\(\int \frac {-2 k+(-1+k) (1+k) x+2 k x^2}{\sqrt [4]{(1-x^2) (1-k^2 x^2)} (1-d+(3+d) k x+(d+3 k^2) x^2+k (-d+k^2) x^3)} \, dx\) [1231]

Optimal result

Integrand size = 77, antiderivative size = 90 \[ \int \frac {-2 k+(-1+k) (1+k) x+2 k x^2}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2+k \left (-d+k^2\right ) x^3\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{1+k x}\right )}{d^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{1+k x}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 k+(-1+k) (1+k) x+2 k x^2}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2+k \left (-d+k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \frac {-2 k-(1-k) (1+k) x+2 k x^2}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2-k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \left (\frac {(-1-k) (1-k) x}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2-k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 k x^2}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2-k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}+\frac {2 k}{\left (-1+d-(3+d) k x-\left (d+3 k^2\right ) x^2+k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \, dx \\ & = ((-1-k) (1-k)) \int \frac {x}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2-k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+(2 k) \int \frac {x^2}{\left (1-d+(3+d) k x+\left (d+3 k^2\right ) x^2-k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+(2 k) \int \frac {1}{\left (-1+d-(3+d) k x-\left (d+3 k^2\right ) x^2+k \left (d-k^2\right ) x^3\right ) \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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