Integrand size = 23, antiderivative size = 121 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\frac {2 \sqrt {1-x^2}}{15 x^3}+\frac {4 \sqrt {1-x^2}}{5 x}-\frac {16 \arcsin (x)}{15}-\frac {\sqrt {1-x^2} \log \left (1-x^2\right )}{5 x^5}-\frac {4 \sqrt {1-x^2} \log \left (1-x^2\right )}{15 x^3}-\frac {8 \sqrt {1-x^2} \log \left (1-x^2\right )}{15 x} \] Output:
2/15*(-x^2+1)^(1/2)/x^3+4/5*(-x^2+1)^(1/2)/x-16/15*arcsin(x)-1/5*(-x^2+1)^ (1/2)*ln(-x^2+1)/x^5-4/15*(-x^2+1)^(1/2)*ln(-x^2+1)/x^3-8/15*(-x^2+1)^(1/2 )*ln(-x^2+1)/x
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\frac {1}{15} \left (-16 \arcsin (x)+\frac {\sqrt {1-x^2} \left (2 \left (x^2+6 x^4\right )-\left (3+4 x^2+8 x^4\right ) \log \left (1-x^2\right )\right )}{x^5}\right ) \] Input:
Integrate[Log[1 - x^2]/(x^6*Sqrt[1 - x^2]),x]
Output:
(-16*ArcSin[x] + (Sqrt[1 - x^2]*(2*(x^2 + 6*x^4) - (3 + 4*x^2 + 8*x^4)*Log [1 - x^2]))/x^5)/15
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.
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}}dx\) |
Input:
Int[Log[1 - x^2]/(x^6*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]
\[\int \frac {\ln \left (-x^{2}+1\right )}{x^{6} \sqrt {-x^{2}+1}}d x\]
Input:
int(ln(-x^2+1)/x^6/(-x^2+1)^(1/2),x)
Output:
int(ln(-x^2+1)/x^6/(-x^2+1)^(1/2),x)
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.58 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\frac {32 \, x^{5} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + {\left (12 \, x^{4} + 2 \, x^{2} - {\left (8 \, x^{4} + 4 \, x^{2} + 3\right )} \log \left (-x^{2} + 1\right )\right )} \sqrt {-x^{2} + 1}}{15 \, x^{5}} \] Input:
integrate(log(-x^2+1)/x^6/(-x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/15*(32*x^5*arctan((sqrt(-x^2 + 1) - 1)/x) + (12*x^4 + 2*x^2 - (8*x^4 + 4 *x^2 + 3)*log(-x^2 + 1))*sqrt(-x^2 + 1))/x^5
\[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\int \frac {\log {\left (1 - x^{2} \right )}}{x^{6} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:
integrate(ln(-x**2+1)/x**6/(-x**2+1)**(1/2),x)
Output:
Integral(log(1 - x**2)/(x**6*sqrt(-(x - 1)*(x + 1))), x)
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.71 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {8 \, \sqrt {-x^{2} + 1}}{x} + \frac {4 \, \sqrt {-x^{2} + 1}}{x^{3}} + \frac {3 \, \sqrt {-x^{2} + 1}}{x^{5}}\right )} \log \left (-x^{2} + 1\right ) + \frac {4 \, \sqrt {-x^{2} + 1}}{5 \, x} + \frac {2 \, \sqrt {-x^{2} + 1}}{15 \, x^{3}} - \frac {16}{15} \, \arcsin \left (x\right ) \] Input:
integrate(log(-x^2+1)/x^6/(-x^2+1)^(1/2),x, algorithm="maxima")
Output:
-1/15*(8*sqrt(-x^2 + 1)/x + 4*sqrt(-x^2 + 1)/x^3 + 3*sqrt(-x^2 + 1)/x^5)*l og(-x^2 + 1) + 4/5*sqrt(-x^2 + 1)/x + 2/15*sqrt(-x^2 + 1)/x^3 - 16/15*arcs in(x)
Time = 0.13 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.61 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\frac {1}{480} \, {\left (\frac {x^{5} {\left (\frac {25 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac {150 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}}{x^{4}} + 3\right )}}{{\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}} - \frac {150 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {25 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - \frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}}{x^{5}}\right )} \log \left (-x^{2} + 1\right ) - \frac {x^{3} {\left (\frac {25 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{60 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}} + \frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{12 \, x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{60 \, x^{3}} - \frac {16}{15} \, \arcsin \left (x\right ) \] Input:
integrate(log(-x^2+1)/x^6/(-x^2+1)^(1/2),x, algorithm="giac")
Output:
1/480*(x^5*(25*(sqrt(-x^2 + 1) - 1)^2/x^2 + 150*(sqrt(-x^2 + 1) - 1)^4/x^4 + 3)/(sqrt(-x^2 + 1) - 1)^5 - 150*(sqrt(-x^2 + 1) - 1)/x - 25*(sqrt(-x^2 + 1) - 1)^3/x^3 - 3*(sqrt(-x^2 + 1) - 1)^5/x^5)*log(-x^2 + 1) - 1/60*x^3*( 25*(sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1)^3 + 5/12*(sqrt(-x^ 2 + 1) - 1)/x + 1/60*(sqrt(-x^2 + 1) - 1)^3/x^3 - 16/15*arcsin(x)
Timed out. \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\int \frac {\ln \left (1-x^2\right )}{x^6\,\sqrt {1-x^2}} \,d x \] Input:
int(log(1 - x^2)/(x^6*(1 - x^2)^(1/2)),x)
Output:
int(log(1 - x^2)/(x^6*(1 - x^2)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.61 \[ \int \frac {\log \left (1-x^2\right )}{x^6 \sqrt {1-x^2}} \, dx=\frac {-16 \mathit {asin} \left (x \right ) x^{5}-8 \sqrt {-x^{2}+1}\, \mathrm {log}\left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right ) x^{4}-4 \sqrt {-x^{2}+1}\, \mathrm {log}\left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right ) x^{2}-3 \sqrt {-x^{2}+1}\, \mathrm {log}\left (\frac {\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}{\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1}\right )+12 \sqrt {-x^{2}+1}\, x^{4}+2 \sqrt {-x^{2}+1}\, x^{2}}{15 x^{5}} \] Input:
int(log(-x^2+1)/x^6/(-x^2+1)^(1/2),x)
Output:
( - 16*asin(x)*x**5 - 8*sqrt( - x**2 + 1)*log((tan(asin(x)/2)**4 - 2*tan(a sin(x)/2)**2 + 1)/(tan(asin(x)/2)**4 + 2*tan(asin(x)/2)**2 + 1))*x**4 - 4* sqrt( - x**2 + 1)*log((tan(asin(x)/2)**4 - 2*tan(asin(x)/2)**2 + 1)/(tan(a sin(x)/2)**4 + 2*tan(asin(x)/2)**2 + 1))*x**2 - 3*sqrt( - x**2 + 1)*log((t an(asin(x)/2)**4 - 2*tan(asin(x)/2)**2 + 1)/(tan(asin(x)/2)**4 + 2*tan(asi n(x)/2)**2 + 1)) + 12*sqrt( - x**2 + 1)*x**4 + 2*sqrt( - x**2 + 1)*x**2)/( 15*x**5)