3.184 \(\int \frac{x \tanh ^{-1}(\frac{x}{\sqrt{2}})}{1-x^2} \, dx\)

Optimal. Leaf size=193 \[ -\frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{x+\sqrt{2}}\right )+\frac{1}{4} \text{PolyLog}\left (2,\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (x+\sqrt{2}\right )}+1\right )+\frac{1}{4} \text{PolyLog}\left (2,1-\frac{4 (x+1)}{\left (2+\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right )+\log \left (\frac{2 \sqrt{2}}{x+\sqrt{2}}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{2} \log \left (-\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{2} \log \left (\frac{4 (x+1)}{\left (2+\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

ArcTanh[x/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + x)] - (ArcTanh[x/Sqrt[2]]*Log[(-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

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Rubi [A]  time = 0.226503, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5992, 5920, 2402, 2315, 2447} \[ -\frac{1}{2} \text{PolyLog}\left (2,1-\frac{2 \sqrt{2}}{x+\sqrt{2}}\right )+\frac{1}{4} \text{PolyLog}\left (2,\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (x+\sqrt{2}\right )}+1\right )+\frac{1}{4} \text{PolyLog}\left (2,1-\frac{4 (x+1)}{\left (2+\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right )+\log \left (\frac{2 \sqrt{2}}{x+\sqrt{2}}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{2} \log \left (-\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{2} \log \left (\frac{4 (x+1)}{\left (2+\sqrt{2}\right ) \left (x+\sqrt{2}\right )}\right ) \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[x/Sqrt[2]]*Log[(2*Sqrt[2])/(Sqrt[2] + x)] - (ArcTanh[x/Sqrt[2]]*Log[(-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

Rule 5992

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])

Rule 5920

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

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[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 2315

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

Rule 2447

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[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{1-x^2} \, dx &=\int \left (-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 (-1+x)}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{2 (1+x)}\right ) \, dx\\ &=-\left (\frac{1}{2} \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{-1+x} \, dx\right )-\frac{1}{2} \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{1+x} \, dx\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+x}\right )-\frac{1}{2} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (\frac{4 (1+x)}{\left (2+\sqrt{2}\right ) \left (\sqrt{2}+x\right )}\right )-2 \frac{\int \frac{\log \left (\frac{2}{1+\frac{x}{\sqrt{2}}}\right )}{1-\frac{x^2}{2}} \, dx}{2 \sqrt{2}}+\frac{\int \frac{\log \left (\frac{\sqrt{2} (-1+x)}{\left (1-\frac{1}{\sqrt{2}}\right ) \left (1+\frac{x}{\sqrt{2}}\right )}\right )}{1-\frac{x^2}{2}} \, dx}{2 \sqrt{2}}+\frac{\int \frac{\log \left (\frac{\sqrt{2} (1+x)}{\left (1+\frac{1}{\sqrt{2}}\right ) \left (1+\frac{x}{\sqrt{2}}\right )}\right )}{1-\frac{x^2}{2}} \, dx}{2 \sqrt{2}}\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+x}\right )-\frac{1}{2} \tanh ^{-1}\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} \tanh ^{-1}\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}{4} \text{Li}_2\left (1+\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (\sqrt{2}+x\right )}\right )+\frac{1}{4} \text{Li}_2\left (1-\frac{4 (1+x)}{\left (2+\sqrt{2}\right ) \left (\sqrt{2}+x\right )}\right )-2 \left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{x}{\sqrt{2}}}\right )\right )\\ &=\tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (\frac{2 \sqrt{2}}{\sqrt{2}+x}\right )-\frac{1}{2} \tanh ^{-1}\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} \tanh ^{-1}\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{Li}_2\left (1-\frac{2 \sqrt{2}}{\sqrt{2}+x}\right )+\frac{1}{4} \text{Li}_2\left (1+\frac{4 (1-x)}{\left (2-\sqrt{2}\right ) \left (\sqrt{2}+x\right )}\right )+\frac{1}{4} \text{Li}_2\left (1-\frac{4 (1+x)}{\left (2+\sqrt{2}\right ) \left (\sqrt{2}+x\right )}\right )\\ \end{align*}

Mathematica [A]  time = 0.247399, size = 232, normalized size = 1.2 \[ \frac{1}{4} \left (-2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}\right )+\text{PolyLog}\left (2,\left (3-2 \sqrt{2}\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}\right )+\text{PolyLog}\left (2,\left (3+2 \sqrt{2}\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}\right )+4 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}+1\right )-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (\left (2 \sqrt{2}-3\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}+1\right )-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \log \left (1-\left (3+2 \sqrt{2}\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}\right )-4 \sinh ^{-1}(1) \tanh ^{-1}(x)+2 \sinh ^{-1}(1) \log \left (\left (2 \sqrt{2}-3\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}+1\right )-2 \sinh ^{-1}(1) \log \left (1-\left (3+2 \sqrt{2}\right ) e^{-2 \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*ArcSinh[1]*ArcTanh[x] + 4*ArcTanh[x/Sqrt[2]]*Log[1 + E^(-2*ArcTanh[x/Sqrt[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*ArcTanh[x/Sqrt[2]]*Log[1 - (3 + 2*Sqr
t[2])/E^(2*ArcTanh[x/Sqrt[2]])] - 2*PolyLog[2, -E^(-2*ArcTanh[x/Sqrt[2]])] + PolyLog[2, (3 - 2*Sqrt[2])/E^(2*A
rcTanh[x/Sqrt[2]])] + PolyLog[2, (3 + 2*Sqrt[2])/E^(2*ArcTanh[x/Sqrt[2]])])/4

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Maple [A]  time = 0.049, size = 251, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ({x}^{2}-1 \right ) }{2}{\it Artanh} \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{\ln \left ({x}^{2}-1 \right ) }{4}\ln \left ({\frac{x\sqrt{2}}{2}}-1 \right ) }+{\frac{1}{4}\ln \left ({\frac{x\sqrt{2}}{2}}-1 \right ) \ln \left ({\frac{\sqrt{2}-x\sqrt{2}}{-2+\sqrt{2}}} \right ) }+{\frac{1}{4}\ln \left ({\frac{x\sqrt{2}}{2}}-1 \right ) \ln \left ({\frac{\sqrt{2}+x\sqrt{2}}{2+\sqrt{2}}} \right ) }+{\frac{1}{4}{\it dilog} \left ({\frac{\sqrt{2}-x\sqrt{2}}{-2+\sqrt{2}}} \right ) }+{\frac{1}{4}{\it dilog} \left ({\frac{\sqrt{2}+x\sqrt{2}}{2+\sqrt{2}}} \right ) }+{\frac{\ln \left ({x}^{2}-1 \right ) }{4}\ln \left ({\frac{x\sqrt{2}}{2}}+1 \right ) }-{\frac{1}{4}\ln \left ({\frac{x\sqrt{2}}{2}}+1 \right ) \ln \left ({\frac{\sqrt{2}-x\sqrt{2}}{2+\sqrt{2}}} \right ) }-{\frac{1}{4}\ln \left ({\frac{x\sqrt{2}}{2}}+1 \right ) \ln \left ({\frac{\sqrt{2}+x\sqrt{2}}{-2+\sqrt{2}}} \right ) }-{\frac{1}{4}{\it dilog} \left ({\frac{\sqrt{2}-x\sqrt{2}}{2+\sqrt{2}}} \right ) }-{\frac{1}{4}{\it dilog} \left ({\frac{\sqrt{2}+x\sqrt{2}}{-2+\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46133, size = 374, normalized size = 1.94 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*(sq
rt(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)))*l
og(x^2 - 1) - log(x + sqrt(2))*log(-(x + sqrt(2))/(sqrt(2) + 1) + 1) + log(x - sqrt(2))*log((x - sqrt(2))/(sqr
t(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))))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x \operatorname{artanh}\left (\frac{1}{2} \, \sqrt{2} x\right )}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x \operatorname{atanh}{\left (\frac{\sqrt{2} x}{2} \right )}}{x^{2} - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x \operatorname{artanh}\left (\frac{1}{2} \, \sqrt{2} x\right )}{x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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