Integrand size = 85, antiderivative size = 224 \[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {-\sqrt {3} \sqrt [3]{d}+\sqrt {3} \sqrt [3]{d} x}{-\sqrt [3]{d}+\sqrt [3]{d} x-2 \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{2/3}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan((-3^(1/2)*d^(1/3)+3^(1/2)*d^(1/3)*x)/(-d^(1/3)+d^(1/3)*x-2* (1+(-k^2-1)*x^2+k^2*x^4)^(1/3)))/d^(2/3)+ln(-d^(1/3)+d^(1/3)*x+(1+(-k^2-1) *x^2+k^2*x^4)^(1/3))/d^(2/3)-1/2*ln(d^(2/3)-2*d^(2/3)*x+d^(2/3)*x^2+(d^(1/ 3)-d^(1/3)*x)*(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^(2/3)
\[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx \]
Integrate[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x ^3)),x]
Integrate[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2)*(1 - k^2*x^2))^(2/3)*(-1 + d + (-1 - 2*d)*x + (d + k^2)*x^2 + 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.
\(\displaystyle \int \frac {k^2 x^4-4 k^2 x^3+\left (k^2+1\right ) x^2+2 \left (k^2+1\right ) x-3}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (x^2 \left (d+k^2\right )+(-2 d-1) x+d+k^2 x^3-1\right )} \, dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {k^2 x^4-4 k^2 x^3+\left (k^2+1\right ) x^2+2 \left (k^2+1\right ) x-3}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (x^2 \left (d+k^2\right )+(-2 d-1) x+d+k^2 x^3-1\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-k^2 x^4+4 k^2 x^3-\left (k^2+1\right ) x^2-2 \left (k^2+1\right ) x+3}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (-x^2 \left (d+k^2\right )+(2 d+1) x-d-k^2 x^3+1\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-x^2 \left (d^2+8 d k^2+6 k^4+2 k^2\right )+x \left (2 d^2+11 d k^2+d-2 k^4+2 k^2\right )-d^2-5 d k^2+d+8 k^2}{k^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (-x^2 \left (d+k^2\right )+(2 d+1) x-d-k^2 x^3+1\right )}+\frac {-\frac {d}{k^2}-5}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}+\frac {x}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (\frac {d}{k^2}+5\right ) x \left (1-x^2\right )^{2/3} \left (1-k^2 x^2\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},\frac {2}{3},\frac {2}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {\left (2 x^2 k^2-k^2-1\right )^2} \left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{2/3}\right ) \sqrt {\frac {2 \sqrt [3]{2} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} k^{4/3}-2^{2/3} \left (k^2-1\right )^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{4/3}}{\left (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 )^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 )}{2^{2/3} k^{2/3} \left (-2 x^2 k^2+k^2+1\right ) \sqrt {\left (-\left (\left (1-2 x^2\right ) k^2\right )-1\right )^2} \sqrt {\frac {\left (k^2-1\right )^{2/3} \left (2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (k^2-1\right )^{2/3}\right )}{\left (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 )^2}}}-\frac {\left (d^2-\left (1-5 k^2\right ) d-8 k^2\right ) \int \frac {1}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{k^2}+\frac {\left (-2 k^4+11 d k^2+2 k^2+2 d^2+d\right ) \int \frac {x}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{k^2}-\frac {\left (6 k^4+8 d k^2+2 k^2+d^2\right ) \int \frac {x^2}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{k^2}\) |
Int[(-3 + 2*(1 + k^2)*x + (1 + k^2)*x^2 - 4*k^2*x^3 + k^2*x^4)/(((1 - x^2) *(1 - k^2*x^2))^(2/3)*(-1 + d + (-1 - 2*d)*x + (d + k^2)*x^2 + k^2*x^3)),x ]
3.26.87.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
\[\int \frac {-3+2 \left (k^{2}+1\right ) x +\left (k^{2}+1\right ) x^{2}-4 k^{2} x^{3}+k^{2} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (-1+d +\left (-1-2 d \right ) x +\left (k^{2}+d \right ) x^{2}+k^{2} x^{3}\right )}d x\]
int((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1)) ^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+k^2*x^3),x)
int((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x^2+1)) ^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+k^2*x^3),x)
Timed out. \[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\text {Timed out} \]
integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x ^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+k^2*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\text {Timed out} \]
integrate((-3+2*(k**2+1)*x+(k**2+1)*x**2-4*k**2*x**3+k**2*x**4)/((-x**2+1) *(-k**2*x**2+1))**(2/3)/(-1+d+(-1-2*d)*x+(k**2+d)*x**2+k**2*x**3),x)
\[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} + {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x ^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+k^2*x^3),x, algorithm="maxima")
integrate((k^2*x^4 - 4*k^2*x^3 + (k^2 + 1)*x^2 + 2*(k^2 + 1)*x - 3)/((k^2* x^3 + (k^2 + d)*x^2 - (2*d + 1)*x + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3) ), x)
\[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{4} - 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} + 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (k^{2} x^{3} + {\left (k^{2} + d\right )} x^{2} - {\left (2 \, d + 1\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate((-3+2*(k^2+1)*x+(k^2+1)*x^2-4*k^2*x^3+k^2*x^4)/((-x^2+1)*(-k^2*x ^2+1))^(2/3)/(-1+d+(-1-2*d)*x+(k^2+d)*x^2+k^2*x^3),x, algorithm="giac")
integrate((k^2*x^4 - 4*k^2*x^3 + (k^2 + 1)*x^2 + 2*(k^2 + 1)*x - 3)/((k^2* x^3 + (k^2 + d)*x^2 - (2*d + 1)*x + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3) ), x)
Timed out. \[ \int \frac {-3+2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2-4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d+(-1-2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=\int \frac {2\,x\,\left (k^2+1\right )-4\,k^2\,x^3+k^2\,x^4+x^2\,\left (k^2+1\right )-3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (d+k^2\,x^3+x^2\,\left (k^2+d\right )-x\,\left (2\,d+1\right )-1\right )} \,d x \]
int((2*x*(k^2 + 1) - 4*k^2*x^3 + k^2*x^4 + x^2*(k^2 + 1) - 3)/(((x^2 - 1)* (k^2*x^2 - 1))^(2/3)*(d + k^2*x^3 + x^2*(d + k^2) - x*(2*d + 1) - 1)),x)