\(\int \frac {(-1+2 k^2) x-2 k^4 x^3+k^4 x^5}{((1-x^2) (1-k^2 x^2))^{2/3} (1-d+(d-2 k^2) x^2+k^4 x^4)} \, dx\) [2676]

Optimal result

Integrand size = 73, antiderivative size = 241 \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{4/3}\right )}{4 \sqrt [3]{d}} \]

[Out]

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

Rubi [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {\$Aborted} \]

[In]

Int[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d + (d - 2*k^2)*x^2 + k^4*x^4
)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 16.78 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (-1+k^2 x^2\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-1+k^2 x^2\right )^{2/3}}{-2 \sqrt [3]{d} \sqrt [3]{-1+x^2}+\left (-1+k^2 x^2\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{d} \sqrt [3]{-1+x^2}+\left (-1+k^2 x^2\right )^{2/3}\right )+\log \left (d^{2/3} \left (-1+x^2\right )^{2/3}-\sqrt [3]{d} \sqrt [3]{-1+x^2} \left (-1+k^2 x^2\right )^{2/3}+\left (-1+k^2 x^2\right )^{4/3}\right )\right )}{4 \sqrt [3]{d} \left (\left (-1+x^2\right ) \left (-1+k^2 x^2\right )\right )^{2/3}} \]

[In]

Integrate[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d + (d - 2*k^2)*x^2 + k
^4*x^4)),x]

[Out]

((-1 + x^2)^(2/3)*(-1 + k^2*x^2)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + k^2*x^2)^(2/3))/(-2*d^(1/3)*(-1 + x^2
)^(1/3) + (-1 + k^2*x^2)^(2/3))] - 2*Log[d^(1/3)*(-1 + x^2)^(1/3) + (-1 + k^2*x^2)^(2/3)] + Log[d^(2/3)*(-1 +
x^2)^(2/3) - d^(1/3)*(-1 + x^2)^(1/3)*(-1 + k^2*x^2)^(2/3) + (-1 + k^2*x^2)^(4/3)]))/(4*d^(1/3)*((-1 + x^2)*(-
1 + k^2*x^2))^(2/3))

Maple [F]

\[\int \frac {\left (2 k^{2}-1\right ) x -2 k^{4} x^{3}+k^{4} x^{5}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d +\left (-2 k^{2}+d \right ) x^{2}+k^{4} x^{4}\right )}d x\]

[In]

int(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2*k^2+d)*x^2+k^4*x^4),x)

[Out]

int(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2*k^2+d)*x^2+k^4*x^4),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2*k^2+d)*x^2+k^4*x^4),x, algori
thm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(((2*k**2-1)*x-2*k**4*x**3+k**4*x**5)/((-x**2+1)*(-k**2*x**2+1))**(2/3)/(1-d+(-2*k**2+d)*x**2+k**4*x*
*4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{5} - 2 \, k^{4} x^{3} + {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2*k^2+d)*x^2+k^4*x^4),x, algori
thm="maxima")

[Out]

integrate((k^4*x^5 - 2*k^4*x^3 + (2*k^2 - 1)*x)/((k^4*x^4 - (2*k^2 - d)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))
^(2/3)), x)

Giac [F]

\[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{5} - 2 \, k^{4} x^{3} + {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2*k^2+d)*x^2+k^4*x^4),x, algori
thm="giac")

[Out]

integrate((k^4*x^5 - 2*k^4*x^3 + (2*k^2 - 1)*x)/((k^4*x^4 - (2*k^2 - d)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))
^(2/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int \frac {k^4\,x^5-2\,k^4\,x^3+x\,\left (2\,k^2-1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (k^4\,x^4-d+x^2\,\left (d-2\,k^2\right )+1\right )} \,d x \]

[In]

int((k^4*x^5 - 2*k^4*x^3 + x*(2*k^2 - 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(k^4*x^4 - d + x^2*(d - 2*k^2) + 1)
),x)

[Out]

int((k^4*x^5 - 2*k^4*x^3 + x*(2*k^2 - 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(k^4*x^4 - d + x^2*(d - 2*k^2) + 1)
), x)