\(\int (\frac {1}{(1-x^2) \arccos (x)^2}-\frac {x}{(1-x^2)^{3/2} \arccos (x)}) \, dx\) [381]

Optimal result

Integrand size = 33, antiderivative size = 17 \[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=-\frac {1}{\sqrt {1-x^2} \arccos (x)} \] Output:

-1/(-x^2+1)^(1/2)/arccos(x)
 

Mathematica [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx \] Input:

Integrate[1/((1 - x^2)*ArcCos[x]^2) - x/((1 - x^2)^(3/2)*ArcCos[x]),x]
 

Output:

Integrate[1/((1 - x^2)*ArcCos[x]^2) - x/((1 - x^2)^(3/2)*ArcCos[x]), x]
 

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.

\(\displaystyle \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{\sqrt {1-x^2} \arccos (x)}-2 \int \frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}dx\)

Input:

Int[1/((1 - x^2)*ArcCos[x]^2) - x/((1 - x^2)^(3/2)*ArcCos[x]),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \left (\frac {1}{\left (-x^{2}+1\right ) \arccos \left (x \right )^{2}}-\frac {x}{\left (-x^{2}+1\right )^{\frac {3}{2}} \arccos \left (x \right )}\right )d x\]

Input:

int(1/(-x^2+1)/arccos(x)^2-x/(-x^2+1)^(3/2)/arccos(x),x)
 

Output:

int(1/(-x^2+1)/arccos(x)^2-x/(-x^2+1)^(3/2)/arccos(x),x)
 

Fricas [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\int { -\frac {x}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )} - \frac {1}{{\left (x^{2} - 1\right )} \arccos \left (x\right )^{2}} \,d x } \] Input:

integrate(1/(-x^2+1)/arccos(x)^2-x/(-x^2+1)^(3/2)/arccos(x),x, algorithm=" 
fricas")
                                                                                    
                                                                                    
 

Output:

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

Sympy [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x \operatorname {acos}{\left (x \right )} - \sqrt {1 - x^{2}}\right )}{\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}} \operatorname {acos}^{2}{\left (x \right )}}\, dx \] Input:

integrate(1/(-x**2+1)/acos(x)**2-x/(-x**2+1)**(3/2)/acos(x),x)
 

Output:

Integral((x - 1)*(x + 1)*(x*acos(x) - sqrt(1 - x**2))/((-(x - 1)*(x + 1))* 
*(5/2)*acos(x)**2), x)
 

Maxima [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\int { -\frac {x}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )} - \frac {1}{{\left (x^{2} - 1\right )} \arccos \left (x\right )^{2}} \,d x } \] Input:

integrate(1/(-x^2+1)/arccos(x)^2-x/(-x^2+1)^(3/2)/arccos(x),x, algorithm=" 
maxima")
 

Output:

-(2*(x^2 - 1)*arctan2(sqrt(x + 1)*sqrt(-x + 1), x)*integrate(sqrt(x + 1)*x 
*sqrt(-x + 1)/((x^4 - 2*x^2 + 1)*arctan2(sqrt(x + 1)*sqrt(-x + 1), x)), x) 
 + sqrt(x + 1)*sqrt(-x + 1))/((x^2 - 1)*arctan2(sqrt(x + 1)*sqrt(-x + 1), 
x))
 

Giac [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\int { -\frac {x}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )} - \frac {1}{{\left (x^{2} - 1\right )} \arccos \left (x\right )^{2}} \,d x } \] Input:

integrate(1/(-x^2+1)/arccos(x)^2-x/(-x^2+1)^(3/2)/arccos(x),x, algorithm=" 
giac")
 

Output:

integrate(-x/((-x^2 + 1)^(3/2)*arccos(x)) - 1/((x^2 - 1)*arccos(x)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=-\int \frac {1}{{\mathrm {acos}\left (x\right )}^2\,\left (x^2-1\right )}+\frac {x}{\mathrm {acos}\left (x\right )\,{\left (1-x^2\right )}^{3/2}} \,d x \] Input:

int(- 1/(acos(x)^2*(x^2 - 1)) - x/(acos(x)*(1 - x^2)^(3/2)),x)
 

Output:

-int(1/(acos(x)^2*(x^2 - 1)) + x/(acos(x)*(1 - x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \left (\frac {1}{\left (1-x^2\right ) \arccos (x)^2}-\frac {x}{\left (1-x^2\right )^{3/2} \arccos (x)}\right ) \, dx=\frac {2 \sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right ) \left (\int \frac {x}{\sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right ) x^{2}-\sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right )}d x \right )+1}{\sqrt {-x^{2}+1}\, \mathit {acos} \left (x \right )} \] Input:

int(1/(-x^2+1)/acos(x)^2-x/(-x^2+1)^(3/2)/acos(x),x)
 

Output:

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