Integrand size = 18, antiderivative size = 239 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{\sqrt {2}+x}\right )}{\sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,1+\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {4 (1+x)}{\left (2+\sqrt {2}\right ) \left (\sqrt {2}+x\right )}\right )}{2 \sqrt {2}} \]
1/2*arctanh(1/2*x*2^(1/2))*ln(-x^2+1)*2^(1/2)-1/2*arctanh(1/2*x*2^(1/2))*l n(-4*(1-x)/(2-2^(1/2))/(x+2^(1/2)))*2^(1/2)-1/2*arctanh(1/2*x*2^(1/2))*ln( 4*(1+x)/(2+2^(1/2))/(x+2^(1/2)))*2^(1/2)+1/4*polylog(2,1+4*(1-x)/(2-2^(1/2 ))/(x+2^(1/2)))*2^(1/2)-1/2*polylog(2,1-2*2^(1/2)/(x+2^(1/2)))*2^(1/2)+1/4 *polylog(2,1-4*(1+x)/(2+2^(1/2))/(x+2^(1/2)))*2^(1/2)+arctanh(1/2*x*2^(1/2 ))*ln(2*2^(1/2)/(x+2^(1/2)))*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.67 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\frac {\log \left (-1+\sqrt {2}\right ) \log (-1+x)-\log \left (1+\sqrt {2}\right ) \log (-1+x)-\log \left (-1+\sqrt {2}\right ) \log (1+x)+\log \left (1+\sqrt {2}\right ) \log (1+x)-\log \left (\sqrt {2}-x\right ) \log \left (1-x^2\right )+\log \left (\sqrt {2}+x\right ) \log \left (1-x^2\right )+\operatorname {PolyLog}\left (2,-\left (\left (-1+\sqrt {2}\right ) (-1+x)\right )\right )-\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) (-1+x)\right )-\operatorname {PolyLog}\left (2,\left (-1+\sqrt {2}\right ) (1+x)\right )+\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) (1+x)\right )\right )}{2 \sqrt {2}} \]
(Log[-1 + Sqrt[2]]*Log[-1 + x] - Log[1 + Sqrt[2]]*Log[-1 + x] - Log[-1 + S qrt[2]]*Log[1 + x] + Log[1 + Sqrt[2]]*Log[1 + x] - Log[Sqrt[2] - x]*Log[1 - x^2] + Log[Sqrt[2] + x]*Log[1 - x^2] + PolyLog[2, -((-1 + Sqrt[2])*(-1 + x))] - PolyLog[2, (1 + Sqrt[2])*(-1 + x)] - PolyLog[2, (-1 + Sqrt[2])*(1 + x)] + PolyLog[2, -((1 + Sqrt[2])*(1 + x))])/(2*Sqrt[2])
Time = 0.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2920, 27, 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.
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx\) |
\(\Big \downarrow \) 2920 |
\(\displaystyle 2 \int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2} \left (1-x^2\right )}dx+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {2} \int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{1-x^2}dx+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}\) |
\(\Big \downarrow \) 6554 |
\(\displaystyle \sqrt {2} \int \left (-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{2 (x-1)}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right )}{2 (x+1)}\right )dx+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \sqrt {2} \left (\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (-\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )-\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {2}}{x+\sqrt {2}}\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,\frac {4 (1-x)}{\left (2-\sqrt {2}\right ) \left (x+\sqrt {2}\right )}+1\right )+\frac {1}{4} \operatorname {PolyLog}\left (2,1-\frac {4 (x+1)}{\left (2+\sqrt {2}\right ) \left (x+\sqrt {2}\right )}\right )\right )+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2}}\right ) \log \left (1-x^2\right )}{\sqrt {2}}\) |
(ArcTanh[x/Sqrt[2]]*Log[1 - x^2])/Sqrt[2] + Sqrt[2]*(ArcTanh[x/Sqrt[2]]*Lo g[(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 + Sq rt[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)
3.4.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.) *(x_)^2), x_Symbol] :> With[{u = IntHide[1/(f + g*x^2), x]}, Simp[u*(a + b* Log[c*(d + e*x^n)^p]), x] - Simp[b*e*n*p Int[u*(x^(n - 1)/(d + e*x^n)), x ], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]
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] )
Time = 1.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (\ln \left (x -\sqrt {2}\right ) \ln \left (-x^{2}+1\right )-\operatorname {dilog}\left (\frac {x +1}{1+\sqrt {2}}\right )-\ln \left (x -\sqrt {2}\right ) \ln \left (\frac {x +1}{1+\sqrt {2}}\right )-\operatorname {dilog}\left (\frac {-1+x}{\sqrt {2}-1}\right )-\ln \left (x -\sqrt {2}\right ) \ln \left (\frac {-1+x}{\sqrt {2}-1}\right )\right ) \sqrt {2}}{4}+\frac {\left (\ln \left (x +\sqrt {2}\right ) \ln \left (-x^{2}+1\right )-\operatorname {dilog}\left (\frac {x +1}{1-\sqrt {2}}\right )-\ln \left (x +\sqrt {2}\right ) \ln \left (\frac {x +1}{1-\sqrt {2}}\right )-\operatorname {dilog}\left (\frac {-1+x}{-1-\sqrt {2}}\right )-\ln \left (x +\sqrt {2}\right ) \ln \left (\frac {-1+x}{-1-\sqrt {2}}\right )\right ) \sqrt {2}}{4}\) | \(194\) |
risch | \(-\frac {\sqrt {2}\, \ln \left (-x^{2}+1\right ) \ln \left (x -\sqrt {2}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right ) \ln \left (\frac {x +1}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x -\sqrt {2}\right ) \ln \left (\frac {-1+x}{\sqrt {2}-1}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {x +1}{1+\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-1+x}{\sqrt {2}-1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (\frac {x +1}{1-\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (\frac {-1+x}{-1-\sqrt {2}}\right )}{4}+\frac {\sqrt {2}\, \ln \left (x +\sqrt {2}\right ) \ln \left (-x^{2}+1\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {-1+x}{-1-\sqrt {2}}\right )}{4}-\frac {\sqrt {2}\, \operatorname {dilog}\left (\frac {x +1}{1-\sqrt {2}}\right )}{4}\) | \(214\) |
parts | \(\frac {\operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right ) \ln \left (-x^{2}+1\right ) \sqrt {2}}{2}+\sqrt {2}\, \left (-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {2}}{2}\right ) \ln \left (x^{2}-1\right )}{2}+\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}+1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-x \sqrt {2}}{2+\sqrt {2}}\right )}{4}-\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}-\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (x^{2}-1\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\ln \left (\frac {x \sqrt {2}}{2}-1\right ) \ln \left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}-x \sqrt {2}}{-2+\sqrt {2}}\right )}{4}+\frac {\operatorname {dilog}\left (\frac {\sqrt {2}+x \sqrt {2}}{2+\sqrt {2}}\right )}{4}\right )\) | \(276\) |
-1/4*(ln(x-2^(1/2))*ln(-x^2+1)-dilog((x+1)/(1+2^(1/2)))-ln(x-2^(1/2))*ln(( x+1)/(1+2^(1/2)))-dilog((-1+x)/(2^(1/2)-1))-ln(x-2^(1/2))*ln((-1+x)/(2^(1/ 2)-1)))*2^(1/2)+1/4*(ln(x+2^(1/2))*ln(-x^2+1)-dilog((x+1)/(1-2^(1/2)))-ln( x+2^(1/2))*ln((x+1)/(1-2^(1/2)))-dilog((-1+x)/(-1-2^(1/2)))-ln(x+2^(1/2))* ln((-1+x)/(-1-2^(1/2))))*2^(1/2)
\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\int { -\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2} \,d x } \]
\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=- \int \frac {\log {\left (1 - x^{2} \right )}}{x^{2} - 2}\, dx \]
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.87 \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\frac {1}{4} \, \sqrt {2} {\left ({\left (\log \left (2 \, x + 2 \, \sqrt {2}\right ) - \log \left (2 \, x - 2 \, \sqrt {2}\right )\right )} \log \left (-x^{2} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} + 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} + 1} + 1\right ) - \log \left (x + \sqrt {2}\right ) \log \left (-\frac {x + \sqrt {2}}{\sqrt {2} - 1} + 1\right ) + \log \left (x - \sqrt {2}\right ) \log \left (\frac {x - \sqrt {2}}{\sqrt {2} - 1} + 1\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} + 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} + 1}\right ) - {\rm Li}_2\left (\frac {x + \sqrt {2}}{\sqrt {2} - 1}\right ) + {\rm Li}_2\left (-\frac {x - \sqrt {2}}{\sqrt {2} - 1}\right )\right )} \]
1/4*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)))
\[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=\int { -\frac {\log \left (-x^{2} + 1\right )}{x^{2} - 2} \,d x } \]
Timed out. \[ \int \frac {\log \left (1-x^2\right )}{2-x^2} \, dx=-\int \frac {\ln \left (1-x^2\right )}{x^2-2} \,d x \]