3.27.17 \(\int \frac {3+(1-2 k^2) x-3 k^2 x^2+k^2 x^3}{\sqrt [3]{(1-x^2) (1-k^2 x^2)} (-1+d-(1+2 d) x+(d+k^2) x^2+k^2 x^3)} \, dx\) [2617]

3.27.17.1 Optimal result

Integrand size = 78, antiderivative size = 229 \[ \int \frac {3+\left (1-2 k^2\right ) x-3 k^2 x^2+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(1+2 d) x+\left (d+k^2\right ) x^2+k^2 x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3}-2 d^{2/3} x+d^{2/3} x^2+\left (\sqrt [3]{d}-\sqrt [3]{d} x\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

output

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

3.27.17.2 Mathematica [F]

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

input

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

output

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

3.27.17.3 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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -(4-d) \int \frac {1}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx-2 \left (d-k^2+1\right ) \int \frac {x}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+\left (d+4 k^2\right ) \int \frac {x^2}{\left (-k^2 x^3-\left (k^2+d\right ) x^2+(2 d+1) x-d+1\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}dx+\frac {x \sqrt [3]{1-x^2} \sqrt [3]{1-k^2 x^2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},\frac {1}{3},\frac {3}{2},x^2,k^2 x^2\right )}{\sqrt [3]{k^2 x^4-\left (k^2+1\right ) x^2+1}}\)

input

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

output

$Aborted
 

3.27.17.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.27.17.4 Maple [F]

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

input

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

output

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

3.27.17.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.27.17.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.27.17.7 Maxima [F]

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

input

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

output

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

3.27.17.8 Giac [F]

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

input

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

output

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

3.27.17.9 Mupad [F(-1)]

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

input

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

output

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