Integrand size = 88, antiderivative size = 190 \[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3}+\sqrt {3} x}{1+x+2 \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt [3]{d}}+\frac {\log \left (1+x-\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{\sqrt [3]{d}}-\frac {\log \left (1+2 x+x^2+\left (\sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
3^(1/2)*arctan((3^(1/2)+x*3^(1/2))/(1+x+2*d^(1/3)*(1+(-k^2-1)*x^2+k^2*x^4) ^(1/3)))/d^(1/3)+ln(1+x-d^(1/3)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3))/d^(1/3)-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^(1/3)
\[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx \]
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 - (2 + d)*x - (1 + d*k^2)*x^2 + d*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 - (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.
| \(\displaystyle \int \frac {k^2 x^4+4 k^2 x^3+\left (k^2+1\right ) x^2-2 \left (k^2+1\right ) x-3}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )} \, dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {k^2 x^4+4 k^2 x^3+\left (k^2+1\right ) x^2-2 \left (k^2+1\right ) x-3}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 d+\frac {1}{k^2}}{d^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}+\frac {x \left (2 d^2 \left (1-k^2\right ) k^2+11 d k^2+d+2\right )-8 d^2 k^2+x^2 \left (2 d^2 \left (3 k^4+k^2\right )+8 d k^2+1\right )-d \left (1-5 k^2\right )+1}{d^2 k^2 \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (d k^2 x^3-x^2 \left (d k^2+1\right )-(d+2) x+d-1\right )}+\frac {x}{d \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (5 d+\frac {1}{k^2}\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 )}{d^2 \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} d 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 (-8 d^2 k^2+5 d k^2-d+1\right ) \int \frac {1}{\left (-d k^2 x^3+\left (d k^2+1\right ) x^2+(d+2) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{d^2 k^2}-\frac {\left (11 d k^2+2 d^2 \left (1-k^2\right ) k^2+d+2\right ) \int \frac {x}{\left (-d k^2 x^3+\left (d k^2+1\right ) x^2+(d+2) x-d+1\right ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{d^2 k^2}-\frac {\left (2 \left (3 k^4+k^2\right ) d^2+8 k^2 d+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 ) \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}}dx}{d^2 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 - (2 + d)*x - (1 + d*k^2)*x^2 + d*k^2*x^3)), x]
3.24.81.3.1 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]
\[\int \frac {-3-2 \left (k^{2}+1\right ) x +\left (k^{2}+1\right ) x^{2}+4 k^{2} x^{3}+k^{2} x^{4}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (-1+d -\left (2+d \right ) x -\left (d \,k^{2}+1\right ) x^{2}+d \,k^{2} x^{3}\right )}d x\]
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-(2+d)*x-(d*k^2+1)*x^2+d*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-(2+d)*x-(d*k^2+1)*x^2+d*k^2*x^3),x)
Timed out. \[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\text {Timed out} \]
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-(2+d)*x-(d*k^2+1)*x^2+d*k^2*x^3),x, algorithm="fricas")
Timed out. \[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\text {Timed out} \]
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-(2+d)*x-(d*k**2+1)*x**2+d*k**2*x**3),x)
\[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (d k^{2} x^{3} - {\left (d k^{2} + 1\right )} x^{2} - {\left (d + 2\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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-(2+d)*x-(d*k^2+1)*x^2+d*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)/((d*k^ 2*x^3 - (d*k^2 + 1)*x^2 - (d + 2)*x + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/ 3)), x)
\[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int { \frac {k^{2} x^{4} + 4 \, k^{2} x^{3} + {\left (k^{2} + 1\right )} x^{2} - 2 \, {\left (k^{2} + 1\right )} x - 3}{{\left (d k^{2} x^{3} - {\left (d k^{2} + 1\right )} x^{2} - {\left (d + 2\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
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-(2+d)*x-(d*k^2+1)*x^2+d*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)/((d*k^ 2*x^3 - (d*k^2 + 1)*x^2 - (d + 2)*x + d - 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/ 3)), x)
Timed out. \[ \int \frac {-3-2 \left (1+k^2\right ) x+\left (1+k^2\right ) x^2+4 k^2 x^3+k^2 x^4}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (-1+d-(2+d) x-\left (1+d k^2\right ) x^2+d k^2 x^3\right )} \, dx=\int -\frac {4\,k^2\,x^3-2\,x\,\left (k^2+1\right )+k^2\,x^4+x^2\,\left (k^2+1\right )-3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (x^2\,\left (d\,k^2+1\right )-d+x\,\left (d+2\right )-d\,k^2\,x^3+1\right )} \,d x \]
int(-(4*k^2*x^3 - 2*x*(k^2 + 1) + k^2*x^4 + x^2*(k^2 + 1) - 3)/(((x^2 - 1) *(k^2*x^2 - 1))^(2/3)*(x^2*(d*k^2 + 1) - d + x*(d + 2) - d*k^2*x^3 + 1)),x )