Integrand size = 18, antiderivative size = 85 \[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\log \left (1-x^2\right ) \log \left (x+\sqrt {-1+x^2}\right )+\log ^2\left (x+\sqrt {-1+x^2}\right )-2 \log \left (x+\sqrt {-1+x^2}\right ) \log \left (1-\left (x+\sqrt {-1+x^2}\right )^2\right )-\operatorname {PolyLog}\left (2,\left (x+\sqrt {-1+x^2}\right )^2\right ) \] Output:
ln(-x^2+1)*ln(x+(x^2-1)^(1/2))+ln(x+(x^2-1)^(1/2))^2-2*ln(x+(x^2-1)^(1/2)) *ln(1-(x+(x^2-1)^(1/2))^2)-polylog(2,(x+(x^2-1)^(1/2))^2)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.60 \[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {i \sqrt {1-x^2} \left (\arcsin \left (\sqrt {1-x^2}\right )^2-2 i \arcsin \left (\sqrt {1-x^2}\right ) \log \left (1-e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )+\log \left (1-x^2\right ) \log \left (\sqrt {x^2}+i \sqrt {1-x^2}\right )+\operatorname {PolyLog}\left (2,e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )}{\sqrt {1-\frac {1}{x^2}} x} \] Input:
Integrate[Log[1 - x^2]/Sqrt[-1 + x^2],x]
Output:
(I*Sqrt[1 - x^2]*(ArcSin[Sqrt[1 - x^2]]^2 - (2*I)*ArcSin[Sqrt[1 - x^2]]*Lo g[1 - E^((-2*I)*ArcSin[Sqrt[1 - x^2]])] + Log[1 - x^2]*Log[Sqrt[x^2] + I*S qrt[1 - x^2]] + PolyLog[2, E^((-2*I)*ArcSin[Sqrt[1 - x^2]])]))/(Sqrt[1 - x ^(-2)]*x)
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.
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{\sqrt {x^2-1}} \, dx\) |
\(\Big \downarrow \) 2923 |
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{\sqrt {x^2-1}}dx\) |
Input:
Int[Log[1 - x^2]/Sqrt[-1 + x^2],x]
Output:
$Aborted
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Unintegrable[(f + g*x^s)^r*(a + b*Log [c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x]
\[\int \frac {\ln \left (-x^{2}+1\right )}{\sqrt {x^{2}-1}}d x\]
Input:
int(ln(-x^2+1)/(x^2-1)^(1/2),x)
Output:
int(ln(-x^2+1)/(x^2-1)^(1/2),x)
\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:
integrate(log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="fricas")
Output:
integral(log(-x^2 + 1)/sqrt(x^2 - 1), x)
\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {\log {\left (1 - x^{2} \right )}}{\sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:
integrate(ln(-x**2+1)/(x**2-1)**(1/2),x)
Output:
Integral(log(1 - x**2)/sqrt((x - 1)*(x + 1)), x)
\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:
integrate(log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="maxima")
Output:
integrate(log(-x^2 + 1)/sqrt(x^2 - 1), x)
\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:
integrate(log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="giac")
Output:
integrate(log(-x^2 + 1)/sqrt(x^2 - 1), x)
Timed out. \[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {\ln \left (1-x^2\right )}{\sqrt {x^2-1}} \,d x \] Input:
int(log(1 - x^2)/(x^2 - 1)^(1/2),x)
Output:
int(log(1 - x^2)/(x^2 - 1)^(1/2), x)
\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {\mathrm {log}\left (-x^{2}+1\right )}{\sqrt {x^{2}-1}}d x \] Input:
int(log(-x^2+1)/(x^2-1)^(1/2),x)
Output:
int(log( - x**2 + 1)/sqrt(x**2 - 1),x)