\(\int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [378]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}-4 a \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 a \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.85 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {\text {arctanh}(a x) \left (\sqrt {1-a^2 x^2} \text {arctanh}(a x)+2 a x \left (-\log \left (1-e^{-\text {arctanh}(a x)}\right )+\log \left (1+e^{-\text {arctanh}(a x)}\right )\right )\right )}{x}+2 a \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-2 a \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right ) \] Input:

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

Output:

-((ArcTanh[a*x]*(Sqrt[1 - a^2*x^2]*ArcTanh[a*x] + 2*a*x*(-Log[1 - E^(-ArcT 
anh[a*x])] + Log[1 + E^(-ArcTanh[a*x])])))/x) + 2*a*PolyLog[2, -E^(-ArcTan 
h[a*x])] - 2*a*PolyLog[2, E^(-ArcTanh[a*x])]
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6570, 6580}

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.

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 6580

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

Input:

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

Output:

-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/x) + 2*a*(-2*ArcTanh[a*x]*ArcTanh[Sqr 
t[1 - a*x]/Sqrt[1 + a*x]] + PolyLog[2, -(Sqrt[1 - a*x]/Sqrt[1 + a*x])] - P 
olyLog[2, Sqrt[1 - a*x]/Sqrt[1 + a*x]])
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}}{x}+2 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(129\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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