\(\int \frac {\text {arctanh}(a x)}{x^3 (1-a^2 x^2)^{3/2}} \, dx\) [394]

Optimal result

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

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

Mathematica [A] (warning: unable to verify)

Time = 1.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02 \[ \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {1}{8} a^2 \left (-\frac {8 a x}{\sqrt {1-a^2 x^2}}+\frac {8 \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {a x \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )}{\sqrt {1-a^2 x^2}}-\text {arctanh}(a x) \text {csch}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+12 \text {arctanh}(a x) \log \left (1-e^{-\text {arctanh}(a x)}\right )-12 \text {arctanh}(a x) \log \left (1+e^{-\text {arctanh}(a x)}\right )+12 \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-12 \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x) \text {sech}^2\left (\frac {1}{2} \text {arctanh}(a x)\right )+2 \tanh \left (\frac {1}{2} \text {arctanh}(a x)\right )\right ) \] Input:

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

Output:

(a^2*((-8*a*x)/Sqrt[1 - a^2*x^2] + (8*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] - (a 
*x*Csch[ArcTanh[a*x]/2]^2)/Sqrt[1 - a^2*x^2] - ArcTanh[a*x]*Csch[ArcTanh[a 
*x]/2]^2 + 12*ArcTanh[a*x]*Log[1 - E^(-ArcTanh[a*x])] - 12*ArcTanh[a*x]*Lo 
g[1 + E^(-ArcTanh[a*x])] + 12*PolyLog[2, -E^(-ArcTanh[a*x])] - 12*PolyLog[ 
2, E^(-ArcTanh[a*x])] - ArcTanh[a*x]*Sech[ArcTanh[a*x]/2]^2 + 2*Tanh[ArcTa 
nh[a*x]/2]))/8
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.43, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6592, 6588, 242, 6580, 6592, 6556, 208, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6588

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 6580

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 6580

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

Input:

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

Output:

-1/2*(a*Sqrt[1 - a^2*x^2])/x - (Sqrt[1 - a^2*x^2]*ArcTanh[a*x])/(2*x^2) + 
(a^2*(-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]]))/ 
2 + a^2*(a^2*(-(x/(a*Sqrt[1 - a^2*x^2])) + ArcTanh[a*x]/(a^2*Sqrt[1 - a^2* 
x^2])) - 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 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 6556

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q 
_.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(2*e*(q 
+ 1))), x] + Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTanh[c* 
x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[c^2*d + e, 0] && 
 GtQ[p, 0] && NeQ[q, -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 6588

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
 + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*A 
rcTanh[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1)))   Int[(f*x)^ 
(m + 1)*((a + b*ArcTanh[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] + Simp[c^2*( 
(m + 2)/(f^2*(m + 1)))   Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/Sqrt[d + 
 e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && G 
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
 

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. \(346\) vs. \(2(151)=302\).

Time = 1.41 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{4} x^{4}-3 \,\operatorname {arctanh}\left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{4} x^{4}+3 \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{4} x^{4}-3 \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) a^{4} x^{4}-\sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \,\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{2}+3 \,\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{2}+3 \,\operatorname {arctanh}\left (a x \right ) a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-3 a^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{2}+3 a^{2} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) x^{2}-a x \sqrt {-a^{2} x^{2}+1}-\sqrt {-a^{2} x^{2}+1}\, \operatorname {arctanh}\left (a x \right )}{2 x^{2} \left (a x +1\right ) \left (a x -1\right )}\)	\(347\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-a^2*x^2 + 1)*arctanh(a*x)/(a^4*x^7 - 2*a^2*x^5 + x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {\mathit {atanh} \left (a x \right )}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{5}-\sqrt {-a^{2} x^{2}+1}\, x^{3}}d x \right ) \] Input:

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

Output:

 - int(atanh(a*x)/(sqrt( - a**2*x**2 + 1)*a**2*x**5 - sqrt( - a**2*x**2 + 
1)*x**3),x)