3.27.75 \(\int \frac {(-2+k^2) x+k^2 x^3}{\sqrt [3]{(1-x^2) (1-k^2 x^2)} (-1+d+(-2 d+k^2) x^2+d x^4)} \, dx\) [2675]

3.27.75.1 Optimal result

Integrand size = 59, antiderivative size = 241 \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{4 \sqrt [3]{d}} \]

output

1/2*3^(1/2)*arctan(3^(1/2)*(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)/(2*d^(1/3)-2*d^( 
1/3)*x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/3)))/d^(1/3)+1/2*ln(-d^(1/3)+d^(1/3)* 
x^2+(1+(-k^2-1)*x^2+k^2*x^4)^(1/3))/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)^(1/3)+(1+(-k^2-1 
)*x^2+k^2*x^4)^(2/3))/d^(1/3)
 

3.27.75.2 Mathematica [A] (verified)

Time = 15.23 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\frac {\sqrt [3]{-1+x^2} \sqrt [3]{-1+k^2 x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+k^2 x^2}}{-2 \sqrt [3]{d} \left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+k^2 x^2}}\right )+2 \log \left (\sqrt [3]{d} \left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+k^2 x^2}\right )-\log \left (d^{2/3} \left (-1+x^2\right )^{4/3}-\sqrt [3]{d} \left (-1+x^2\right )^{2/3} \sqrt [3]{-1+k^2 x^2}+\left (-1+k^2 x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{d} \sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]

input

Integrate[((-2 + k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + 
d + (-2*d + k^2)*x^2 + d*x^4)),x]
 

output

((-1 + x^2)^(1/3)*(-1 + k^2*x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + k^ 
2*x^2)^(1/3))/(-2*d^(1/3)*(-1 + x^2)^(2/3) + (-1 + k^2*x^2)^(1/3))] + 2*Lo 
g[d^(1/3)*(-1 + x^2)^(2/3) + (-1 + k^2*x^2)^(1/3)] - Log[d^(2/3)*(-1 + x^2 
)^(4/3) - d^(1/3)*(-1 + x^2)^(2/3)*(-1 + k^2*x^2)^(1/3) + (-1 + k^2*x^2)^( 
2/3)]))/(4*d^(1/3)*((-1 + x^2)*(-1 + k^2*x^2))^(1/3))
 

3.27.75.3 Rubi [F]

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.

\(\displaystyle \int \frac {k^2 x^3+\left (k^2-2\right ) x}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (x^2 \left (k^2-2 d\right )+d x^4+d-1\right )} \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \frac {x \left (k^2 x^2+k^2-2\right )}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (x^2 \left (k^2-2 d\right )+d x^4+d-1\right )}dx\)

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {x \left (k^2 x^2+k^2-2\right )}{\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1} \left (x^2 \left (k^2-2 d\right )+d x^4+d-1\right )}dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {1}{2} \int \frac {-x^2 k^2-k^2+2}{\left (-d x^4+\left (2 d-k^2\right ) x^2-d+1\right ) \sqrt [3]{k^2 x^4-\left (k^2+1\right ) x^2+1}}dx^2\)

\(\Big \downarrow \) 1375

\(\displaystyle \frac {1}{2} \int \frac {-x^2 k^2-k^2+2}{\left (-d x^4+\left (2 d-k^2\right ) x^2-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx^2\)

input

Int[((-2 + k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d + (- 
2*d + k^2)*x^2 + d*x^4)),x]
 

output

$Aborted
 

3.27.75.3.1 Defintions of rubi rules used

rule 1375

Int[((g_.) + (h_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Unintegrable[(g + h*x)*(a + b 
*x + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, p, 
q}, x]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2048

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]
 

rule 7266

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]
 

3.27.75.4 Maple [F]

\[\int \frac {\left (k^{2}-2\right ) x +k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (-1+d +\left (k^{2}-2 d \right ) x^{2}+d \,x^{4}\right )}d x\]

input

int(((k^2-2)*x+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d+(k^2-2*d)*x^2+ 
d*x^4),x)
 

output

int(((k^2-2)*x+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d+(k^2-2*d)*x^2+ 
d*x^4),x)
 

3.27.75.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\text {Timed out} \]

input

integrate(((k^2-2)*x+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d+(k^2-2*d 
)*x^2+d*x^4),x, algorithm="fricas")
 

output

Timed out
 

3.27.75.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\text {Timed out} \]

input

integrate(((k**2-2)*x+k**2*x**3)/((-x**2+1)*(-k**2*x**2+1))**(1/3)/(-1+d+( 
k**2-2*d)*x**2+d*x**4),x)
 

output

Timed out
 

3.27.75.7 Maxima [F]

\[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int { \frac {k^{2} x^{3} + {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(((k^2-2)*x+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d+(k^2-2*d 
)*x^2+d*x^4),x, algorithm="maxima")
 

output

integrate((k^2*x^3 + (k^2 - 2)*x)/((d*x^4 + (k^2 - 2*d)*x^2 + d - 1)*((k^2 
*x^2 - 1)*(x^2 - 1))^(1/3)), x)
 

3.27.75.8 Giac [F]

\[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int { \frac {k^{2} x^{3} + {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate(((k^2-2)*x+k^2*x^3)/((-x^2+1)*(-k^2*x^2+1))^(1/3)/(-1+d+(k^2-2*d 
)*x^2+d*x^4),x, algorithm="giac")
 

output

integrate((k^2*x^3 + (k^2 - 2)*x)/((d*x^4 + (k^2 - 2*d)*x^2 + d - 1)*((k^2 
*x^2 - 1)*(x^2 - 1))^(1/3)), x)
 

3.27.75.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int \frac {x\,\left (k^2-2\right )+k^2\,x^3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (d\,x^4+\left (k^2-2\,d\right )\,x^2+d-1\right )} \,d x \]

input

int((x*(k^2 - 2) + k^2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/3)*(d - x^2*(2*d 
 - k^2) + d*x^4 - 1)),x)
 

output

int((x*(k^2 - 2) + k^2*x^3)/(((x^2 - 1)*(k^2*x^2 - 1))^(1/3)*(d - x^2*(2*d 
 - k^2) + d*x^4 - 1)), x)