Integrand size = 76, antiderivative size = 223 \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2-2 k x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}-\frac {\log \left (-1+k x+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}+\frac {\log \left (1-2 k x+k^2 x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} k x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \] 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*k*x+d^ (1/3)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)))/d^(2/3)-ln(-1+k*x+d^(1/3)*(1+(-k^2- 1)*x^2+k^2*x^4)^(1/3))/d^(2/3)+1/2*ln(1-2*k*x+k^2*x^2+(d^(1/3)-d^(1/3)*k*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)
\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx \] Input:
Integrate[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^ 2))^(1/3)*(1 - d - (2 + d)*k*x + (d + k^2)*x^2 + d*k*x^3)),x]
Output:
Integrate[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^ 2))^(1/3)*(1 - d - (2 + d)*k*x + (d + k^2)*x^2 + d*k*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.
| \(\displaystyle \int \frac {k^2 x^3+\left (k^2-2\right ) x-3 k x^2+3 k}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (x^2 \left (d+k^2\right )+d k x^3-(d+2) k x-d+1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {k^2 x^3+\left (k^2-2\right ) x-3 k x^2+3 k}{\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (x^2 \left (d+k^2\right )+d k x^3-(d+2) k x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {k^2 x^3-\left (2-k^2\right ) x-3 k x^2+3 k}{\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (x^2 \left (d+k^2\right )+d k x^3-(d+2) k x-d+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {k}{d \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}-\frac {k x^2 \left (4 d+k^2\right )-2 x \left (k^2-d \left (1-k^2\right )\right )-4 d k+k}{d \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (x^2 \left (d+k^2\right )+d k x^3-(d+2) k x-d+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(1-4 d) k \int \frac {1}{\left (d k x^3+\left (k^2+d\right ) x^2-(d+2) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}+\frac {2 \left (k^2-d \left (1-k^2\right )\right ) \int \frac {x}{\left (d k x^3+\left (k^2+d\right ) x^2-(d+2) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}-\frac {k \left (4 d+k^2\right ) \int \frac {x^2}{\left (d k x^3+\left (k^2+d\right ) x^2-(d+2) k x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx}{d}+\frac {k x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{d \sqrt [3]{k^2 x^4-\left (k^2+1\right ) x^2+1}}\) |
Input:
Int[(3*k + (-2 + k^2)*x - 3*k*x^2 + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1 /3)*(1 - d - (2 + d)*k*x + (d + k^2)*x^2 + d*k*x^3)),x]
Output:
$Aborted
\[\int \frac {3 k +\left (k^{2}-2\right ) x -3 k \,x^{2}+k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (1-d -\left (2+d \right ) k x +\left (k^{2}+d \right ) x^{2}+d k \,x^{3}\right )}d x\]
Input:
int((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1-d-(2+ d)*k*x+(k^2+d)*x^2+d*k*x^3),x)
Output:
int((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1-d-(2+ d)*k*x+(k^2+d)*x^2+d*k*x^3),x)
Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1 -d-(2+d)*k*x+(k^2+d)*x^2+d*k*x^3),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((3*k+(k**2-2)*x-3*k*x**2+k**2*x**3)/((-x**2+1)*(-k**2*x**2+1))** (1/3)/(1-d-(2+d)*k*x+(k**2+d)*x**2+d*k*x**3),x)
Output:
Timed out
\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int { \frac {k^{2} x^{3} - 3 \, k x^{2} + {\left (k^{2} - 2\right )} x + 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{2} + d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1 -d-(2+d)*k*x+(k^2+d)*x^2+d*k*x^3),x, algorithm="maxima")
Output:
integrate((k^2*x^3 - 3*k*x^2 + (k^2 - 2)*x + 3*k)/((d*k*x^3 - (d + 2)*k*x + (k^2 + d)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(1/3)), x)
\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int { \frac {k^{2} x^{3} - 3 \, k x^{2} + {\left (k^{2} - 2\right )} x + 3 \, k}{{\left (d k x^{3} - {\left (d + 2\right )} k x + {\left (k^{2} + d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \] Input:
integrate((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1 -d-(2+d)*k*x+(k^2+d)*x^2+d*k*x^3),x, algorithm="giac")
Output:
integrate((k^2*x^3 - 3*k*x^2 + (k^2 - 2)*x + 3*k)/((d*k*x^3 - (d + 2)*k*x + (k^2 + d)*x^2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(1/3)), x)
Timed out. \[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx=\int \frac {3\,k+x\,\left (k^2-2\right )+k^2\,x^3-3\,k\,x^2}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (d\,k\,x^3+\left (k^2+d\right )\,x^2-k\,\left (d+2\right )\,x-d+1\right )} \,d x \] Input:
int((3*k + x*(k^2 - 2) + k^2*x^3 - 3*k*x^2)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/ 3)*(x^2*(d + k^2) - d - k*x*(d + 2) + d*k*x^3 + 1)),x)
Output:
int((3*k + x*(k^2 - 2) + k^2*x^3 - 3*k*x^2)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/ 3)*(x^2*(d + k^2) - d - k*x*(d + 2) + d*k*x^3 + 1)), x)
\[ \int \frac {3 k+\left (-2+k^2\right ) x-3 k x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d-(2+d) k x+\left (d+k^2\right ) x^2+d k x^3\right )} \, dx =\text {Too large to display} \] Input:
int((3*k+(k^2-2)*x-3*k*x^2+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(1-d-(2+ d)*k*x+(k^2+d)*x^2+d*k*x^3),x)
Output:
int(x**3/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x**3 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)* *(1/3)*d*x**2 - (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)*k**2*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x)*k**2 - 3*i nt(x**2/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x**3 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)** (1/3)*d*x**2 - (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)*k**2*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x)*k + int(x/( (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x**3 - (k**2*x**4 - k**2*x** 2 - x**2 + 1)**(1/3)*d*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*x **2 - (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)*k**2*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3 )*k*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*x**3 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*k*x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*d*x**2 - (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)*k**2*x**2 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)*k* x + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(1/3)),x) + 3*int(1/((k**2*x**4...