\(\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx\) [444]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x}-2 a \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-4 a \text {arctanh}(a x) \text {arctanh}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 i a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )-2 i a \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )+2 a \operatorname {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \operatorname {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i a \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )+2 i a \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{x^2} \, dx=a \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)^2}{a x}+2 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )+i \text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-i \text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )+2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+2 i \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-2 i \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

a*(-((Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/(a*x)) + 2*ArcTanh[a*x]*Log[1 - E^ 
(-ArcTanh[a*x])] + I*ArcTanh[a*x]^2*Log[1 - I/E^ArcTanh[a*x]] - I*ArcTanh[ 
a*x]^2*Log[1 + I/E^ArcTanh[a*x]] - 2*ArcTanh[a*x]*Log[1 + E^(-ArcTanh[a*x] 
)] + 2*PolyLog[2, -E^(-ArcTanh[a*x])] + (2*I)*ArcTanh[a*x]*PolyLog[2, (-I) 
/E^ArcTanh[a*x]] - (2*I)*ArcTanh[a*x]*PolyLog[2, I/E^ArcTanh[a*x]] - 2*Pol 
yLog[2, E^(-ArcTanh[a*x])] + (2*I)*PolyLog[3, (-I)/E^ArcTanh[a*x]] - (2*I) 
*PolyLog[3, I/E^ArcTanh[a*x]])
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6576, 6514, 3042, 4668, 3011, 2720, 6570, 6580, 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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6514

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 6580

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

\(\Big \downarrow \) 7143

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

Input:

Int[(Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^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]]) - a*(2*ArcTan[E^ArcTanh[a*x]]*ArcT 
anh[a*x]^2 + (2*I)*(-(ArcTanh[a*x]*PolyLog[2, (-I)*E^ArcTanh[a*x]]) + Poly 
Log[3, (-I)*E^ArcTanh[a*x]]) - (2*I)*(-(ArcTanh[a*x]*PolyLog[2, I*E^ArcTan 
h[a*x]]) + PolyLog[3, I*E^ArcTanh[a*x]]))
 

Defintions of rubi rules used

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 4668

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

rule 6514

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

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 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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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