\(\int \frac {x^2 \log (1-x^2)}{\sqrt {-1+x^2}} \, dx\) [657]

Optimal result

Integrand size = 21, antiderivative size = 194 \[ \int \frac {x^2 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {1}{8 \left (x+\sqrt {-1+x^2}\right )^2}-\frac {1}{8} \left (x+\sqrt {-1+x^2}\right )^2-\frac {\log \left (1-x^2\right )}{8 \left (x+\sqrt {-1+x^2}\right )^2}+\frac {1}{8} \left (x+\sqrt {-1+x^2}\right )^2 \log \left (1-x^2\right )-\frac {1}{2} \log \left (x+\sqrt {-1+x^2}\right )+\frac {1}{2} \log \left (1-x^2\right ) \log \left (x+\sqrt {-1+x^2}\right )+\frac {1}{2} \log ^2\left (x+\sqrt {-1+x^2}\right )-\log \left (x+\sqrt {-1+x^2}\right ) \log \left (1-\left (x+\sqrt {-1+x^2}\right )^2\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\left (x+\sqrt {-1+x^2}\right )^2\right ) \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-\frac {x \sqrt {1-x^2} \left (-\sqrt {x^2-x^4}-i \arcsin \left (\sqrt {1-x^2}\right )^2-\arcsin \left (\sqrt {1-x^2}\right ) \left (1+2 \log \left (1-e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )+\sqrt {x^2-x^4} \log \left (1-x^2\right )-i \log \left (1-x^2\right ) \log \left (\sqrt {x^2}+i \sqrt {1-x^2}\right )-i \operatorname {PolyLog}\left (2,e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )}{2 \sqrt {x^2} \sqrt {-1+x^2}} \] Input:

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

Output:

-1/2*(x*Sqrt[1 - x^2]*(-Sqrt[x^2 - x^4] - I*ArcSin[Sqrt[1 - x^2]]^2 - ArcS 
in[Sqrt[1 - x^2]]*(1 + 2*Log[1 - E^((-2*I)*ArcSin[Sqrt[1 - x^2]])]) + Sqrt 
[x^2 - x^4]*Log[1 - x^2] - I*Log[1 - x^2]*Log[Sqrt[x^2] + I*Sqrt[1 - x^2]] 
 - I*PolyLog[2, E^((-2*I)*ArcSin[Sqrt[1 - x^2]])]))/(Sqrt[x^2]*Sqrt[-1 + x 
^2])
 

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.

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

\(\Big \downarrow \) 2929

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

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

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

Output:

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