\(\int \frac {(1-a^2 x^2)^{3/2} \text {arctanh}(a x)}{x^2} \, dx\) [453]

Optimal result

Integrand size = 22, antiderivative size = 179 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^2} \, dx=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+3 a \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {3}{2} i a \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:

-1/2*a*(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)*arctanh(a*x)/x-1/2*a^2*x*(-a^ 
2*x^2+1)^(1/2)*arctanh(a*x)+3*a*arctan((-a*x+1)^(1/2)/(a*x+1)^(1/2))*arcta 
nh(a*x)-a*arctanh((-a^2*x^2+1)^(1/2))+3/2*I*a*polylog(2,-I*(-a*x+1)^(1/2)/ 
(a*x+1)^(1/2))-3/2*I*a*polylog(2,I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^2} \, dx=\frac {1}{2} \left (-a \sqrt {1-a^2 x^2}-\frac {2 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}-a^2 x \sqrt {1-a^2 x^2} \text {arctanh}(a x)+3 i a \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )-3 i a \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-2 a \log \left (\cosh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+2 a \log \left (\sinh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+3 i a \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-3 i a \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

(-(a*Sqrt[1 - a^2*x^2]) - (2*Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x - a^2*x*Sqr 
t[1 - a^2*x^2]*ArcTanh[a*x] + (3*I)*a*ArcTanh[a*x]*Log[1 - I/E^ArcTanh[a*x 
]] - (3*I)*a*ArcTanh[a*x]*Log[1 + I/E^ArcTanh[a*x]] - 2*a*Log[Cosh[ArcTanh 
[a*x]/2]] + 2*a*Log[Sinh[ArcTanh[a*x]/2]] + (3*I)*a*PolyLog[2, (-I)/E^ArcT 
anh[a*x]] - (3*I)*a*PolyLog[2, I/E^ArcTanh[a*x]])/2
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.63, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6576, 6504, 6512, 6576, 6512, 6570, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6504

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/x) - a*ArcTanh[Sqrt[1 - a^2*x^2]] - a^2 
*((-2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - (I*PolyLog[2, 
((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*Sqrt[1 - a*x])/S 
qrt[1 + a*x]])/a) - a^2*(Sqrt[1 - a^2*x^2]/(2*a) + (x*Sqrt[1 - a^2*x^2]*Ar 
cTanh[a*x])/2 + ((-2*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]]*ArcTanh[a*x])/a - 
 (I*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a + (I*PolyLog[2, (I*S 
qrt[1 - a*x])/Sqrt[1 + a*x]])/a)/2)
 

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6504

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

rule 6512

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol 
] :> Simp[-2*(a + b*ArcTanh[c*x])*(ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]]/(c*S 
qrt[d])), x] + (-Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/( 
c*Sqrt[d])), x] + Simp[I*b*(PolyLog[2, I*(Sqrt[1 - c*x]/Sqrt[1 + c*x])]/(c* 
Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 
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 6576

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\left (a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+a x +2 \,\operatorname {arctanh}\left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 x}-a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\frac {3 i a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 i a \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 i a \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 i a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\)	\(203\)

Input:

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

Output:

-1/2*(a^2*x^2*arctanh(a*x)+a*x+2*arctanh(a*x))*(-a^2*x^2+1)^(1/2)/x-a*ln(1 
+(a*x+1)/(-a^2*x^2+1)^(1/2))+a*ln((a*x+1)/(-a^2*x^2+1)^(1/2)-1)+3/2*I*a*ar 
ctanh(a*x)*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))+3/2*I*a*dilog(1+I*(a*x+1)/(- 
a^2*x^2+1)^(1/2))-3/2*I*a*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))-3/2*I*a*ar 
ctanh(a*x)*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-a^2 x^2\right )^{3/2} \text {arctanh}(a x)}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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