\(\int \frac {x^2 \log (1-x^2)}{\sqrt {1-x^2}} \, dx\) [643]

Optimal result

Integrand size = 23, antiderivative size = 374 \[ \int \frac {x^2 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {\sqrt {-1+x^2}}{8 \sqrt {1-x^2} \left (x+\sqrt {-1+x^2}\right )^2}-\frac {\sqrt {-1+x^2} \left (x+\sqrt {-1+x^2}\right )^2}{8 \sqrt {1-x^2}}-\frac {\sqrt {-1+x^2} \log \left (1-x^2\right )}{8 \sqrt {1-x^2} \left (x+\sqrt {-1+x^2}\right )^2}+\frac {\sqrt {-1+x^2} \left (x+\sqrt {-1+x^2}\right )^2 \log \left (1-x^2\right )}{8 \sqrt {1-x^2}}-\frac {\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right )}{2 \sqrt {1-x^2}}+\frac {\sqrt {-1+x^2} \log \left (1-x^2\right ) \log \left (x+\sqrt {-1+x^2}\right )}{2 \sqrt {1-x^2}}+\frac {\sqrt {-1+x^2} \log ^2\left (x+\sqrt {-1+x^2}\right )}{2 \sqrt {1-x^2}}-\frac {\sqrt {-1+x^2} \log \left (x+\sqrt {-1+x^2}\right ) \log \left (1-\left (x+\sqrt {-1+x^2}\right )^2\right )}{\sqrt {1-x^2}}-\frac {\sqrt {-1+x^2} \operatorname {PolyLog}\left (2,\left (x+\sqrt {-1+x^2}\right )^2\right )}{2 \sqrt {1-x^2}} \] Output:

1/8*(x^2-1)^(1/2)/(-x^2+1)^(1/2)/(x+(x^2-1)^(1/2))^2-1/8*(x^2-1)^(1/2)*(x+ 
(x^2-1)^(1/2))^2/(-x^2+1)^(1/2)-1/8*(x^2-1)^(1/2)*ln(-x^2+1)/(-x^2+1)^(1/2 
)/(x+(x^2-1)^(1/2))^2+1/8*(x^2-1)^(1/2)*(x+(x^2-1)^(1/2))^2*ln(-x^2+1)/(-x 
^2+1)^(1/2)-1/2*(x^2-1)^(1/2)*ln(x+(x^2-1)^(1/2))/(-x^2+1)^(1/2)+1/2*(x^2- 
1)^(1/2)*ln(-x^2+1)*ln(x+(x^2-1)^(1/2))/(-x^2+1)^(1/2)+1/2*(x^2-1)^(1/2)*l 
n(x+(x^2-1)^(1/2))^2/(-x^2+1)^(1/2)-(x^2-1)^(1/2)*ln(x+(x^2-1)^(1/2))*ln(1 
-(x+(x^2-1)^(1/2))^2)/(-x^2+1)^(1/2)-1/2*(x^2-1)^(1/2)*polylog(2,(x+(x^2-1 
)^(1/2))^2)/(-x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {x^2 \sqrt {1-x^2}+i \sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right )^2+\sqrt {x^2} \arcsin \left (\sqrt {1-x^2}\right ) \left (1+2 \log \left (1-e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )-x^2 \sqrt {1-x^2} \log \left (1-x^2\right )+i \sqrt {x^2} \log \left (1-x^2\right ) \log \left (\sqrt {x^2}+i \sqrt {1-x^2}\right )+i \sqrt {x^2} \operatorname {PolyLog}\left (2,e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )}{2 x} \] Input:

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

Output:

(x^2*Sqrt[1 - x^2] + I*Sqrt[x^2]*ArcSin[Sqrt[1 - x^2]]^2 + Sqrt[x^2]*ArcSi 
n[Sqrt[1 - x^2]]*(1 + 2*Log[1 - E^((-2*I)*ArcSin[Sqrt[1 - x^2]])]) - x^2*S 
qrt[1 - x^2]*Log[1 - x^2] + I*Sqrt[x^2]*Log[1 - x^2]*Log[Sqrt[x^2] + I*Sqr 
t[1 - x^2]] + I*Sqrt[x^2]*PolyLog[2, E^((-2*I)*ArcSin[Sqrt[1 - x^2]])])/(2 
*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.

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

\(\Big \downarrow \) 2929

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2929

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((h_.)* 
(x_))^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Unintegrable[(h*x) 
^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e 
, f, g, h, m, n, p, q, r, s}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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