\(\int \frac {(1-a^2 x^2) \text {arctanh}(a x)^2}{x^2} \, dx\) [178]

Optimal result

Integrand size = 20, antiderivative size = 93 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^2} \, dx=-\frac {\text {arctanh}(a x)^2}{x}-a^2 x \text {arctanh}(a x)^2+2 a \text {arctanh}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )+a \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-a \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.10 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^2} \, dx=-a \text {arctanh}(a x) \left (-\text {arctanh}(a x)+a x \text {arctanh}(a x)-2 \log \left (1+e^{-2 \text {arctanh}(a x)}\right )\right )-a \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(a x)}\right )+a \left (\text {arctanh}(a x) \left (\text {arctanh}(a x)-\frac {\text {arctanh}(a x)}{a x}+2 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.46, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6576, 6436, 6452, 6546, 6470, 2849, 2752, 6550, 6494, 2897}

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.

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

\(\Big \downarrow \) 6576

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

\(\Big \downarrow \) 6436

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

-(ArcTanh[a*x]^2/x) - a^2*(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)) + 2*a*(ArcTanh[a*x]^2/2 + ArcTanh[a*x]*Log[2 - 2/(1 + a*x)] - PolyLog 
[2, -1 + 2/(1 + a*x)]/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 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 6452

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : 
> Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m 
+ 1))   Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x 
], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 
] && IntegerQ[m])) && NeQ[m, -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 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 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 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]
 

rule 6550

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

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 = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(a \left (-\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right )^{2}}{a x}+2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )-\operatorname {dilog}\left (a x \right )-\operatorname {dilog}\left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {\ln \left (a x -1\right )^{2}}{2}+2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x +1\right )^{2}}{2}-\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right )\)	\(156\)
default	\(a \left (-\operatorname {arctanh}\left (a x \right )^{2} a x -\frac {\operatorname {arctanh}\left (a x \right )^{2}}{a x}+2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )-2 \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )-\operatorname {dilog}\left (a x \right )-\operatorname {dilog}\left (a x +1\right )-\ln \left (a x \right ) \ln \left (a x +1\right )-\frac {\ln \left (a x -1\right )^{2}}{2}+2 \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x +1\right )^{2}}{2}-\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right )\)	\(156\)
parts	\(-a^{2} x \operatorname {arctanh}\left (a x \right )^{2}-\frac {\operatorname {arctanh}\left (a x \right )^{2}}{x}+2 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-2 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )-2 a \,\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )+2 a \left (-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{4}+\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )\)	\(159\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.63 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)^2}{x^2} \, dx=\frac {1}{2} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 2 \, a {\left (\log \left (a x + 1\right ) + \log \left (a x - 1\right ) - \log \left (x\right )\right )} \operatorname {artanh}\left (a x\right ) - {\left (a^{2} x + \frac {1}{x}\right )} \operatorname {artanh}\left (a x\right )^{2} \] Input:

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

Output:

1/2*a^2*((log(a*x + 1)^2 - 2*log(a*x + 1)*log(a*x - 1) - log(a*x - 1)^2)/a 
 + 4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 2*(log( 
a*x + 1)*log(x) + dilog(-a*x))/a + 2*(log(-a*x + 1)*log(x) + dilog(a*x))/a 
) - 2*a*(log(a*x + 1) + log(a*x - 1) - log(x))*arctanh(a*x) - (a^2*x + 1/x 
)*arctanh(a*x)^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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