\(\int \frac {x^4 \log (1-x^2)}{\sqrt {-1+x^2}} \, dx\) [656]

Optimal result

Integrand size = 21, antiderivative size = 280 \[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\frac {1}{128 \left (x+\sqrt {-1+x^2}\right )^4}+\frac {5}{32 \left (x+\sqrt {-1+x^2}\right )^2}-\frac {5}{32} \left (x+\sqrt {-1+x^2}\right )^2-\frac {1}{128} \left (x+\sqrt {-1+x^2}\right )^4-\frac {\log \left (1-x^2\right )}{64 \left (x+\sqrt {-1+x^2}\right )^4}-\frac {\log \left (1-x^2\right )}{8 \left (x+\sqrt {-1+x^2}\right )^2}+\frac {1}{8} \left (x+\sqrt {-1+x^2}\right )^2 \log \left (1-x^2\right )+\frac {1}{64} \left (x+\sqrt {-1+x^2}\right )^4 \log \left (1-x^2\right )-\frac {9}{16} \log \left (x+\sqrt {-1+x^2}\right )+\frac {3}{8} \log \left (1-x^2\right ) \log \left (x+\sqrt {-1+x^2}\right )+\frac {3}{8} \log ^2\left (x+\sqrt {-1+x^2}\right )-\frac {3}{4} \log \left (x+\sqrt {-1+x^2}\right ) \log \left (1-\left (x+\sqrt {-1+x^2}\right )^2\right )-\frac {3}{8} \operatorname {PolyLog}\left (2,\left (x+\sqrt {-1+x^2}\right )^2\right ) \] Output:

1/128/(x+(x^2-1)^(1/2))^4+5/32/(x+(x^2-1)^(1/2))^2-5/32*(x+(x^2-1)^(1/2))^ 
2-1/128*(x+(x^2-1)^(1/2))^4-1/64*ln(-x^2+1)/(x+(x^2-1)^(1/2))^4-1/8*ln(-x^ 
2+1)/(x+(x^2-1)^(1/2))^2+1/8*(x+(x^2-1)^(1/2))^2*ln(-x^2+1)+1/64*(x+(x^2-1 
)^(1/2))^4*ln(-x^2+1)-9/16*ln(x+(x^2-1)^(1/2))+3/8*ln(-x^2+1)*ln(x+(x^2-1) 
^(1/2))+3/8*ln(x+(x^2-1)^(1/2))^2-3/4*ln(x+(x^2-1)^(1/2))*ln(1-(x+(x^2-1)^ 
(1/2))^2)-3/8*polylog(2,(x+(x^2-1)^(1/2))^2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=-\frac {x \sqrt {1-x^2} \left (-2 \left (x^2\right )^{3/2} \sqrt {1-x^2}-9 \sqrt {x^2-x^4}-6 i \arcsin \left (\sqrt {1-x^2}\right )^2-3 \arcsin \left (\sqrt {1-x^2}\right ) \left (3+4 \log \left (1-e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )+4 \left (x^2\right )^{3/2} \sqrt {1-x^2} \log \left (1-x^2\right )+6 \sqrt {x^2-x^4} \log \left (1-x^2\right )-6 i \log \left (1-x^2\right ) \log \left (\sqrt {x^2}+i \sqrt {1-x^2}\right )-6 i \operatorname {PolyLog}\left (2,e^{-2 i \arcsin \left (\sqrt {1-x^2}\right )}\right )\right )}{16 \sqrt {x^2} \sqrt {-1+x^2}} \] Input:

Integrate[(x^4*Log[1 - x^2])/Sqrt[-1 + x^2],x]
 

Output:

-1/16*(x*Sqrt[1 - x^2]*(-2*(x^2)^(3/2)*Sqrt[1 - x^2] - 9*Sqrt[x^2 - x^4] - 
 (6*I)*ArcSin[Sqrt[1 - x^2]]^2 - 3*ArcSin[Sqrt[1 - x^2]]*(3 + 4*Log[1 - E^ 
((-2*I)*ArcSin[Sqrt[1 - x^2]])]) + 4*(x^2)^(3/2)*Sqrt[1 - x^2]*Log[1 - x^2 
] + 6*Sqrt[x^2 - x^4]*Log[1 - x^2] - (6*I)*Log[1 - x^2]*Log[Sqrt[x^2] + I* 
Sqrt[1 - x^2]] - (6*I)*PolyLog[2, E^((-2*I)*ArcSin[Sqrt[1 - x^2]])]))/(Sqr 
t[x^2]*Sqrt[-1 + x^2])
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {x^2-1}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {x^2-1}}dx\)

Input:

Int[(x^4*Log[1 - x^2])/Sqrt[-1 + x^2],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2929

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]
 

Maple [F]

\[\int \frac {x^{4} \ln \left (-x^{2}+1\right )}{\sqrt {x^{2}-1}}d x\]

Input:

int(x^4*ln(-x^2+1)/(x^2-1)^(1/2),x)
 

Output:

int(x^4*ln(-x^2+1)/(x^2-1)^(1/2),x)
 

Fricas [F]

\[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {x^{4} \log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:

integrate(x^4*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="fricas")
 

Output:

integral(x^4*log(-x^2 + 1)/sqrt(x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {x^{4} \log {\left (1 - x^{2} \right )}}{\sqrt {\left (x - 1\right ) \left (x + 1\right )}}\, dx \] Input:

integrate(x**4*ln(-x**2+1)/(x**2-1)**(1/2),x)
 

Output:

Integral(x**4*log(1 - x**2)/sqrt((x - 1)*(x + 1)), x)
 

Maxima [F]

\[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {x^{4} \log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:

integrate(x^4*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^4*log(-x^2 + 1)/sqrt(x^2 - 1), x)
 

Giac [F]

\[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int { \frac {x^{4} \log \left (-x^{2} + 1\right )}{\sqrt {x^{2} - 1}} \,d x } \] Input:

integrate(x^4*log(-x^2+1)/(x^2-1)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^4*log(-x^2 + 1)/sqrt(x^2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {x^4\,\ln \left (1-x^2\right )}{\sqrt {x^2-1}} \,d x \] Input:

int((x^4*log(1 - x^2))/(x^2 - 1)^(1/2),x)
 

Output:

int((x^4*log(1 - x^2))/(x^2 - 1)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \log \left (1-x^2\right )}{\sqrt {-1+x^2}} \, dx=\int \frac {\mathrm {log}\left (-x^{2}+1\right ) x^{4}}{\sqrt {x^{2}-1}}d x \] Input:

int(x^4*log(-x^2+1)/(x^2-1)^(1/2),x)
 

Output:

int((log( - x**2 + 1)*x**4)/sqrt(x**2 - 1),x)