\(\int \frac {x^2}{\sqrt {(1-x^2) (1-k^2 x^2)} (-1+k^2 x^4)} \, dx\) [1306]

Optimal result

Integrand size = 37, antiderivative size = 94 \[ \int \frac {x^2}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=-\frac {\arctan \left (\frac {(-1+k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{4 (-1+k) k}+\frac {\arctan \left (\frac {(1+k) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 k (1+k)} \] Output:

-1/4*arctan((-1+k)*x/(1+(-k^2-1)*x^2+k^2*x^4)^(1/2))/(-1+k)/k+1/2*arctan(( 
1+k)*x/(1+k*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/2)))/k/(1+k)
 

Mathematica [A] (verified)

Time = 5.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\frac {-\frac {\arctan \left (\frac {(-1+k) x}{\sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}}\right )}{-1+k}+\frac {2 \arctan \left (\frac {(1+k) x}{1+k x^2+\sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}}\right )}{1+k}}{4 k} \] Input:

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

Output:

(-(ArcTan[((-1 + k)*x)/Sqrt[(-1 + x^2)*(-1 + k^2*x^2)]]/(-1 + k)) + (2*Arc 
Tan[((1 + k)*x)/(1 + k*x^2 + Sqrt[(-1 + x^2)*(-1 + k^2*x^2)])])/(1 + k))/( 
4*k)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2048, 7276, 2009}

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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {(k+1) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{4 k (k+1)}-\frac {\arctan \left (\frac {(1-k) x}{\sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{4 (1-k) k}\)

Input:

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

Output:

-1/4*ArcTan[((1 - k)*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/((1 - k)*k) + A 
rcTan[((1 + k)*x)/Sqrt[1 - (1 + k^2)*x^2 + k^2*x^4]]/(4*k*(1 + k))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 3.92 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (1+k \right )}\right )}{4 k \left (1+k \right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{4 k \left (-1+k \right )}\right ) \sqrt {2}}{2}\)	\(93\)
default	\(\frac {-\frac {\sqrt {-\left (1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )}{2}-\frac {\sqrt {-\left (1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )}{2}+\sqrt {-\left (-1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-x \left (1+k \right )^{2}}{k \,x^{2}+1}\right )+\ln \left (2\right ) \left (\sqrt {-\left (-1+k \right )^{2}}-\sqrt {-\left (1+k \right )^{2}}\right )}{4 \sqrt {-\left (1+k \right )^{2}}\, \sqrt {-\left (-1+k \right )^{2}}\, k}\)	\(268\)
pseudoelliptic	\(\frac {-\frac {\sqrt {-\left (1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )}{2}-\frac {\sqrt {-\left (1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )}{2}+\sqrt {-\left (-1+k \right )^{2}}\, \ln \left (\frac {\sqrt {-\left (1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-x \left (1+k \right )^{2}}{k \,x^{2}+1}\right )+\ln \left (2\right ) \left (\sqrt {-\left (-1+k \right )^{2}}-\sqrt {-\left (1+k \right )^{2}}\right )}{4 \sqrt {-\left (1+k \right )^{2}}\, \sqrt {-\left (-1+k \right )^{2}}\, k}\)	\(268\)

Input:

int(x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^4-1),x,method=_RETURNVERBOSE)
 

Output:

1/2*(-1/4/k*2^(1/2)/(1+k)*arctan(((-x^2+1)*(-k^2*x^2+1))^(1/2)/x/(1+k))+1/ 
4/k*2^(1/2)/(-1+k)*arctan(((-x^2+1)*(-k^2*x^2+1))^(1/2)/x/(-1+k)))*2^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\frac {{\left (k - 1\right )} \arctan \left (\frac {{\left (k + 1\right )} x}{\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}\right ) - {\left (k + 1\right )} \arctan \left (\frac {{\left (k - 1\right )} x}{\sqrt {k^{2} x^{4} - {\left (k^{2} + 1\right )} x^{2} + 1}}\right )}{4 \, {\left (k^{3} - k\right )}} \] Input:

integrate(x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^4-1),x, algorithm="fric 
as")
 

Output:

1/4*((k - 1)*arctan((k + 1)*x/sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)) - (k + 1) 
*arctan((k - 1)*x/sqrt(k^2*x^4 - (k^2 + 1)*x^2 + 1)))/(k^3 - k)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^2/((-x^2+1)*(-k^2*x^2+1))^(1/2)/(k^2*x^4-1),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+k^2 x^4\right )} \, dx=\int \frac {\sqrt {k^{2} x^{2}-1}\, \sqrt {x^{2}-1}\, x^{2}}{k^{4} x^{8}-k^{4} x^{6}-k^{2} x^{6}+k^{2} x^{2}+x^{2}-1}d x \] Input:

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

Output:

int((sqrt(k**2*x**2 - 1)*sqrt(x**2 - 1)*x**2)/(k**4*x**8 - k**4*x**6 - k** 
2*x**6 + k**2*x**2 + x**2 - 1),x)