\(\int \frac {(1-3 k^2) x+2 k^2 x^3}{\sqrt [4]{(1-x^2) (1-k^2 x^2)} (-1+d+(-d+3 k^2) x^2-3 k^4 x^4+k^6 x^6)} \, dx\) [1363]

Optimal result

Integrand size = 74, antiderivative size = 98 \[ \int \frac {\left (1-3 k^2\right ) x+2 k^2 x^3}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-d+3 k^2\right ) x^2-3 k^4 x^4+k^6 x^6\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+k^2 x^2}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{-1+k^2 x^2}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F(-1)]

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

[In]

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

[Out]

$Aborted

Rubi steps Aborted

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

integrate(((-3*k^2+1)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(3*k^2-d)*x^2-3*k^4*x^4+k^6*x^6),x, alg
orithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate(((-3*k^2+1)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(3*k^2-d)*x^2-3*k^4*x^4+k^6*x^6),x, alg
orithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(((-3*k^2+1)*x+2*k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/4)/(-1+d+(3*k^2-d)*x^2-3*k^4*x^4+k^6*x^6),x, alg
orithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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