\(\int \frac {\log (-1+x^2)}{\sqrt {-1+x^2}} \, dx\) [633]

Optimal result

Integrand size = 16, antiderivative size = 83 \[ \int \frac {\log \left (-1+x^2\right )}{\sqrt {-1+x^2}} \, dx=\log \left (-1+x^2\right ) \log \left (x+\sqrt {-1+x^2}\right )+\log ^2\left (x+\sqrt {-1+x^2}\right )-2 \log \left (x+\sqrt {-1+x^2}\right ) \log \left (1-\left (x+\sqrt {-1+x^2}\right )^2\right )-\operatorname {PolyLog}\left (2,\left (x+\sqrt {-1+x^2}\right )^2\right ) \] 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)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int \frac {\log \left (-1+x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {\sqrt {-1+x^2} \left (-\text {arcsinh}\left (\sqrt {-1+x^2}\right )^2-2 \text {arcsinh}\left (\sqrt {-1+x^2}\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-1+x^2}\right )}\right )+\log \left (-1+x^2\right ) \log \left (\sqrt {x^2}+\sqrt {-1+x^2}\right )+\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-1+x^2}\right )}\right )\right )}{x \sqrt {\frac {-1+x^2}{x^2}}} \] Input:

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

Output:

(Sqrt[-1 + x^2]*(-ArcSinh[Sqrt[-1 + x^2]]^2 - 2*ArcSinh[Sqrt[-1 + x^2]]*Lo 
g[1 - E^(-2*ArcSinh[Sqrt[-1 + x^2]])] + Log[-1 + x^2]*Log[Sqrt[x^2] + Sqrt 
[-1 + x^2]] + PolyLog[2, E^(-2*ArcSinh[Sqrt[-1 + x^2]])]))/(x*Sqrt[(-1 + x 
^2)/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.

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

\(\Big \downarrow \) 2923

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

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

Sympy [F]

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

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

Output:

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

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

Output:

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