Integrand size = 80, antiderivative size = 212 \[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2+2 x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}+\frac {\log \left (1+x-\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}-\frac {\log \left (1+2 x+x^2+\left (\sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \] Output:
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)
\[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx \] Input:
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]
Output:
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {k^2 x^3+3 k^2 x^2+\left (1-2 k^2\right ) x-3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {k^2 x^3+3 k^2 x^2+\left (1-2 k^2\right ) x-3}{\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{d \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}+\frac {x^2 \left (4 d k^2+1\right )+2 x \left (d \left (1-k^2\right )+1\right )-4 d+1}{d \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(1-4 d) \int \frac {1}{\left (-d k^2 x^3+\left (d k^2+1\right ) x^2+(d+2) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}-\frac {2 \left (-d k^2+d+1\right ) \int \frac {x}{\left (-d k^2 x^3+\left (d k^2+1\right ) x^2+(d+2) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}-\frac {\left (4 d k^2+1\right ) \int \frac {x^2}{\left (-d k^2 x^3+\left (d k^2+1\right ) x^2+(d+2) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}+\frac {x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{d \sqrt [3]{k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
Input:
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]
Output:
$Aborted
\[\int \frac {-3+\left (-2 k^{2}+1\right ) x +3 k^{2} x^{2}+k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (-1+d -\left (2+d \right ) x -\left (d \,k^{2}+1\right ) x^{2}+d \,k^{2} x^{3}\right )}d x\]
Input:
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)
Output:
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)
Timed out. \[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\text {Timed out} \] Input:
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")
Output:
Timed out
Timed out. \[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\text {Timed out} \] Input:
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)
Output:
Timed out
\[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{3} + 3 \, k^{2} x^{2} - {\left (2 \, k^{2} - 1\right )} x - 3}{{\left (d k^{2} x^{3} - {\left (d k^{2} + 1\right )} x^{2} - {\left (d + 2\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \] Input:
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")
Output:
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)
\[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{3} + 3 \, k^{2} x^{2} - {\left (2 \, k^{2} - 1\right )} x - 3}{{\left (d k^{2} x^{3} - {\left (d k^{2} + 1\right )} x^{2} - {\left (d + 2\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \] Input:
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")
Output:
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)
Timed out. \[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int -\frac {3\,k^2\,x^2+k^2\,x^3-x\,\left (2\,k^2-1\right )-3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (x^2\,\left (d\,k^2+1\right )-d+x\,\left (d+2\right )-d\,k^2\,x^3+1\right )} \,d x \] Input:
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)
Output:
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)
\[ \int \frac {-3+\left (1-2 k^2\right ) x+3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx =\text {Too large to display} \] Input:
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)
Output:
int(x**3/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**3 - (k**2*x* *4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x **2 + 1)**(1/3)*d*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d - (k**2* x**4 - k**2*x**2 - x**2 + 1)**(1/3)*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*x - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x)*k**2 + 3*in t(x**2/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**3 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x** 2 + 1)**(1/3)*d*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d - (k**2*x* *4 - k**2*x**2 - x**2 + 1)**(1/3)*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*x - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x)*k**2 - 2*int( x/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**3 - (k**2*x**4 - k* *2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x**2 + 1 )**(1/3)*d*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)** (1/3)*x - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x)*k**2 + int(x/((k** 2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**3 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3 )*d*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d - (k**2*x**4 - k**2*x* *2 - x**2 + 1)**(1/3)*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*x - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x) - 3*int(1/((k**2*x**4 ...