\(\int \frac {x^2 \text {arctanh}(a x)^2}{(1-a^2 x^2)^{3/2}} \, dx\) [397]

Optimal result

Integrand size = 24, antiderivative size = 171 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {2 x}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \text {arctanh}(a x)}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \text {arctanh}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \arctan \left (e^{\text {arctanh}(a x)}\right ) \text {arctanh}(a x)^2}{a^3}+\frac {2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arctanh}(a x)}\right )}{a^3}-\frac {2 i \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arctanh}(a x)}\right )}{a^3}-\frac {2 i \operatorname {PolyLog}\left (3,-i e^{\text {arctanh}(a x)}\right )}{a^3}+\frac {2 i \operatorname {PolyLog}\left (3,i e^{\text {arctanh}(a x)}\right )}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {i \left (-\frac {2 i a x}{\sqrt {1-a^2 x^2}}+\frac {2 i \text {arctanh}(a x)}{\sqrt {1-a^2 x^2}}-\frac {i a x \text {arctanh}(a x)^2}{\sqrt {1-a^2 x^2}}+\text {arctanh}(a x)^2 \log \left (1-i e^{-\text {arctanh}(a x)}\right )-\text {arctanh}(a x)^2 \log \left (1+i e^{-\text {arctanh}(a x)}\right )+2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,-i e^{-\text {arctanh}(a x)}\right )-2 \text {arctanh}(a x) \operatorname {PolyLog}\left (2,i e^{-\text {arctanh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arctanh}(a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arctanh}(a x)}\right )\right )}{a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6590, 6514, 3042, 4668, 3011, 2720, 6524, 208, 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.

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

\(\Big \downarrow \) 6590

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

\(\Big \downarrow \) 6514

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

\(\displaystyle \frac {\int \frac {\text {arctanh}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}-\frac {-2 i \int \text {arctanh}(a x) \log \left (1-i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 i \int \text {arctanh}(a x) \log \left (1+i e^{\text {arctanh}(a x)}\right )d\text {arctanh}(a x)+2 \text {arctanh}(a x)^2 \arctan \left (e^{\text {arctanh}(a x)}\right )}{a^3}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6524

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

\(\Big \downarrow \) 208

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 6524

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[(-b)*p*((a + b*ArcTanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2]) 
), x] + (Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x] + Simp[b^2 
*p*(p - 1)   Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; 
 FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 1]
 

rule 6590

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*A 
rcTanh[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcT 
anh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && In 
tegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && 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 [F]

\[\int \frac {x^{2} \operatorname {arctanh}\left (a x \right )^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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