\(\int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx\) [235]

Optimal result

Integrand size = 22, antiderivative size = 75 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {\text {arctanh}(a x)^2}{a^3}-\frac {x \text {arctanh}(a x)^2}{a^2}+\frac {\text {arctanh}(a x)^3}{3 a^3}+\frac {2 \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {-\frac {1}{3} \text {arctanh}(a x) \left (-3 a x \text {arctanh}(a x)+\text {arctanh}(a x) (3+\text {arctanh}(a x))+6 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )}{a^3} \] Input:

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

Output:

-((-1/3*(ArcTanh[a*x]*(-3*a*x*ArcTanh[a*x] + ArcTanh[a*x]*(3 + ArcTanh[a*x 
]) + 6*Log[1 + E^(-2*ArcTanh[a*x])])) + PolyLog[2, -E^(-2*ArcTanh[a*x])])/ 
a^3)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6542, 6436, 6510, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 6542

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6510

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6436

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a 
 + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p   Int[x^n*((a + b*ArcTanh[c*x^n]) 
^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] 
 && (EqQ[n, 1] || EqQ[p, 1])
 

rule 6470

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

rule 6510

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

rule 6542

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

rule 6546

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 10.53 (sec) , antiderivative size = 5330, normalized size of antiderivative = 71.07

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5330\)
default	\(\text {Expression too large to display}\)	\(5330\)
parts	\(\text {Expression too large to display}\)	\(5580\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (70) = 140\).

Time = 0.03 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.67 \[ \int \frac {x^2 \text {arctanh}(a x)^2}{1-a^2 x^2} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {artanh}\left (a x\right )^{2} - \frac {\frac {3 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right )^{2} - \log \left (a x + 1\right )^{3} + \log \left (a x - 1\right )^{3} - 3 \, {\left (\log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) + 6 \, \log \left (a x - 1\right )^{2}}{a} - \frac {24 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a}}{24 \, a^{2}} + \frac {{\left (2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )\right )} \operatorname {artanh}\left (a x\right )}{4 \, a^{3}} \] Input:

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

Output:

-1/2*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3)*arctanh(a*x)^2 - 1/24 
*((3*(log(a*x - 1) - 2)*log(a*x + 1)^2 - log(a*x + 1)^3 + log(a*x - 1)^3 - 
 3*(log(a*x - 1)^2 - 4*log(a*x - 1))*log(a*x + 1) + 6*log(a*x - 1)^2)/a - 
24*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a)/a^2 + 1/4* 
(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^2 - log(a*x - 1)^2 - 4*l 
og(a*x - 1))*arctanh(a*x)/a^3
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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