\(\int \frac {\text {arctanh}(a x)^2}{x (1-a^2 x^2)^{3/2}} \, dx\) [400]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2}{\sqrt {1-a^2 x^2}}-\frac {2 a x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}+\frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}-2 \text {arctanh}\left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{\text {arctanh}(a x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.25 \[ \int \frac {\text {arctanh}(a x)^2}{x \left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2}{\sqrt {1-a^2 x^2}}-\frac {2 a x \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}+\frac {\text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+\text {arctanh}(a x)^2 \log \left (1-e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \log \left (1+e^{-\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,e^{-\text {arctanh}(a x)}\right ) \] Input:

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

Output:

2/Sqrt[1 - a^2*x^2] - (2*a*x*ArcTanh[a*x])/Sqrt[1 - a^2*x^2] + ArcTanh[a*x 
]^2/Sqrt[1 - a^2*x^2] + ArcTanh[a*x]^2*Log[1 - E^(-ArcTanh[a*x])] - ArcTan 
h[a*x]^2*Log[1 + E^(-ArcTanh[a*x])] + 2*ArcTanh[a*x]*PolyLog[2, -E^(-ArcTa 
nh[a*x])] - 2*ArcTanh[a*x]*PolyLog[2, E^(-ArcTanh[a*x])] + 2*PolyLog[3, -E 
^(-ArcTanh[a*x])] - 2*PolyLog[3, E^(-ArcTanh[a*x])]
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.18 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6592, 6556, 6520, 6582, 3042, 26, 4670, 3011, 2720, 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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 6520

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

\(\Big \downarrow \) 6582

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 4670

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

rule 6520

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb 
ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* 
Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
 

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 6582

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

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 [A] (verified)

Time = 0.51 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\left (\operatorname {arctanh}\left (a x \right )^{2}-2 \,\operatorname {arctanh}\left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (\operatorname {arctanh}\left (a x \right )^{2}+2 \,\operatorname {arctanh}\left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}+\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\operatorname {arctanh}\left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(232\)

Input:

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

Output:

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

Fricas [F]

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

integrate(arctanh(a*x)^2/x/(-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^5 - 2*a^2*x^3 + x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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