Integrand size = 73, antiderivative size = 241 \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x^2+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{4/3}\right )}{4 \sqrt [3]{d}} \]
-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)*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) *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)*x^2+ d^(2/3)*x^4+(d^(1/3)-d^(1/3)*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 = 16.78 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\frac {\left (-1+x^2\right )^{2/3} \left (-1+k^2 x^2\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \left (-1+k^2 x^2\right )^{2/3}}{-2 \sqrt [3]{d} \sqrt [3]{-1+x^2}+\left (-1+k^2 x^2\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{d} \sqrt [3]{-1+x^2}+\left (-1+k^2 x^2\right )^{2/3}\right )+\log \left (d^{2/3} \left (-1+x^2\right )^{2/3}-\sqrt [3]{d} \sqrt [3]{-1+x^2} \left (-1+k^2 x^2\right )^{2/3}+\left (-1+k^2 x^2\right )^{4/3}\right )\right )}{4 \sqrt [3]{d} \left (\left (-1+x^2\right ) \left (-1+k^2 x^2\right )\right )^{2/3}} \]
Integrate[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2) )^(2/3)*(1 - d + (d - 2*k^2)*x^2 + k^4*x^4)),x]
((-1 + x^2)^(2/3)*(-1 + k^2*x^2)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + k ^2*x^2)^(2/3))/(-2*d^(1/3)*(-1 + x^2)^(1/3) + (-1 + k^2*x^2)^(2/3))] - 2*L og[d^(1/3)*(-1 + x^2)^(1/3) + (-1 + k^2*x^2)^(2/3)] + Log[d^(2/3)*(-1 + x^ 2)^(2/3) - d^(1/3)*(-1 + x^2)^(1/3)*(-1 + k^2*x^2)^(2/3) + (-1 + k^2*x^2)^ (4/3)]))/(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.
| \(\displaystyle \int \frac {k^4 x^5-2 k^4 x^3+\left (2 k^2-1\right ) x}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (x^2 \left (d-2 k^2\right )-d+k^4 x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \int \frac {x \left (k^4 x^4-2 k^4 x^2+2 k^2-1\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (x^2 \left (d-2 k^2\right )-d+k^4 x^4+1\right )}dx\) |
| \(\Big \downarrow \) 2048 |
| \(\displaystyle \int \frac {x \left (k^4 x^4-2 k^4 x^2+2 k^2-1\right )}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (x^2 \left (d-2 k^2\right )-d+k^4 x^4+1\right )}dx\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {1}{2} \int -\frac {-x^4 k^4+2 x^2 k^4-2 k^2+1}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3} \left (k^4 x^4+\left (d-2 k^2\right ) x^2-d+1\right )}dx^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {-x^4 k^4+2 x^2 k^4-2 k^2+1}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3} \left (k^4 x^4+\left (d-2 k^2\right ) x^2-d+1\right )}dx^2\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {-2 k^2+\left (d-2 k^2 \left (1-k^2\right )\right ) x^2-d+2}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3} \left (k^4 x^4+\left (d-2 k^2\right ) x^2-d+1\right )}-\frac {1}{\left (k^2 x^4-\left (k^2+1\right ) x^2+1\right )^{2/3}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {-2 k^2+\left (d-2 k^2 \left (1-k^2\right )\right ) x^2-d+2}{\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3} \left (k^4 x^4+\left (d-2 k^2\right ) x^2-d+1\right )}dx^2-\frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {\left (2 k^2 x^2-k^2-1\right )^2} \left (\left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right ) \sqrt {\frac {\left (k^2-1\right )^{4/3}+2 \sqrt [3]{2} k^{4/3} \left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3}-2^{2/3} \left (k^2-1\right )^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\left (\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1-\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}{2^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} k^{2/3}+\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}}\right ),-7-4 \sqrt {3}\right )}{k^{2/3} \left (-2 k^2 x^2+k^2+1\right ) \sqrt {\left (-\left (k^2 \left (1-2 x^2\right )\right )-1\right )^2} \sqrt {\frac {\left (k^2-1\right )^{2/3} \left (\left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )}{\left (\left (1+\sqrt {3}\right ) \left (k^2-1\right )^{2/3}+2^{2/3} k^{2/3} \sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}\right )^2}}}\right )\) |
Int[((-1 + 2*k^2)*x - 2*k^4*x^3 + k^4*x^5)/(((1 - x^2)*(1 - k^2*x^2))^(2/3 )*(1 - d + (d - 2*k^2)*x^2 + k^4*x^4)),x]
3.27.76.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
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[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
\[\int \frac {\left (2 k^{2}-1\right ) x -2 k^{4} x^{3}+k^{4} x^{5}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {2}{3}} \left (1-d +\left (-2 k^{2}+d \right ) x^{2}+k^{4} x^{4}\right )}d x\]
int(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2 *k^2+d)*x^2+k^4*x^4),x)
int(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1-d+(-2 *k^2+d)*x^2+k^4*x^4),x)
Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]
integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1 -d+(-2*k^2+d)*x^2+k^4*x^4),x, algorithm="fricas")
Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\text {Timed out} \]
integrate(((2*k**2-1)*x-2*k**4*x**3+k**4*x**5)/((-x**2+1)*(-k**2*x**2+1))* *(2/3)/(1-d+(-2*k**2+d)*x**2+k**4*x**4),x)
\[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{5} - 2 \, k^{4} x^{3} + {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1 -d+(-2*k^2+d)*x^2+k^4*x^4),x, algorithm="maxima")
integrate((k^4*x^5 - 2*k^4*x^3 + (2*k^2 - 1)*x)/((k^4*x^4 - (2*k^2 - d)*x^ 2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)
\[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int { \frac {k^{4} x^{5} - 2 \, k^{4} x^{3} + {\left (2 \, k^{2} - 1\right )} x}{{\left (k^{4} x^{4} - {\left (2 \, k^{2} - d\right )} x^{2} - d + 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {2}{3}}} \,d x } \]
integrate(((2*k^2-1)*x-2*k^4*x^3+k^4*x^5)/((-x^2+1)*(-k^2*x^2+1))^(2/3)/(1 -d+(-2*k^2+d)*x^2+k^4*x^4),x, algorithm="giac")
integrate((k^4*x^5 - 2*k^4*x^3 + (2*k^2 - 1)*x)/((k^4*x^4 - (2*k^2 - d)*x^ 2 - d + 1)*((k^2*x^2 - 1)*(x^2 - 1))^(2/3)), x)
Timed out. \[ \int \frac {\left (-1+2 k^2\right ) x-2 k^4 x^3+k^4 x^5}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{2/3} \left (1-d+\left (d-2 k^2\right ) x^2+k^4 x^4\right )} \, dx=\int \frac {k^4\,x^5-2\,k^4\,x^3+x\,\left (2\,k^2-1\right )}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{2/3}\,\left (k^4\,x^4-d+x^2\,\left (d-2\,k^2\right )+1\right )} \,d x \]
int((k^4*x^5 - 2*k^4*x^3 + x*(2*k^2 - 1))/(((x^2 - 1)*(k^2*x^2 - 1))^(2/3) *(k^4*x^4 - d + x^2*(d - 2*k^2) + 1)),x)