\(\int \frac {\log (1-x^2)}{\sqrt {1-x^2}} \, dx\) [630]

Optimal result

Integrand size = 20, antiderivative size = 107 \[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\arcsin (x) \log \left (1-x^2\right )-i \log ^2\left (i x+\sqrt {1-x^2}\right )+2 i \log \left (i x+\sqrt {1-x^2}\right ) \log \left (1+\left (i x+\sqrt {1-x^2}\right )^2\right )+i \operatorname {PolyLog}\left (2,-\left (i x+\sqrt {1-x^2}\right )^2\right ) \] Output:

arcsin(x)*ln(-x^2+1)-I*ln(I*x+(-x^2+1)^(1/2))^2+2*I*ln(I*x+(-x^2+1)^(1/2)) 
*ln(1+(I*x+(-x^2+1)^(1/2))^2)+I*polylog(2,-(I*x+(-x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.79 \[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=-2 i \pi \arcsin (x)+i \arcsin (x)^2-4 \pi \log \left (1+e^{-i \arcsin (x)}\right )-\pi \log \left (1-i e^{i \arcsin (x)}\right )-2 \arcsin (x) \log \left (1-i e^{i \arcsin (x)}\right )+\pi \log \left (1+i e^{i \arcsin (x)}\right )-2 \arcsin (x) \log \left (1+i e^{i \arcsin (x)}\right )+\arcsin (x) \log \left (1-x^2\right )+4 \pi \log \left (\cos \left (\frac {\arcsin (x)}{2}\right )\right )-\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (x))\right )\right )+\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (x))\right )\right )+2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (x)}\right )+2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (x)}\right ) \] Input:

Integrate[Log[1 - x^2]/Sqrt[1 - x^2],x]
 

Output:

(-2*I)*Pi*ArcSin[x] + I*ArcSin[x]^2 - 4*Pi*Log[1 + E^((-I)*ArcSin[x])] - P 
i*Log[1 - I*E^(I*ArcSin[x])] - 2*ArcSin[x]*Log[1 - I*E^(I*ArcSin[x])] + Pi 
*Log[1 + I*E^(I*ArcSin[x])] - 2*ArcSin[x]*Log[1 + I*E^(I*ArcSin[x])] + Arc 
Sin[x]*Log[1 - x^2] + 4*Pi*Log[Cos[ArcSin[x]/2]] - Pi*Log[-Cos[(Pi + 2*Arc 
Sin[x])/4]] + Pi*Log[Sin[(Pi + 2*ArcSin[x])/4]] + (2*I)*PolyLog[2, (-I)*E^ 
(I*ArcSin[x])] + (2*I)*PolyLog[2, I*E^(I*ArcSin[x])]
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2923

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

Input:

Int[Log[1 - x^2]/Sqrt[1 - x^2],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2923

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]
 

Maple [F]

\[\int \frac {\ln \left (-x^{2}+1\right )}{\sqrt {-x^{2}+1}}d x\]

Input:

int(ln(-x^2+1)/(-x^2+1)^(1/2),x)
 

Output:

int(ln(-x^2+1)/(-x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int \frac {\log {\left (1 - x^{2} \right )}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:

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

Output:

Integral(log(1 - x**2)/sqrt(-(x - 1)*(x + 1)), x)
 

Maxima [F]

\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1}} \,d x } \] 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)
 

Giac [F]

\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1}} \,d x } \] 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)
 

Mupad [F(-1)]

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

int(log(1 - x^2)/(1 - x^2)^(1/2),x)
 

Output:

int(log(1 - x^2)/(1 - x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int \frac {\mathrm {log}\left (-x^{2}+1\right )}{\sqrt {-x^{2}+1}}d x \] Input:

int(log(-x^2+1)/(-x^2+1)^(1/2),x)
 

Output:

int(log( - x**2 + 1)/sqrt( - x**2 + 1),x)