*x)^2/(-a^2*x^2+1)^(1/2)-6*a*arctanh((a*x+1)/(-a^2*x^2+1)^(1/2))*arctan 
h(a*x)^2+a^2*x*arctanh(a*x)^3/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)*arctan 
h(a*x)^3/x-6*a*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*arc 
tanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*a*polylog(3,-(a*x+1)/(-a 
^2*x^2+1)^(1/2))-6*a*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Mathematica [A] (warning: unable to verify)

Time = 1.40 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.44 \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\frac {6 a}{\sqrt {1-a^2 x^2}}+\frac {6 a^2 x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {3 a \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {a^2 x \text {arctanh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{4 \sqrt {1-a^2 x^2}}+3 a \text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )-3 a \text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-6 a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+6 a \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-6 a \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right )+\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3 \sinh ^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{x} \] Input:

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

Output:

(-6*a)/Sqrt[1 - a^2*x^2] + (6*a^2*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (3*a 
*ArcTanh[a*x]^2)/Sqrt[1 - a^2*x^2] + (a^2*x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x 
^2] - (a^2*x*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2)/(4*Sqrt[1 - a^2*x^2]) 
+ 3*a*ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - 3*a*ArcTanh[a*x]^2*Log[1 
 + E^(-ArcTanh[a*x])] + 6*a*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTanh[a*x])] - 
6*a*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 6*a*PolyLog[3, -E^(-ArcTa 
nh[a*x])] - 6*a*PolyLog[3, E^(-ArcTanh[a*x])] + (Sqrt[1 - a^2*x^2]*ArcTanh 
[a*x]^3*Sinh[ArcTanh[a*x]/2]^2)/x
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6592, 6524, 6520, 6570, 6582, 3042, 26, 4670, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6592

\(\displaystyle a^2 \int \frac {\text {arctanh}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}}dx+\int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 6524

\(\displaystyle a^2 \left (6 \int \frac {\text {arctanh}(a x)}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}\right )+\int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx\)

\(\Big \downarrow \) 6520

\(\displaystyle \int \frac {\text {arctanh}(a x)^3}{x^2 \sqrt {1-a^2 x^2}}dx+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 6570

\(\displaystyle 3 a \int \frac {\text {arctanh}(a x)^2}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 6582

\(\displaystyle 3 a \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 a \int i \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 3 i a \int \text {arctanh}(a x)^2 \csc (i \text {arctanh}(a x))d\text {arctanh}(a x)-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 4670

\(\displaystyle 3 i a \left (2 i \int \text {arctanh}(a x) \log \left (1-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-2 i \int \text {arctanh}(a x) \log \left (1+e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle 3 i a \left (-2 i \left (\int \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle 3 i a \left (-2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\int e^{-\text {arctanh}(a x)} \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )de^{\text {arctanh}(a x)}-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{x}+a^2 \left (\frac {x \text {arctanh}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {3 \text {arctanh}(a x)^2}{a \sqrt {1-a^2 x^2}}+6 \left (\frac {x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {1}{a \sqrt {1-a^2 x^2}}\right )\right )+3 i a \left (-2 i \left (\operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+2 i \left (\operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2\right )\)

Input:

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

Output:

-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x) + a^2*((-3*ArcTanh[a*x]^2)/(a*Sqrt 
[1 - a^2*x^2]) + (x*ArcTanh[a*x]^3)/Sqrt[1 - a^2*x^2] + 6*(-(1/(a*Sqrt[1 - 
 a^2*x^2])) + (x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2])) + (3*I)*a*((2*I)*ArcTan 
h[E^ArcTanh[a*x]]*ArcTanh[a*x]^2 - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, -E^Arc 
Tanh[a*x]]) + PolyLog[3, -E^ArcTanh[a*x]]) + (2*I)*(-(ArcTanh[a*x]*PolyLog 
[2, E^ArcTanh[a*x]]) + PolyLog[3, E^ArcTanh[a*x]]))
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6520

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb 
ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* 
Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
 

rule 6524

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) 
), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 
*p*(p - 1)   Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; 
 FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
 

rule 6570

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a 
+ b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1))   Int[(f*x)^(m 
+ 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
 && NeQ[m, -1]
 

rule 6582

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[1/Sqrt[d]   Subst[Int[(a + b*x)^p*Csch[x], x], x, Arc 
Tanh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 
 0] && GtQ[d, 0]
 

rule 6592

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTanh 
[c*x])^p, x], x] - Simp[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] && Integers 
Q[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(235)=470\).

Time = 0.70 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.61

method	result	size
default	\(-\frac {3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-3 \operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+2 \operatorname {arctanh}\left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-6 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+6 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+6 \,\operatorname {arctanh}\left (a x \right ) a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-3 \operatorname {arctanh}\left (a x \right )^{2} a x \sqrt {-a^{2} x^{2}+1}-3 \operatorname {arctanh}\left (a x \right )^{2} a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x +3 \operatorname {arctanh}\left (a x \right )^{2} a \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x -\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{3}-6 \,\operatorname {arctanh}\left (a x \right ) a \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x +6 \,\operatorname {arctanh}\left (a x \right ) a \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x -6 a x \sqrt {-a^{2} x^{2}+1}+6 a \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x -6 a \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x}{\left (a^{2} x^{2}-1\right ) x}\)	\(488\)

Input:

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

Output:

-(3*arctanh(a*x)^2*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3-3*arctanh(a*x) 
^2*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3+2*arctanh(a*x)^3*(-a^2*x^2+1)^ 
(1/2)*a^2*x^2+6*arctanh(a*x)*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^ 
3-6*arctanh(a*x)*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3-6*polylog(3 
,-(a*x+1)/(-a^2*x^2+1)^(1/2))*a^3*x^3+6*polylog(3,(a*x+1)/(-a^2*x^2+1)^(1/ 
2))*a^3*x^3+6*arctanh(a*x)*a^2*x^2*(-a^2*x^2+1)^(1/2)-3*arctanh(a*x)^2*a*x 
*(-a^2*x^2+1)^(1/2)-3*arctanh(a*x)^2*a*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))*x+ 
3*arctanh(a*x)^2*a*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))*x-(-a^2*x^2+1)^(1/2)*a 
rctanh(a*x)^3-6*arctanh(a*x)*a*polylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))*x+6* 
arctanh(a*x)*a*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))*x-6*a*x*(-a^2*x^2+1)^ 
(1/2)+6*a*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2))*x-6*a*polylog(3,(a*x+1)/( 
-a^2*x^2+1)^(1/2))*x)/(a^2*x^2-1)/x
 

Fricas [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

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

Output:

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^4*x^6 - 2*a^2*x^4 + x^2), x)
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^3}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{4}-\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x \right ) \] Input:

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

Output:

 - int(atanh(a*x)**3/(sqrt( - a**2*x**2 + 1)*a**2*x**4 - sqrt( - a**2*x**2 
 + 1)*x**2),x)