Integrand size = 66, antiderivative size = 256 \[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} k^2 x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} k^2 x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} k^2 x^2+d^{2/3} k^4 x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} k^2 x^2\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{4/3}\right )}{4 \sqrt [3]{d}} \] Output:
-1/2*3^(1/2)*arctan(3^(1/2)*(1+(-k^2-1)*x^2+k^2*x^4)^(2/3)/(2*d^(1/3)-2*d^ (1/3)*k^2*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(2/3)))/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)^(2/3))/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)^(2/3)+(1+(-k^2-1)*x^2+k^2*x^4)^(4/3))/d^(1/3)
Time = 14.73 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.77 \[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (-1+k^2 x^2\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-1+x^2\right )^{2/3}}{\left (-1+x^2\right )^{2/3}-2 \sqrt [3]{d} \sqrt [3]{-1+k^2 x^2}}\right )-2 \log \left (\left (-1+x^2\right )^{2/3}+\sqrt [3]{d} \sqrt [3]{-1+k^2 x^2}\right )+\log \left (\left (-1+x^2\right )^{4/3}-\sqrt [3]{d} \left (-1+x^2\right )^{2/3} \sqrt [3]{-1+k^2 x^2}+d^{2/3} \left (-1+k^2 x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{d} \left (\left (-1+x^2\right ) \left (-1+k^2 x^2\right )\right )^{2/3}} \] Input:
Integrate[((2 - k^2)*x - 2*x^3 + k^2*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3) *(1 - d + (-2 + d*k^2)*x^2 + x^4)),x]
Output:
((-1 + x^2)^(2/3)*(-1 + k^2*x^2)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x ^2)^(2/3))/((-1 + x^2)^(2/3) - 2*d^(1/3)*(-1 + k^2*x^2)^(1/3))] - 2*Log[(- 1 + x^2)^(2/3) + d^(1/3)*(-1 + k^2*x^2)^(1/3)] + Log[(-1 + x^2)^(4/3) - d^ (1/3)*(-1 + x^2)^(2/3)*(-1 + k^2*x^2)^(1/3) + d^(2/3)*(-1 + k^2*x^2)^(2/3) ]))/(4*d^(1/3)*((-1 + x^2)*(-1 + k^2*x^2))^(2/3))
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.
| \(\displaystyle \int \frac {k^2 x^5+\left (2-k^2\right ) x-2 x^3}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (x^2 \left (d k^2-2\right )-d+x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (k^2 x^4-k^2-2 x^2+2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (x^2 \left (d k^2-2\right )-d+x^4+1\right )}dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {x \left (k^2 x^4-k^2-2 x^2+2\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (x^2 \left (d k^2-2\right )-d+x^4+1\right )}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {1}{2} \int \frac {k^2 x^4-2 x^2-k^2+2}{\left (x^4-\left (2-d k^2\right ) x^2-d+1\right ) \left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}dx^2\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {k^2}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}+\frac {-\left ((2-d) k^2\right )-\left (d k^4-2 k^2+2\right ) x^2+2}{\left (x^4+\left (d k^2-2\right ) x^2-d+1\right ) \left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {-\left ((2-d) k^2\right )+\left (-d k^4+2 k^2-2\right ) x^2+2}{\left (x^4+\left (d k^2-2\right ) x^2-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx^2-\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} k^{4/3} \sqrt {\left (2 k^2 x^2-k^2-1\right )^2} \left (\left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right ) \sqrt {\frac {\left (k^2-1\right )^{4/3}+2 \sqrt [3]{2} k^{4/3} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}-2^{2/3} \left (k^2-1\right )^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\left (\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1-\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}{2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}\right ),-7-4 \sqrt {3}\right )}{\left (-2 k^2 x^2+k^2+1\right ) \sqrt {\left (-\left (k^2 \left (1-2 x^2\right )\right )-1\right )^2} \sqrt {\frac {\left (k^2-1\right )^{2/3} \left (\left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )}{\left (\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}}}\right )\) |
Input:
Int[((2 - k^2)*x - 2*x^3 + k^2*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(1 - d + (-2 + d*k^2)*x^2 + x^4)),x]
Output:
$Aborted
\[\int \frac {\left (-k^{2}+2\right ) x -2 x^{3}+k^{2} x^{5}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d +\left (d \,k^{2}-2\right ) x^{2}+x^{4}\right )}d x\]
Input:
int(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d*k^2-2 )*x^2+x^4),x)
Output:
int(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d*k^2-2 )*x^2+x^4),x)
Timed out. \[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d *k^2-2)*x^2+x^4),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(((-k**2+2)*x-2*x**3+k**2*x**5)/((-x**2+1)*(-k**2*x**2+1))**(2/3) /(1-d+(d*k**2-2)*x**2+x**4),x)
Output:
Timed out
\[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\int { \frac {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (x^{4} + {\left (d k^{2} - 2\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \] Input:
integrate(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d *k^2-2)*x^2+x^4),x, algorithm="maxima")
Output:
integrate((k^2*x^5 - 2*x^3 - (k^2 - 2)*x)/((x^4 + (d*k^2 - 2)*x^2 - d + 1) *((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)
\[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\int { \frac {k^{2} x^{5} - 2 \, x^{3} - {\left (k^{2} - 2\right )} x}{{\left (x^{4} + {\left (d k^{2} - 2\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \] Input:
integrate(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d *k^2-2)*x^2+x^4),x, algorithm="giac")
Output:
integrate((k^2*x^5 - 2*x^3 - (k^2 - 2)*x)/((x^4 + (d*k^2 - 2)*x^2 - d + 1) *((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)
Timed out. \[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=-\int \frac {x\,\left (k^2-2\right )-k^2\,x^5+2\,x^3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (x^4+\left (d\,k^2-2\right )\,x^2-d+1\right )} \,d x \] Input:
int(-(x*(k^2 - 2) - k^2*x^5 + 2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(x^2 *(d*k^2 - 2) - d + x^4 + 1)),x)
Output:
-int((x*(k^2 - 2) - k^2*x^5 + 2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3)*(x^2 *(d*k^2 - 2) - d + x^4 + 1)), x)
\[ \int \frac {\left (2-k^2\right ) x-2 x^3+k^2 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (-2+d k^2\right ) x^2+x^4\right )} \, dx=\left (\int \frac {x^{5}}{\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d \,k^{2} x^{2}-\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d +\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{4}-2 \left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{2}+\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}}}d x \right ) k^{2}-2 \left (\int \frac {x^{3}}{\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d \,k^{2} x^{2}-\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d +\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{4}-2 \left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{2}+\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}}}d x \right )-\left (\int \frac {x}{\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d \,k^{2} x^{2}-\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d +\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{4}-2 \left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{2}+\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}}}d x \right ) k^{2}+2 \left (\int \frac {x}{\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d \,k^{2} x^{2}-\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} d +\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{4}-2 \left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}} x^{2}+\left (k^{2} x^{4}-k^{2} x^{2}-x^{2}+1\right )^{\frac {2}{3}}}d x \right ) \] Input:
int(((-k^2+2)*x-2*x^3+k^2*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(d*k^2-2 )*x^2+x^4),x)
Output:
int(x**5/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*d*k**2*x**2 - (k**2*x* *4 - k**2*x**2 - x**2 + 1)**(2/3)*d + (k**2*x**4 - k**2*x**2 - x**2 + 1)** (2/3)*x**4 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**2 + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)),x)*k**2 - 2*int(x**3/((k**2*x**4 - k**2*x **2 - x**2 + 1)**(2/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x**2 + 1)**( 2/3)*d + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**4 - 2*(k**2*x**4 - k **2*x**2 - x**2 + 1)**(2/3)*x**2 + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/ 3)),x) - int(x/((k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*d*k**2*x**2 - (k **2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*d + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**4 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**2 + (k** 2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)),x)*k**2 + 2*int(x/((k**2*x**4 - k** 2*x**2 - x**2 + 1)**(2/3)*d*k**2*x**2 - (k**2*x**4 - k**2*x**2 - x**2 + 1) **(2/3)*d + (k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**4 - 2*(k**2*x**4 - k**2*x**2 - x**2 + 1)**(2/3)*x**2 + (k**2*x**4 - k**2*x**2 - x**2 + 1)** (2/3)),x)