\(\int \frac {(-3+k^2) x+2 k^2 x^3}{\sqrt [4]{(1-x^2) (1-k^2 x^2)} (-1+d+(3-d k^2) x^2-3 x^4+x^6)} \, dx\) [1248]

Optimal result

Integrand size = 64, antiderivative size = 90 \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+x^2}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+x^2}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F(-1)]

Timed out. \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\text {\$Aborted} \]

[In]

Int[((-3 + k^2)*x + 2*k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d + (3 - d*k^2)*x^2 - 3*x^4 + x^6)),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [A] (verified)

Time = 20.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\frac {\sqrt [4]{-1+x^2} \sqrt [4]{-1+k^2 x^2} \left (\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{-1+k^2 x^2}}{\left (-1+x^2\right )^{3/4}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{-1+k^2 x^2}}{\left (-1+x^2\right )^{3/4}}\right )\right )}{d^{3/4} \sqrt [4]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]

[In]

Integrate[((-3 + k^2)*x + 2*k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/4)*(-1 + d + (3 - d*k^2)*x^2 - 3*x^4 + x^6)
),x]

[Out]

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

Maple [F]

\[\int \frac {\left (k^{2}-3\right ) x +2 k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{4}} \left (-1+d +\left (-d \,k^{2}+3\right ) x^{2}-3 x^{4}+x^{6}\right )}d x\]

[In]

int(((k^2-3)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(-d*k^2+3)*x^2-3*x^4+x^6),x)

[Out]

int(((k^2-3)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(-d*k^2+3)*x^2-3*x^4+x^6),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate(((k^2-3)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(-d*k^2+3)*x^2-3*x^4+x^6),x, algorithm="fr
icas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\text {Timed out} \]

[In]

integrate(((k**2-3)*x+2*k**2*x**3)/((-x**2+1)*(-k**2*x**2+1))**(1/4)/(-1+d+(-d*k**2+3)*x**2-3*x**4+x**6),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\int { \frac {2 \, k^{2} x^{3} + {\left (k^{2} - 3\right )} x}{{\left (x^{6} - 3 \, x^{4} - {\left (d k^{2} - 3\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}} \,d x } \]

[In]

integrate(((k^2-3)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(-d*k^2+3)*x^2-3*x^4+x^6),x, algorithm="ma
xima")

[Out]

integrate((2*k^2*x^3 + (k^2 - 3)*x)/((x^6 - 3*x^4 - (d*k^2 - 3)*x^2 + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(1/4)),
x)

Giac [F]

\[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=\int { \frac {2 \, k^{2} x^{3} + {\left (k^{2} - 3\right )} x}{{\left (x^{6} - 3 \, x^{4} - {\left (d k^{2} - 3\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}} \,d x } \]

[In]

integrate(((k^2-3)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(-d*k^2+3)*x^2-3*x^4+x^6),x, algorithm="gi
ac")

[Out]

integrate((2*k^2*x^3 + (k^2 - 3)*x)/((x^6 - 3*x^4 - (d*k^2 - 3)*x^2 + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(1/4)),
x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-3+k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (3-d k^2\right ) x^2-3 x^4+x^6\right )} \, dx=-\int \frac {x\,\left (k^2-3\right )+2\,k^2\,x^3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/4}\,\left (-x^6+3\,x^4+\left (d\,k^2-3\right )\,x^2-d+1\right )} \,d x \]

[In]

int(-(x*(k^2 - 3) + 2*k^2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/4)*(x^2*(d*k^2 - 3) - d + 3*x^4 - x^6 + 1)),x)

[Out]

-int((x*(k^2 - 3) + 2*k^2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/4)*(x^2*(d*k^2 - 3) - d + 3*x^4 - x^6 + 1)), x)