\(\int \frac {3+(1-2 k^2) x-3 k^2 x^2+k^2 x^3}{\sqrt [3]{(1-x^2) (1-k^2 x^2)} (1-d-(2+d) x+(1+d k^2) x^2+d k^2 x^3)} \, dx\) [2545]

Optimal result

Integrand size = 81, antiderivative size = 214 \[ \int \frac {3+\left (1-2 k^2\right ) x-3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2-2 x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}-\frac {\log \left (-1+x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}+\frac {\log \left (1-2 x+x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(x*(1 - x^2)^(1/3)*(1 - k^2*x^2)^(1/3)*AppellF1[1/2, 1/3, 1/3, 3/2, x^2, k^2*x^2])/(d*(1 - (1 + k^2)*x^2 + k^2
*x^4)^(1/3)) + ((1 - 4*d)*Defer[Int][1/((-1 + d + (2 + d)*x - (1 + d*k^2)*x^2 - d*k^2*x^3)*(1 + (-1 - k^2)*x^2
+ k^2*x^4)^(1/3)), x])/d + (2*(1 + d - d*k^2)*Defer[Int][x/((1 - d - (2 + d)*x + (1 + d*k^2)*x^2 + d*k^2*x^3)
*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)), x])/d - ((1 + 4*d*k^2)*Defer[Int][x^2/((1 - d - (2 + d)*x + (1 + d*k^2
)*x^2 + d*k^2*x^3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(1/3)), x])/d

Rubi steps \begin{align*} \text {integral}& = \int \frac {3+\left (1-2 k^2\right ) x-3 k^2 x^2+k^2 x^3}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {1}{d \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}-\frac {1-4 d-2 \left (1+d \left (1-k^2\right )\right ) x+\left (1+4 d k^2\right ) x^2}{d \left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = \frac {\int \frac {1}{\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d}-\frac {\int \frac {1-4 d-2 \left (1+d \left (1-k^2\right )\right ) x+\left (1+4 d k^2\right ) x^2}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d} \\ & = -\frac {\int \frac {1-4 d-2 \left (1+d-d k^2\right ) x+\left (1+4 d k^2\right ) x^2}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d}+\frac {\left (\sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}\right ) \int \frac {1}{\sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2}} \, dx}{d \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \\ & = \frac {x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{d \sqrt [3]{1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {\int \left (\frac {4 \left (1-\frac {1}{4 d}\right ) d}{\left (-1+d+(2+d) x-\left (1+d k^2\right ) x^2-d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 \left (-1-d \left (1-k^2\right )\right ) x}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {\left (1+4 d k^2\right ) x^2}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx}{d} \\ & = \frac {x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{d \sqrt [3]{1-\left (1+k^2\right ) x^2+k^2 x^4}}-\frac {(-1+4 d) \int \frac {1}{\left (-1+d+(2+d) x-\left (1+d k^2\right ) x^2-d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d}-\frac {\left (1+4 d k^2\right ) \int \frac {x^2}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d}-\frac {\left (2 \left (-1-d \left (1-k^2\right )\right )\right ) \int \frac {x}{\left (1-d-(2+d) x+\left (1+d k^2\right ) x^2+d k^2 x^3\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx}{d} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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