\(\int \frac {(1-2 k^2) x+k^2 x^3}{\sqrt [3]{(1-x^2) (1-k^2 x^2)} (-1+d+(1-2 d k^2) x^2+d k^4 x^4)} \, dx\) [2748]

Optimal result

Integrand size = 65, antiderivative size = 256 \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} k^2 x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} k^2 x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{4 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (1-2 k^2+k^2 x^2\right )}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx \\ & = \int \frac {x \left (1-2 k^2+k^2 x^2\right )}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1-2 k^2+k^2 x}{\sqrt [3]{1+\left (-1-k^2\right ) x+k^2 x^2} \left (-1+d+\left (1-2 d k^2\right ) x+d k^4 x^2\right )} \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 20.69 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.78 \[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx=\frac {\sqrt [3]{-1+x^2} \sqrt [3]{-1+k^2 x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{\sqrt [3]{-1+x^2}-2 \sqrt [3]{d} \left (-1+k^2 x^2\right )^{2/3}}\right )+2 \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{d} \left (-1+k^2 x^2\right )^{2/3}\right )-\log \left (\left (-1+x^2\right )^{2/3}-\sqrt [3]{d} \sqrt [3]{-1+x^2} \left (-1+k^2 x^2\right )^{2/3}+d^{2/3} \left (-1+k^2 x^2\right )^{4/3}\right )\right )}{4 \sqrt [3]{d} \sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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