3.4.86 \(\int \frac {\text {arctanh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx\) [386]

3.4.86.1 Optimal result

Integrand size = 24, antiderivative size = 267 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=-\frac {3 a \sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{2 x}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}-a^2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3-6 a^2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+\frac {3}{2} a^2 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-3 a^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+3 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-3 a^2 \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right ) \]

output

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

3.4.86.2 Mathematica [A] (verified)

Time = 6.62 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.13 \[ \int \frac {\text {arctanh}(a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx=\frac {1}{16} a \left (a \pi ^4-2 a \text {arctanh}(a x)^4-12 a \text {arctanh}(a x)^2 \coth \left (\frac {1}{2} \text {arctanh}(a x)\right )-2 a \text {arctanh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+48 a \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-48 a \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )-8 a \text {arctanh}(a x)^3 \log \left (1+e^{-\text {arctanh}(a x)}\right )+8 a \text {arctanh}(a x)^3 \log \left (1-e^{\text {arctanh}(a x)}\right )+24 a \left (2+\text {arctanh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-48 a \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+24 a \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+48 a \text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-48 a \text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )+48 a \operatorname {PolyLog}\left (4,-e^{-\text {arctanh}(a x)}\right )+48 a \operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )+12 a \text {arctanh}(a x)^2 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )-\frac {4 \sqrt {1-a^2 x^2} \text {arctanh}(a x)^3 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )}{x}\right ) \]

input

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

output

(a*(a*Pi^4 - 2*a*ArcTanh[a*x]^4 - 12*a*ArcTanh[a*x]^2*Coth[ArcTanh[a*x]/2] 
 - 2*a*ArcTanh[a*x]^3*Csch[ArcTanh[a*x]/2]^2 + 48*a*ArcTanh[a*x]*Log[1 - E 
^(-ArcTanh[a*x])] - 48*a*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] - 8*a*Arc 
Tanh[a*x]^3*Log[1 + E^(-ArcTanh[a*x])] + 8*a*ArcTanh[a*x]^3*Log[1 - E^ArcT 
anh[a*x]] + 24*a*(2 + ArcTanh[a*x]^2)*PolyLog[2, -E^(-ArcTanh[a*x])] - 48* 
a*PolyLog[2, E^(-ArcTanh[a*x])] + 24*a*ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh 
[a*x]] + 48*a*ArcTanh[a*x]*PolyLog[3, -E^(-ArcTanh[a*x])] - 48*a*ArcTanh[a 
*x]*PolyLog[3, E^ArcTanh[a*x]] + 48*a*PolyLog[4, -E^(-ArcTanh[a*x])] + 48* 
a*PolyLog[4, E^ArcTanh[a*x]] + 12*a*ArcTanh[a*x]^2*Tanh[ArcTanh[a*x]/2] - 
(4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3*Tanh[ArcTanh[a*x]/2])/x))/16
 

3.4.86.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.57 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6588, 6570, 6580, 6582, 3042, 26, 4670, 3011, 7163, 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.

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

\(\Big \downarrow \) 6588

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 6580

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

\(\Big \downarrow \) 6582

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

\(\displaystyle \frac {3}{2} a \left (2 a \left (-2 \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}\right )+\frac {1}{2} i a^2 \left (-3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )\right )+3 i \left (2 \left (\text {arctanh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arctanh}(a x)}\right )\right )-\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )\right )+2 i \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^3\right )-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^3}{2 x^2}\)

input

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

output

-1/2*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^3)/x^2 + (3*a*(-((Sqrt[1 - a^2*x^2]*A 
rcTanh[a*x]^2)/x) + 2*a*(-2*ArcTanh[a*x]*ArcTanh[Sqrt[1 - a*x]/Sqrt[1 + a* 
x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - PolyLog[2, Sqrt[1 - a*x 
]/Sqrt[1 + a*x]])))/2 + (I/2)*a^2*((2*I)*ArcTanh[E^ArcTanh[a*x]]*ArcTanh[a 
*x]^3 - (3*I)*(-(ArcTanh[a*x]^2*PolyLog[2, -E^ArcTanh[a*x]]) + 2*(ArcTanh[ 
a*x]*PolyLog[3, -E^ArcTanh[a*x]] - PolyLog[4, -E^ArcTanh[a*x]])) + (3*I)*( 
-(ArcTanh[a*x]^2*PolyLog[2, E^ArcTanh[a*x]]) + 2*(ArcTanh[a*x]*PolyLog[3, 
E^ArcTanh[a*x]] - PolyLog[4, E^ArcTanh[a*x]])))
 

3.4.86.3.1 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 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 6580

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

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 6588

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*A 
rcTanh[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1)))   Int[(f*x)^ 
(m + 1)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[c^2*( 
(m + 2)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + 
 e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && G 
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
 

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]
 

rule 7163

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

3.4.86.4 Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \operatorname {arctanh}\left (a x \right )^{2} \left (3 a x +\operatorname {arctanh}\left (a x \right )\right )}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a^{2} \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-3 a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {a^{2} \operatorname {arctanh}\left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{2} \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+3 a^{2} \operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 a^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(386\)

input

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

output

-1/2*(-(a*x-1)*(a*x+1))^(1/2)*arctanh(a*x)^2*(3*a*x+arctanh(a*x))/x^2+1/2* 
a^2*arctanh(a*x)^3*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3/2*a^2*arctanh(a*x)^2 
*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*arctanh(a*x)*polylog(3,(a*x+1 
)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(4,(a*x+1)/(-a^2*x^2+1)^(1/2))-1/2*a^2* 
arctanh(a*x)^3*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-3/2*a^2*arctanh(a*x)^2*pol 
ylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arctanh(a*x)*polylog(3,-(a*x+1)/ 
(-a^2*x^2+1)^(1/2))-3*a^2*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*arc 
tanh(a*x)*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2))+3*a^2*polylog(2,(a*x+1)/(-a^2*x 
^2+1)^(1/2))-3*a^2*arctanh(a*x)*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))-3*a^2*pol 
ylog(2,-(a*x+1)/(-a^2*x^2+1)^(1/2))
 

3.4.86.5 Fricas [F]

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

input

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

output

integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)^3/(a^2*x^5 - x^3), x)
 

3.4.86.6 Sympy [F]

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

input

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

output

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

3.4.86.7 Maxima [F]

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

input

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

output

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

3.4.86.8 Giac [F]

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

input

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

output

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

3.4.86.9 Mupad [F(-1)]

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

input

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

output

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