\(\int \frac {x \text {arctanh}(\frac {x}{\sqrt {2}})}{1-x^2} \, dx\) [184]

Optimal result

Integrand size = 19, antiderivative size = 193 \[ \int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right ) \] Output:

arctanh(1/2*x*2^(1/2))*ln(2*2^(1/2)/(2^(1/2)+x))-1/2*arctanh(1/2*x*2^(1/2) 
)*ln((-4+4*x)/(2-2^(1/2))/(2^(1/2)+x))-1/2*arctanh(1/2*x*2^(1/2))*ln(4*(1+ 
x)/(2+2^(1/2))/(2^(1/2)+x))-1/2*polylog(2,1-2*2^(1/2)/(2^(1/2)+x))+1/4*pol 
ylog(2,1+4*(1-x)/(2-2^(1/2))/(2^(1/2)+x))+1/4*polylog(2,1-4*(1+x)/(2+2^(1/ 
2))/(2^(1/2)+x))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.20 \[ \int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx=\frac {1}{4} \left (-4 \text {arcsinh}(1) \text {arctanh}(x)+4 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1+e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )+2 \text {arcsinh}(1) \log \left (1+\left (-3+2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1+\left (-3+2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \text {arcsinh}(1) \log \left (1-\left (3+2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-\left (3+2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )-2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )+\operatorname {PolyLog}\left (2,\left (3-2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )+\operatorname {PolyLog}\left (2,\left (3+2 \sqrt {2}\right ) e^{-2 \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}\right )\right ) \] Input:

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

Output:

(-4*ArcSinh[1]*ArcTanh[x] + 4*ArcTanh[x/Sqrt[2]]*Log[1 + E^(-2*ArcTanh[x/S 
qrt[2]])] + 2*ArcSinh[1]*Log[1 + (-3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]]) 
] - 2*ArcTanh[x/Sqrt[2]]*Log[1 + (-3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]]) 
] - 2*ArcSinh[1]*Log[1 - (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*Arc 
Tanh[x/Sqrt[2]]*Log[1 - (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*Poly 
Log[2, -E^(-2*ArcTanh[x/Sqrt[2]])] + PolyLog[2, (3 - 2*Sqrt[2])/E^(2*ArcTa 
nh[x/Sqrt[2]])] + PolyLog[2, (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])])/4
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6554, 2009}

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.

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

\(\Big \downarrow \) 6554

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

\(\Big \downarrow \) 2009

\(\displaystyle \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )\)

Input:

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

Output:

ArcTanh[x/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + x)] - (ArcTanh[x/Sqrt[2]]*Lo 
g[(-4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + x))])/2 - (ArcTanh[x/Sqrt[2]]*Log 
[(4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))])/2 - PolyLog[2, 1 - (2*Sqrt[2] 
)/(Sqrt[2] + x)]/2 + PolyLog[2, 1 + (4*(1 - x))/((2 - Sqrt[2])*(Sqrt[2] + 
x))]/4 + PolyLog[2, 1 - (4*(1 + x))/((2 + Sqrt[2])*(Sqrt[2] + x))]/4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6554

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Int[ExpandIntegrand[a + b*ArcTanh[c*x], x^m/(d + e*x^2), x], 
x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a, 0] 
)
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.09

method	result	size
parts	\(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right ) \ln \left (-1+x \right )}{2}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right ) \ln \left (1+x \right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}\, \ln \left (-1+x \right ) \ln \left (\frac {\sqrt {2}-x}{\sqrt {2}-1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (-1+x \right ) \ln \left (\frac {\sqrt {2}+x}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {\sqrt {2}-x}{\sqrt {2}-1}\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {\sqrt {2}+x}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (1+x \right ) \ln \left (\frac {\sqrt {2}-x}{1+\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \ln \left (1+x \right ) \ln \left (\frac {\sqrt {2}+x}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {\sqrt {2}-x}{1+\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {\sqrt {2}+x}{\sqrt {2}-1}\right )}{4}\right )}{2}\)	\(210\)
derivativedivides	\(-\frac {\ln \left (x^{2}-1\right ) \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )}{2}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (\frac {\sqrt {2}-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (\frac {\sqrt {2}+\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (\frac {\sqrt {2}-\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (\frac {\sqrt {2}+\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}\)	\(251\)
default	\(-\frac {\ln \left (x^{2}-1\right ) \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )}{2}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (\frac {\sqrt {2}-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}+1\right ) \ln \left (\frac {\sqrt {2}+\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (\frac {\sqrt {2}-\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {\sqrt {2}\, x}{2}-1\right ) \ln \left (\frac {\sqrt {2}+\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-\sqrt {2}\, x}{-2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}\)	\(251\)
risch	\(\frac {\left (\ln \left (1-\frac {\sqrt {2}\, x}{2}\right )-\ln \left (\frac {2-\sqrt {2}\, x}{2+\sqrt {2}}\right )\right ) \ln \left (\frac {\sqrt {2}+\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {2-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}+\frac {\left (\ln \left (1-\frac {\sqrt {2}\, x}{2}\right )-\ln \left (\frac {2-\sqrt {2}\, x}{2-\sqrt {2}}\right )\right ) \ln \left (\frac {\sqrt {2}\, x -\sqrt {2}}{2-\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {2-\sqrt {2}\, x}{2-\sqrt {2}}\right )}{4}-\frac {\left (\ln \left (\frac {\sqrt {2}\, x}{2}+1\right )-\ln \left (\frac {\sqrt {2}\, x +2}{2+\sqrt {2}}\right )\right ) \ln \left (\frac {\sqrt {2}-\sqrt {2}\, x}{2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}\, x +2}{2+\sqrt {2}}\right )}{4}-\frac {\left (\ln \left (\frac {\sqrt {2}\, x}{2}+1\right )-\ln \left (\frac {\sqrt {2}\, x +2}{2-\sqrt {2}}\right )\right ) \ln \left (\frac {-\sqrt {2}\, x -\sqrt {2}}{2-\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}\, x +2}{2-\sqrt {2}}\right )}{4}\)	\(300\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.44 \[ \int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2} \, dx=-\frac {1}{2} \, \operatorname {artanh}\left (\frac {1}{2} \, \sqrt {2} x\right ) \log \left (x^{2} - 1\right ) - \frac {1}{4} \, \log \left (x^{2} - 1\right ) \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (\sqrt {2} \log \left (x^{2} - 1\right ) \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + \sqrt {2} {\left ({\left (\log \left (2 \, x + 2 \, \sqrt {2}\right ) - \log \left (2 \, x - 2 \, \sqrt {2}\right )\right )} \log \left (x^{2} - 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} + 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} + 1} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} - 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} - 1} + 1\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} + 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} + 1}\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} - 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} - 1}\right )\right )}\right )} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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