Integrand size = 21, antiderivative size = 129 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-2 \sqrt {-1+x^2}-\frac {2}{3} \left (-1+x^2\right )^{3/2}-\frac {6}{25} \left (-1+x^2\right )^{5/2}-\frac {2}{49} \left (-1+x^2\right )^{7/2}+\sqrt {-1+x^2} \log \left (1-x^2\right )+\left (-1+x^2\right )^{3/2} \log \left (1-x^2\right )+\frac {3}{5} \left (-1+x^2\right )^{5/2} \log \left (1-x^2\right )+\frac {1}{7} \left (-1+x^2\right )^{7/2} \log \left (1-x^2\right ) \] Output:
-2*(x^2-1)^(1/2)-2/3*(x^2-1)^(3/2)-6/25*(x^2-1)^(5/2)-2/49*(x^2-1)^(7/2)+( x^2-1)^(1/2)*ln(-x^2+1)+(x^2-1)^(3/2)*ln(-x^2+1)+3/5*(x^2-1)^(5/2)*ln(-x^2 +1)+1/7*(x^2-1)^(7/2)*ln(-x^2+1)
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.47 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {\sqrt {-1+x^2} \left (-2 \left (2816+568 x^2+216 x^4+75 x^6\right )+105 \left (16+8 x^2+6 x^4+5 x^6\right ) \log \left (1-x^2\right )\right )}{3675} \] Input:
Integrate[(x^7*Log[1 - x^2])/Sqrt[-1 + x^2],x]
Output:
(Sqrt[-1 + x^2]*(-2*(2816 + 568*x^2 + 216*x^4 + 75*x^6) + 105*(16 + 8*x^2 + 6*x^4 + 5*x^6)*Log[1 - x^2]))/3675
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.
\(\displaystyle \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {x^2-1}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {x^2-1}}dx\) |
Input:
Int[(x^7*Log[1 - x^2])/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]
Time = 1.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.48
method | result | size |
risch | \(\frac {\left (5 x^{6}+6 x^{4}+8 x^{2}+16\right ) \sqrt {x^{2}-1}\, \ln \left (-x^{2}+1\right )}{35}-\frac {2 \left (75 x^{6}+216 x^{4}+568 x^{2}+2816\right ) \sqrt {x^{2}-1}}{3675}\) | \(62\) |
orering | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (975 x^{8}+1401 x^{6}+2736 x^{4}+14600 x^{2}-19712\right ) \ln \left (-x^{2}+1\right )}{3675 x^{2} \sqrt {x^{2}-1}}-\frac {\left (75 x^{6}+216 x^{4}+568 x^{2}+2816\right ) \left (x -1\right )^{2} \left (1+x \right )^{2} \left (\frac {7 x^{6} \ln \left (-x^{2}+1\right )}{\sqrt {x^{2}-1}}-\frac {2 x^{8}}{\left (-x^{2}+1\right ) \sqrt {x^{2}-1}}-\frac {x^{8} \ln \left (-x^{2}+1\right )}{\left (x^{2}-1\right )^{\frac {3}{2}}}\right )}{3675 x^{8}}\) | \(144\) |
Input:
int(x^7*ln(-x^2+1)/(x^2-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/35*(5*x^6+6*x^4+8*x^2+16)*(x^2-1)^(1/2)*ln(-x^2+1)-2/3675*(75*x^6+216*x^ 4+568*x^2+2816)*(x^2-1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.41 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-\frac {1}{3675} \, {\left (150 \, x^{6} + 432 \, x^{4} + 1136 \, x^{2} - 105 \, {\left (5 \, x^{6} + 6 \, x^{4} + 8 \, x^{2} + 16\right )} \log \left (-x^{2} + 1\right ) + 5632\right )} \sqrt {x^{2} - 1} \] Input:
integrate(x^7*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="fricas")
Output:
-1/3675*(150*x^6 + 432*x^4 + 1136*x^2 - 105*(5*x^6 + 6*x^4 + 8*x^2 + 16)*l og(-x^2 + 1) + 5632)*sqrt(x^2 - 1)
Time = 8.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {x^{6} \sqrt {x^{2} - 1} \log {\left (1 - x^{2} \right )}}{7} - \frac {2 x^{6} \sqrt {x^{2} - 1}}{49} + \frac {6 x^{4} \sqrt {x^{2} - 1} \log {\left (1 - x^{2} \right )}}{35} - \frac {144 x^{4} \sqrt {x^{2} - 1}}{1225} + \frac {8 x^{2} \sqrt {x^{2} - 1} \log {\left (1 - x^{2} \right )}}{35} - \frac {1136 x^{2} \sqrt {x^{2} - 1}}{3675} + \frac {16 \sqrt {x^{2} - 1} \log {\left (1 - x^{2} \right )}}{35} - \frac {5632 \sqrt {x^{2} - 1}}{3675} \] Input:
integrate(x**7*ln(-x**2+1)/(x**2-1)**(1/2),x)
Output:
x**6*sqrt(x**2 - 1)*log(1 - x**2)/7 - 2*x**6*sqrt(x**2 - 1)/49 + 6*x**4*sq rt(x**2 - 1)*log(1 - x**2)/35 - 144*x**4*sqrt(x**2 - 1)/1225 + 8*x**2*sqrt (x**2 - 1)*log(1 - x**2)/35 - 1136*x**2*sqrt(x**2 - 1)/3675 + 16*sqrt(x**2 - 1)*log(1 - x**2)/35 - 5632*sqrt(x**2 - 1)/3675
Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-\frac {2}{49} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} x^{4} - \frac {194}{1225} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} x^{2} + \frac {1}{35} \, {\left (5 \, \sqrt {x^{2} - 1} x^{6} + 6 \, \sqrt {x^{2} - 1} x^{4} + 8 \, \sqrt {x^{2} - 1} x^{2} + 16 \, \sqrt {x^{2} - 1}\right )} \log \left (-x^{2} + 1\right ) - \frac {1718}{3675} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{2} - 1} \] Input:
integrate(x^7*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="maxima")
Output:
-2/49*(x^2 - 1)^(3/2)*x^4 - 194/1225*(x^2 - 1)^(3/2)*x^2 + 1/35*(5*sqrt(x^ 2 - 1)*x^6 + 6*sqrt(x^2 - 1)*x^4 + 8*sqrt(x^2 - 1)*x^2 + 16*sqrt(x^2 - 1)) *log(-x^2 + 1) - 1718/3675*(x^2 - 1)^(3/2) - 2*sqrt(x^2 - 1)
Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.65 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-\frac {2}{49} \, {\left (x^{2} - 1\right )}^{\frac {7}{2}} - \frac {6}{25} \, {\left (x^{2} - 1\right )}^{\frac {5}{2}} + \frac {1}{35} \, {\left (5 \, {\left (x^{2} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (x^{2} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {x^{2} - 1}\right )} \log \left (-x^{2} + 1\right ) - \frac {2}{3} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {x^{2} - 1} \] Input:
integrate(x^7*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="giac")
Output:
-2/49*(x^2 - 1)^(7/2) - 6/25*(x^2 - 1)^(5/2) + 1/35*(5*(x^2 - 1)^(7/2) + 2 1*(x^2 - 1)^(5/2) + 35*(x^2 - 1)^(3/2) + 35*sqrt(x^2 - 1))*log(-x^2 + 1) - 2/3*(x^2 - 1)^(3/2) - 2*sqrt(x^2 - 1)
Timed out. \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {x^7\,\ln \left (1-x^2\right )}{\sqrt {x^2-1}} \,d x \] Input:
int((x^7*log(1 - x^2))/(x^2 - 1)^(1/2),x)
Output:
int((x^7*log(1 - x^2))/(x^2 - 1)^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.57 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {\sqrt {x^{2}-1}\, \left (525 \,\mathrm {log}\left (-x^{2}+1\right ) x^{6}+630 \,\mathrm {log}\left (-x^{2}+1\right ) x^{4}+840 \,\mathrm {log}\left (-x^{2}+1\right ) x^{2}+1680 \,\mathrm {log}\left (-x^{2}+1\right )-150 x^{6}-432 x^{4}-1136 x^{2}-5632\right )}{3675} \] Input:
int(x^7*log(-x^2+1)/(x^2-1)^(1/2),x)
Output:
(sqrt(x**2 - 1)*(525*log( - x**2 + 1)*x**6 + 630*log( - x**2 + 1)*x**4 + 8 40*log( - x**2 + 1)*x**2 + 1680*log( - x**2 + 1) - 150*x**6 - 432*x**4 - 1 136*x**2 - 5632))/3675