Integrand size = 23, antiderivative size = 146 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=2 \sqrt {1-x^2}-\frac {2}{3} \left (1-x^2\right )^{3/2}+\frac {6}{25} \left (1-x^2\right )^{5/2}-\frac {2}{49} \left (1-x^2\right )^{7/2}-\sqrt {1-x^2} \log \left (1-x^2\right )+\left (1-x^2\right )^{3/2} \log \left (1-x^2\right )-\frac {3}{5} \left (1-x^2\right )^{5/2} \log \left (1-x^2\right )+\frac {1}{7} \left (1-x^2\right )^{7/2} \log \left (1-x^2\right ) \] 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)*l n(-x^2+1)+1/7*(-x^2+1)^(7/2)*ln(-x^2+1)
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.42 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {\sqrt {1-x^2} \left (2 \left (2816+568 x^2+216 x^4+75 x^6\right )-105 \left (16+8 x^2+6 x^4+5 x^6\right ) \log \left (1-x^2\right )\right )}{3675} \] 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.
\(\displaystyle \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx\) |
\(\Big \downarrow \) 2929 |
\(\displaystyle \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}}dx\) |
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 = 1.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {\left (5 x^{6}+6 x^{4}+8 x^{2}+16\right ) \left (x^{2}-1\right ) \ln \left (-x^{2}+1\right )}{35 \sqrt {-x^{2}+1}}-\frac {2 \left (75 x^{6}+216 x^{4}+568 x^{2}+2816\right ) \left (x^{2}-1\right )}{3675 \sqrt {-x^{2}+1}}\) | \(76\) |
orering | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (975 x^{8}+1401 x^{6}+2736 x^{4}+14600 x^{2}-19712\right ) \ln \left (-x^{2}+1\right )}{3675 x^{2} \sqrt {-x^{2}+1}}-\frac {\left (75 x^{6}+216 x^{4}+568 x^{2}+2816\right ) \left (x -1\right )^{2} \left (1+x \right )^{2} \left (\frac {7 x^{6} \ln \left (-x^{2}+1\right )}{\sqrt {-x^{2}+1}}-\frac {2 x^{8}}{\left (-x^{2}+1\right )^{\frac {3}{2}}}+\frac {x^{8} \ln \left (-x^{2}+1\right )}{\left (-x^{2}+1\right )^{\frac {3}{2}}}\right )}{3675 x^{8}}\) | \(142\) |
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)/(-x^2+1)^(1/2)*ln(-x^2+1)-2/3675*(75*x ^6+216*x^4+568*x^2+2816)*(x^2-1)/(-x^2+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.38 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {1}{3675} \, {\left (150 \, x^{6} + 432 \, x^{4} + 1136 \, x^{2} - 105 \, {\left (5 \, x^{6} + 6 \, x^{4} + 8 \, x^{2} + 16\right )} \log \left (-x^{2} + 1\right ) + 5632\right )} \sqrt {-x^{2} + 1} \] 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)*lo g(-x^2 + 1) + 5632)*sqrt(-x^2 + 1)
Time = 130.48 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.48 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\sqrt {1 - x^{2}} \left (- \frac {x^{6}}{7} - \frac {6 x^{4}}{35} - \frac {8 x^{2}}{35} - \frac {16}{35}\right ) \log {\left (1 - x^{2} \right )} + 2 \sqrt {1 - x^{2}} \left (\frac {x^{6}}{49} + \frac {72 x^{4}}{1225} + \frac {568 x^{2}}{3675} + \frac {2816}{3675}\right ) \] Input:
integrate(x**7*ln(-x**2+1)/(-x**2+1)**(1/2),x)
Output:
sqrt(1 - x**2)*(-x**6/7 - 6*x**4/35 - 8*x**2/35 - 16/35)*log(1 - x**2) + 2 *sqrt(1 - x**2)*(x**6/49 + 72*x**4/1225 + 568*x**2/3675 + 2816/3675)
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.81 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {2}{49} \, \sqrt {-x^{2} + 1} x^{6} + \frac {144}{1225} \, \sqrt {-x^{2} + 1} x^{4} + \frac {1136}{3675} \, \sqrt {-x^{2} + 1} x^{2} - \frac {1}{35} \, {\left (5 \, \sqrt {-x^{2} + 1} x^{6} + 6 \, \sqrt {-x^{2} + 1} x^{4} + 8 \, \sqrt {-x^{2} + 1} x^{2} + 16 \, \sqrt {-x^{2} + 1}\right )} \log \left (-x^{2} + 1\right ) + \frac {5632}{3675} \, \sqrt {-x^{2} + 1} \] Input:
integrate(x^7*log(-x^2+1)/(-x^2+1)^(1/2),x, algorithm="maxima")
Output:
2/49*sqrt(-x^2 + 1)*x^6 + 144/1225*sqrt(-x^2 + 1)*x^4 + 1136/3675*sqrt(-x^ 2 + 1)*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) + 5632/3675*sqrt(-x^2 + 1)
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {2}{49} \, {\left (x^{2} - 1\right )}^{3} \sqrt {-x^{2} + 1} + \frac {6}{25} \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1} - \frac {1}{35} \, {\left (5 \, {\left (x^{2} - 1\right )}^{3} \sqrt {-x^{2} + 1} + 21 \, {\left (x^{2} - 1\right )}^{2} \sqrt {-x^{2} + 1} - 35 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {-x^{2} + 1}\right )} \log \left (-x^{2} + 1\right ) - \frac {2}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {-x^{2} + 1} \] Input:
integrate(x^7*log(-x^2+1)/(-x^2+1)^(1/2),x, algorithm="giac")
Output:
2/49*(x^2 - 1)^3*sqrt(-x^2 + 1) + 6/25*(x^2 - 1)^2*sqrt(-x^2 + 1) - 1/35*( 5*(x^2 - 1)^3*sqrt(-x^2 + 1) + 21*(x^2 - 1)^2*sqrt(-x^2 + 1) - 35*(-x^2 + 1)^(3/2) + 35*sqrt(-x^2 + 1))*log(-x^2 + 1) - 2/3*(-x^2 + 1)^(3/2) + 2*sqr t(-x^2 + 1)
Timed out. \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\int \frac {x^7\,\ln \left (1-x^2\right )}{\sqrt {1-x^2}} \,d x \] Input:
int((x^7*log(1 - x^2))/(1 - x^2)^(1/2),x)
Output:
int((x^7*log(1 - x^2))/(1 - x^2)^(1/2), x)
Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.52 \[ \int \frac {x^7 \log \left (1-x^2\right )}{\sqrt {1-x^2}} \, dx=\frac {\sqrt {-x^{2}+1}\, \left (-525 \,\mathrm {log}\left (-x^{2}+1\right ) x^{6}-630 \,\mathrm {log}\left (-x^{2}+1\right ) x^{4}-840 \,\mathrm {log}\left (-x^{2}+1\right ) x^{2}-1680 \,\mathrm {log}\left (-x^{2}+1\right )+150 x^{6}+432 x^{4}+1136 x^{2}+5632\right )}{3675} \] 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 - 840*log( - x**2 + 1)*x**2 - 1680*log( - x**2 + 1) + 150*x**6 + 432*x* *4 + 1136*x**2 + 5632))/3675