Integrand size = 23, antiderivative size = 77 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\frac {2 \sqrt {1-x^2}}{3 x}-\frac {4 \arcsin (x)}{3}-\frac {\sqrt {1-x^2} \log \left (1-x^2\right )}{3 x^3}-\frac {2 \sqrt {1-x^2} \log \left (1-x^2\right )}{3 x} \] Output:
2/3*(-x^2+1)^(1/2)/x-4/3*arcsin(x)-1/3*(-x^2+1)^(1/2)*ln(-x^2+1)/x^3-2/3*( -x^2+1)^(1/2)*ln(-x^2+1)/x
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\frac {1}{3} \left (-4 \arcsin (x)+\frac {\sqrt {1-x^2} \left (2 x^2-\left (1+2 x^2\right ) \log \left (1-x^2\right )\right )}{x^3}\right ) \] Input:
Integrate[Log[1 - x^2]/(x^4*Sqrt[1 - x^2]),x]
Output:
(-4*ArcSin[x] + (Sqrt[1 - x^2]*(2*x^2 - (1 + 2*x^2)*Log[1 - x^2]))/x^3)/3
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.
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}}dx\) |
Input:
Int[Log[1 - x^2]/(x^4*Sqrt[1 - x^2]),x]
Output:
$Aborted
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]
\[\int \frac {\ln \left (-x^{2}+1\right )}{x^{4} \sqrt {-x^{2}+1}}d x\]
Input:
int(ln(-x^2+1)/x^4/(-x^2+1)^(1/2),x)
Output:
int(ln(-x^2+1)/x^4/(-x^2+1)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\frac {8 \, x^{3} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + {\left (2 \, x^{2} - {\left (2 \, x^{2} + 1\right )} \log \left (-x^{2} + 1\right )\right )} \sqrt {-x^{2} + 1}}{3 \, x^{3}} \] Input:
integrate(log(-x^2+1)/x^4/(-x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/3*(8*x^3*arctan((sqrt(-x^2 + 1) - 1)/x) + (2*x^2 - (2*x^2 + 1)*log(-x^2 + 1))*sqrt(-x^2 + 1))/x^3
\[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\int \frac {\log {\left (1 - x^{2} \right )}}{x^{4} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:
integrate(ln(-x**2+1)/x**4/(-x**2+1)**(1/2),x)
Output:
Integral(log(1 - x**2)/(x**4*sqrt(-(x - 1)*(x + 1))), x)
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.74 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=-\frac {1}{3} \, {\left (\frac {2 \, \sqrt {-x^{2} + 1}}{x} + \frac {\sqrt {-x^{2} + 1}}{x^{3}}\right )} \log \left (-x^{2} + 1\right ) + \frac {2 \, \sqrt {-x^{2} + 1}}{3 \, x} - \frac {4}{3} \, \arcsin \left (x\right ) \] Input:
integrate(log(-x^2+1)/x^4/(-x^2+1)^(1/2),x, algorithm="maxima")
Output:
-1/3*(2*sqrt(-x^2 + 1)/x + sqrt(-x^2 + 1)/x^3)*log(-x^2 + 1) + 2/3*sqrt(-x ^2 + 1)/x - 4/3*arcsin(x)
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\frac {1}{24} \, {\left (\frac {x^{3} {\left (\frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} - \frac {9 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \log \left (-x^{2} + 1\right ) - \frac {x}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {\sqrt {-x^{2} + 1} - 1}{3 \, x} - \frac {4}{3} \, \arcsin \left (x\right ) \] Input:
integrate(log(-x^2+1)/x^4/(-x^2+1)^(1/2),x, algorithm="giac")
Output:
1/24*(x^3*(9*(sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1)^3 - 9*(s qrt(-x^2 + 1) - 1)/x - (sqrt(-x^2 + 1) - 1)^3/x^3)*log(-x^2 + 1) - 1/3*x/( sqrt(-x^2 + 1) - 1) + 1/3*(sqrt(-x^2 + 1) - 1)/x - 4/3*arcsin(x)
Timed out. \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\int \frac {\ln \left (1-x^2\right )}{x^4\,\sqrt {1-x^2}} \,d x \] Input:
int(log(1 - x^2)/(x^4*(1 - x^2)^(1/2)),x)
Output:
int(log(1 - x^2)/(x^4*(1 - x^2)^(1/2)), x)
Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.68 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {1-x^2}} \, dx=\frac {-4 \mathit {asin} \left (x \right ) x^{3}-2 \sqrt {-x^{2}+1}\, \mathrm {log}\left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right ) x^{2}-\sqrt {-x^{2}+1}\, \mathrm {log}\left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right )+2 \sqrt {-x^{2}+1}\, x^{2}}{3 x^{3}} \] Input:
int(log(-x^2+1)/x^4/(-x^2+1)^(1/2),x)
Output:
( - 4*asin(x)*x**3 - 2*sqrt( - x**2 + 1)*log((tan(asin(x)/2)**4 - 2*tan(as in(x)/2)**2 + 1)/(tan(asin(x)/2)**4 + 2*tan(asin(x)/2)**2 + 1))*x**2 - sqr t( - x**2 + 1)*log((tan(asin(x)/2)**4 - 2*tan(asin(x)/2)**2 + 1)/(tan(asin (x)/2)**4 + 2*tan(asin(x)/2)**2 + 1)) + 2*sqrt( - x**2 + 1)*x**2)/(3*x**3)