Integrand size = 21, antiderivative size = 81 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=-\frac {2 \sqrt {-1+x^2}}{3 x}-\frac {4}{3} \text {arctanh}\left (\frac {x}{\sqrt {-1+x^2}}\right )+\frac {\sqrt {-1+x^2} \log \left (1-x^2\right )}{3 x^3}+\frac {2 \sqrt {-1+x^2} \log \left (1-x^2\right )}{3 x} \] Output:
-2/3*(x^2-1)^(1/2)/x-4/3*arctanh(x/(x^2-1)^(1/2))+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 = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=-\frac {2 \sqrt {-1+x^2}}{3 x}+\frac {\sqrt {-1+x^2} \left (1+2 x^2\right ) \log \left (1-x^2\right )}{3 x^3}-\frac {4}{3} \log \left (x+\sqrt {-1+x^2}\right ) \] Input:
Integrate[Log[1 - x^2]/(x^4*Sqrt[-1 + x^2]),x]
Output:
(-2*Sqrt[-1 + x^2])/(3*x) + (Sqrt[-1 + x^2]*(1 + 2*x^2)*Log[1 - x^2])/(3*x ^3) - (4*Log[x + Sqrt[-1 + x^2]])/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.
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {x^2-1}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {x^2-1}}dx\) |
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 = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=\frac {4 \, x^{3} \log \left (-x + \sqrt {x^{2} - 1}\right ) - 2 \, x^{3} - {\left (2 \, x^{2} - {\left (2 \, x^{2} + 1\right )} \log \left (-x^{2} + 1\right )\right )} \sqrt {x^{2} - 1}}{3 \, x^{3}} \] Input:
integrate(log(-x^2+1)/x^4/(x^2-1)^(1/2),x, algorithm="fricas")
Output:
1/3*(4*x^3*log(-x + sqrt(x^2 - 1)) - 2*x^3 - (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 = 66, normalized size of antiderivative = 0.81 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, \sqrt {x^{2} - 1}}{x} + \frac {\sqrt {x^{2} - 1}}{x^{3}}\right )} \log \left (-x^{2} + 1\right ) - \frac {2 \, \sqrt {x + 1} \sqrt {x - 1}}{3 \, x} + \frac {4}{3} \, \log \left (-\frac {1}{3} \, x + \frac {1}{3} \, \sqrt {x^{2} - 1}\right ) \] 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 + 1 )*sqrt(x - 1)/x + 4/3*log(-1/3*x + 1/3*sqrt(x^2 - 1))
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=-\frac {4}{3 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}} + \frac {4 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )} \log \left (-x^{2} + 1\right )}{3 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}^{3}} + \frac {2}{3} \, \log \left ({\left (x - \sqrt {x^{2} - 1}\right )}^{2}\right ) - \frac {2}{3} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:
integrate(log(-x^2+1)/x^4/(x^2-1)^(1/2),x, algorithm="giac")
Output:
-4/3/((x - sqrt(x^2 - 1))^2 + 1) + 4/3*(3*(x - sqrt(x^2 - 1))^2 + 1)*log(- x^2 + 1)/((x - sqrt(x^2 - 1))^2 + 1)^3 + 2/3*log((x - sqrt(x^2 - 1))^2) - 2/3*log(abs(x + 1)) - 2/3*log(abs(x - 1))
Timed out. \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=\int \frac {\ln \left (1-x^2\right )}{x^4\,\sqrt {x^2-1}} \,d x \] Input:
int(log(1 - x^2)/(x^4*(x^2 - 1)^(1/2)),x)
Output:
int(log(1 - x^2)/(x^4*(x^2 - 1)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {\log \left (1-x^2\right )}{x^4 \sqrt {-1+x^2}} \, dx=\frac {6 \sqrt {x^{2}-1}\, \mathrm {log}\left (-x^{2}+1\right ) x^{2}+3 \sqrt {x^{2}-1}\, \mathrm {log}\left (-x^{2}+1\right )-6 \sqrt {x^{2}-1}\, x^{2}+6 \,\mathrm {log}\left (-x^{2}+1\right ) x^{3}-12 \,\mathrm {log}\left (\sqrt {x^{2}-1}+x -1\right ) x^{3}-12 \,\mathrm {log}\left (\sqrt {x^{2}-1}+x +1\right ) x^{3}+2 x^{3}}{9 x^{3}} \] Input:
int(log(-x^2+1)/x^4/(x^2-1)^(1/2),x)
Output:
(6*sqrt(x**2 - 1)*log( - x**2 + 1)*x**2 + 3*sqrt(x**2 - 1)*log( - x**2 + 1 ) - 6*sqrt(x**2 - 1)*x**2 + 6*log( - x**2 + 1)*x**3 - 12*log(sqrt(x**2 - 1 ) + x - 1)*x**3 - 12*log(sqrt(x**2 - 1) + x + 1)*x**3 + 2*x**3)/(9*x**3)