\(\int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2} \, dx\) [432]

Optimal result

Integrand size = 22, antiderivative size = 130 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x}+2 a \arctan \left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \text {arctanh}(a x)-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+i a \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-i a \operatorname {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{x^2} \, dx=a \left (-\frac {\sqrt {1-a^2 x^2} \text {arctanh}(a x)}{a x}+i \text {arctanh}(a x) \log \left (1-i e^{-\text {arctanh}(a x)}\right )-i \text {arctanh}(a x) \log \left (1+i e^{-\text {arctanh}(a x)}\right )-\log \left (\cosh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6512

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

\(\Big \downarrow \) 6570

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(Sqrt[1 - a^2*x^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)
 

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 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 = 0.92 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.43

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )}{x}+i a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )+i a \operatorname {dilog}\left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-i a \operatorname {dilog}\left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-i a \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )\)	\(186\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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