\(\int \frac {\text {arctanh}(a x)^2}{x^2 (1-a^2 x^2)^{3/2}} \, dx\) [401]

Optimal result

Integrand size = 24, antiderivative size = 171 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2 a^2 x}{\sqrt {1-a^2 x^2}}-\frac {2 a \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}-4 a \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 a \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:

2*a^2*x/(-a^2*x^2+1)^(1/2)-2*a*arctanh(a*x)/(-a^2*x^2+1)^(1/2)+a^2*x*arcta 
nh(a*x)^2/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)*arctanh(a*x)^2/x-4*a*arcta 
nh(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] (warning: unable to verify)

Time = 0.78 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.26 \[ \int \frac {\text {arctanh}(a x)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {a \left (4 a x-4 \text {arctanh}(a x)+2 a x \text {arctanh}(a x)^2-\frac {1}{2} a x \text {arctanh}(a x)^2 \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+4 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-4 \sqrt {1-a^2 x^2} \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+4 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-4 \sqrt {1-a^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )-\frac {2 \left (-1+a^2 x^2\right ) \text {arctanh}(a x)^2 \sinh ^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{a x}\right )}{2 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(a*(4*a*x - 4*ArcTanh[a*x] + 2*a*x*ArcTanh[a*x]^2 - (a*x*ArcTanh[a*x]^2*Cs 
ch[ArcTanh[a*x]/2]^2)/2 + 4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 - E^(-Arc 
Tanh[a*x])] - 4*Sqrt[1 - a^2*x^2]*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x])] 
+ 4*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^(-ArcTanh[a*x])] - 4*Sqrt[1 - a^2*x^2] 
*PolyLog[2, E^(-ArcTanh[a*x])] - (2*(-1 + a^2*x^2)*ArcTanh[a*x]^2*Sinh[Arc 
Tanh[a*x]/2]^2)/(a*x)))/(2*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6592, 6524, 208, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6524

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 6580

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(335\) vs. \(2(151)=302\).

Time = 0.51 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.96

method	result	size
default	\(-\frac {2 \operatorname {arctanh}\left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-2 \,\operatorname {arctanh}\left (a x \right ) a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x -2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+2 \,\operatorname {arctanh}\left (a x \right ) a \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x +2 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}-2 a \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x -2 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{3} x^{3}+2 a \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x +2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 x \,\operatorname {arctanh}\left (a x \right ) a \sqrt {-a^{2} x^{2}+1}-\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )^{2}}{\left (a^{2} x^{2}-1\right ) x}\)	\(336\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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