\(\int \frac {\text {arctanh}(a x)}{x^3 (1-a^2 x^2)} \, dx\) [233]

Optimal result

Integrand size = 20, antiderivative size = 84 \[ \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx=-\frac {a}{2 x}+\frac {1}{2} a^2 \text {arctanh}(a x)-\frac {\text {arctanh}(a x)}{2 x^2}+\frac {1}{2} a^2 \text {arctanh}(a x)^2+a^2 \text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx=-\frac {1}{2} a^2 \left (\frac {1}{a x}-\text {arctanh}(a x) \left (1-\frac {1}{a^2 x^2}+\text {arctanh}(a x)+2 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6544, 6452, 264, 219, 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.

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

\(\Big \downarrow \) 6544

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

\(\Big \downarrow \) 6452

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

Input:

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

Output:

-1/2*ArcTanh[a*x]/x^2 + (a*(-x^(-1) + a*ArcTanh[a*x]))/2 + a^2*(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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 264

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] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

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 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 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 6544

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(74)=148\).

Time = 0.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.02

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {1}{2 a x}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\)	\(170\)
default	\(a^{2} \left (-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{2 a^{2} x^{2}}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {1}{2 a x}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\)	\(170\)
risch	\(-\frac {a^{2} \ln \left (a x +1\right )^{2}}{8}-\frac {a^{2} \operatorname {dilog}\left (a x +1\right )}{2}-\frac {a^{2} \ln \left (a x \right )}{4}-\frac {a}{2 x}+\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {\ln \left (a x +1\right )}{4 x^{2}}-\frac {a^{2} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4}+\frac {a^{2} \operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {a^{2} \ln \left (-a x +1\right )^{2}}{8}+\frac {a^{2} \operatorname {dilog}\left (-a x +1\right )}{2}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}+\frac {a^{2} \ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4}-\frac {a^{2} \operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\)	\(184\)
parts	\(-\frac {\operatorname {arctanh}\left (a x \right )}{2 x^{2}}+\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (x \right )-\frac {\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right ) a^{2} \ln \left (a x -1\right )}{2}-\frac {a \left (-\frac {a \ln \left (a x +1\right )}{2}+\frac {1}{x}+\frac {a \ln \left (a x -1\right )}{2}+a \left (-\frac {\ln \left (a x +1\right )^{2}}{4}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )+a \left (\frac {\ln \left (a x -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}\right )-2 a^{2} \left (-\frac {\operatorname {dilog}\left (a x +1\right )}{2 a}-\frac {\ln \left (x \right ) \ln \left (a x +1\right )}{2 a}+\frac {\left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{2 a}-\frac {\operatorname {dilog}\left (a x \right )}{2 a}\right )\right )}{2}\)	\(220\)

Input:

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

Output:

a^2*(-1/2*arctanh(a*x)*ln(a*x-1)-1/2*arctanh(a*x)*ln(a*x+1)-1/2*arctanh(a* 
x)/a^2/x^2+arctanh(a*x)*ln(a*x)-1/2/a/x-1/4*ln(a*x-1)+1/4*ln(a*x+1)-1/8*ln 
(a*x-1)^2+1/2*dilog(1/2*a*x+1/2)+1/4*ln(a*x-1)*ln(1/2*a*x+1/2)+1/8*ln(a*x+ 
1)^2-1/4*(ln(a*x+1)-ln(1/2*a*x+1/2))*ln(-1/2*a*x+1/2)-1/2*dilog(a*x)-1/2*d 
ilog(a*x+1)-1/2*ln(a*x)*ln(a*x+1))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-Integral(atanh(a*x)/(a**2*x**5 - x**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (73) = 146\).

Time = 0.04 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.93 \[ \int \frac {\text {arctanh}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx=\frac {1}{8} \, {\left (4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} - 4}{x}\right )} a - \frac {1}{2} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

1/8*(4*(log(a*x - 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))*a - 4*(lo 
g(a*x + 1)*log(x) + dilog(-a*x))*a + 4*(log(-a*x + 1)*log(x) + dilog(a*x)) 
*a + 2*a*log(a*x + 1) - 2*a*log(a*x - 1) + (a*x*log(a*x + 1)^2 - 2*a*x*log 
(a*x + 1)*log(a*x - 1) - a*x*log(a*x - 1)^2 - 4)/x)*a - 1/2*(a^2*log(a^2*x 
^2 - 1) - a^2*log(x^2) + 1/x^2)*arctanh(a*x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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