\(\int \frac {\log (1-x^2)}{x^3 \sqrt {1-x^2}} \, dx\) [641]

Optimal result

Integrand size = 23, antiderivative size = 99 \[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\text {arctanh}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2} \log \left (1-x^2\right )}{2 x^2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1-x^2}\right ) \log \left (1-x^2\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-\sqrt {1-x^2}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,\sqrt {1-x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\frac {1}{4} \left (-\frac {2 \sqrt {1-x^2} \log \left (1-x^2\right )}{x^2}-2 \log \left (1-\sqrt {1-x^2}\right )+\log \left (1-x^2\right ) \log \left (1-\sqrt {1-x^2}\right )+2 \log \left (1+\sqrt {1-x^2}\right )-\log \left (1-x^2\right ) \log \left (1+\sqrt {1-x^2}\right )-2 \operatorname {PolyLog}\left (2,-\sqrt {1-x^2}\right )+2 \operatorname {PolyLog}\left (2,\sqrt {1-x^2}\right )\right ) \] Input:

Integrate[Log[1 - x^2]/(x^3*Sqrt[1 - x^2]),x]
 

Output:

((-2*Sqrt[1 - x^2]*Log[1 - x^2])/x^2 - 2*Log[1 - Sqrt[1 - x^2]] + Log[1 - 
x^2]*Log[1 - Sqrt[1 - x^2]] + 2*Log[1 + Sqrt[1 - x^2]] - Log[1 - x^2]*Log[ 
1 + Sqrt[1 - x^2]] - 2*PolyLog[2, -Sqrt[1 - x^2]] + 2*PolyLog[2, Sqrt[1 - 
x^2]])/4
 

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.

\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}}dx\)

Input:

Int[Log[1 - x^2]/(x^3*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]

\[\int \frac {\ln \left (-x^{2}+1\right )}{x^{3} \sqrt {-x^{2}+1}}d x\]

Input:

int(ln(-x^2+1)/x^3/(-x^2+1)^(1/2),x)
 

Output:

int(ln(-x^2+1)/x^3/(-x^2+1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(log(-x^2+1)/x^3/(-x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-x^2 + 1)*log(-x^2 + 1)/(x^5 - x^3), x)
 

Sympy [F]

\[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int \frac {\log {\left (1 - x^{2} \right )}}{x^{3} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:

integrate(ln(-x**2+1)/x**3/(-x**2+1)**(1/2),x)
 

Output:

Integral(log(1 - x**2)/(x**3*sqrt(-(x - 1)*(x + 1))), x)
 

Maxima [F]

\[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(log(-x^2+1)/x^3/(-x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(log(-x^2 + 1)/(sqrt(-x^2 + 1)*x^3), x)
 

Giac [F]

\[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int { \frac {\log \left (-x^{2} + 1\right )}{\sqrt {-x^{2} + 1} x^{3}} \,d x } \] Input:

integrate(log(-x^2+1)/x^3/(-x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(log(-x^2 + 1)/(sqrt(-x^2 + 1)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int \frac {\ln \left (1-x^2\right )}{x^3\,\sqrt {1-x^2}} \,d x \] Input:

int(log(1 - x^2)/(x^3*(1 - x^2)^(1/2)),x)
 

Output:

int(log(1 - x^2)/(x^3*(1 - x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\log \left (1-x^2\right )}{x^3 \sqrt {1-x^2}} \, dx=\int \frac {\mathrm {log}\left (-x^{2}+1\right )}{\sqrt {-x^{2}+1}\, x^{3}}d x \] Input:

int(log(-x^2+1)/x^3/(-x^2+1)^(1/2),x)
 

Output:

int(log( - x**2 + 1)/(sqrt( - x**2 + 1)*x**3),x)