\(\int \frac {\text {arctanh}(a x)}{x (1-a^2 x^2)^2} \, dx\) [263]

Optimal result

Integrand size = 20, antiderivative size = 91 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=-\frac {a x}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \text {arctanh}(a x)+\frac {\text {arctanh}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \text {arctanh}(a x)^2+\text {arctanh}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \] Output:

-1/4*a*x/(-a^2*x^2+1)-1/4*arctanh(a*x)+arctanh(a*x)/(-2*a^2*x^2+2)+1/2*arc 
tanh(a*x)^2+arctanh(a*x)*ln(2-2/(a*x+1))-1/2*polylog(2,-1+2/(a*x+1))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.69 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{8} \left (4 \text {arctanh}(a x)^2+2 \text {arctanh}(a x) \left (\cosh (2 \text {arctanh}(a x))+4 \log \left (1-e^{-2 \text {arctanh}(a x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(a x)}\right )-\sinh (2 \text {arctanh}(a x))\right ) \] Input:

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

Output:

(4*ArcTanh[a*x]^2 + 2*ArcTanh[a*x]*(Cosh[2*ArcTanh[a*x]] + 4*Log[1 - E^(-2 
*ArcTanh[a*x])]) - 4*PolyLog[2, E^(-2*ArcTanh[a*x])] - Sinh[2*ArcTanh[a*x] 
])/8
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6592, 6550, 6494, 2897, 6556, 215, 219}

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.

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

\(\Big \downarrow \) 6592

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

\(\Big \downarrow \) 6550

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

\(\Big \downarrow \) 6494

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

\(\Big \downarrow \) 2897

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

\(\Big \downarrow \) 6556

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 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 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 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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(80)=160\).

Time = 0.46 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.09

method	result	size
derivativedivides	\(\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}\)	\(190\)
default	\(\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}+\operatorname {arctanh}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {dilog}\left (a x \right )}{2}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8}\)	\(190\)
risch	\(-\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\operatorname {dilog}\left (a x +1\right )}{2}+\frac {\ln \left (a x +1\right )}{8 a x +8}+\frac {1}{8 a x +8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )}{16}-\frac {\ln \left (a x +1\right ) \left (a x +1\right )}{16 \left (a x -1\right )}+\frac {\ln \left (-a x +1\right )^{2}}{8}+\frac {\operatorname {dilog}\left (-a x +1\right )}{2}-\frac {\ln \left (-a x +1\right )}{8 \left (-a x +1\right )}-\frac {1}{8 \left (-a x +1\right )}+\frac {\left (\ln \left (-a x +1\right )-\ln \left (-\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\operatorname {dilog}\left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (-a x -1\right )}{16}+\frac {\left (-a x +1\right ) \ln \left (-a x +1\right )}{-16 a x -16}\)	\(220\)
parts	\(\operatorname {arctanh}\left (a x \right ) \ln \left (x \right )+\frac {\operatorname {arctanh}\left (a x \right )}{4 a x +4}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\operatorname {arctanh}\left (a x \right )}{4 \left (a x -1\right )}-\frac {\operatorname {arctanh}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {a \left (\frac {2 \operatorname {dilog}\left (a x +1\right )}{a}+\frac {2 \ln \left (x \right ) \ln \left (a x +1\right )}{a}-\frac {2 \left (\ln \left (x \right )-\ln \left (a x \right )\right ) \ln \left (-a x +1\right )}{a}+\frac {2 \operatorname {dilog}\left (a x \right )}{a}+\frac {\ln \left (a x -1\right )^{2}}{2 a}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{a}+\frac {-\frac {\ln \left (a x +1\right )^{2}}{2}+\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )-\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {1}{2 \left (a x +1\right ) a}+\frac {\ln \left (a x +1\right )}{2 a}-\frac {1}{2 a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{2 a}\right )}{4}\)	\(256\)

Input:

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

Output:

1/4*arctanh(a*x)/(a*x+1)-1/2*arctanh(a*x)*ln(a*x+1)-1/4*arctanh(a*x)/(a*x- 
1)-1/2*arctanh(a*x)*ln(a*x-1)+arctanh(a*x)*ln(a*x)-1/2*dilog(a*x)-1/2*dilo 
g(a*x+1)-1/2*ln(a*x)*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)+1/8/(a*x-1)+1/8*ln(a*x-1)+1/8/(a*x+1)-1/8*ln(a*x+1)
 

Fricas [F]

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

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

Output:

integral(arctanh(a*x)/(a^4*x^5 - 2*a^2*x^3 + x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (78) = 156\).

Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.26 \[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\frac {1}{8} \, a {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 2 \, a x}{a^{3} x^{2} - a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{2} x^{2} - 1} + \log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \] Input:

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

Output:

1/8*a*(((a^2*x^2 - 1)*log(a*x + 1)^2 - 2*(a^2*x^2 - 1)*log(a*x + 1)*log(a* 
x - 1) - (a^2*x^2 - 1)*log(a*x - 1)^2 + 2*a*x)/(a^3*x^2 - a) + 4*(log(a*x 
- 1)*log(1/2*a*x + 1/2) + dilog(-1/2*a*x + 1/2))/a - 4*(log(a*x + 1)*log(x 
) + dilog(-a*x))/a + 4*(log(-a*x + 1)*log(x) + dilog(a*x))/a - log(a*x + 1 
)/a + log(a*x - 1)/a) - 1/2*(1/(a^2*x^2 - 1) + log(a^2*x^2 - 1) - log(x^2) 
)*arctanh(a*x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx=\int \frac {\mathit {atanh} \left (a x \right )}{a^{4} x^{5}-2 a^{2} x^{3}+x}d x \] Input:

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

Output:

int(atanh(a*x)/(a**4*x**5 - 2*a**2*x**3 + x),x)