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

Optimal. Leaf size=86 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}{x+1}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}{x+1}\right )}{d^{3/4}} \]

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Rubi [F]  time = 5.40, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2-(-1+k) (1+k) x+2 k^2 x^2}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(3+d) x-\left (3+d k^2\right ) x^2+\left (-1+d k^2\right ) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 5.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2-(-1+k) (1+k) x+2 k^2 x^2}{\sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d-(3+d) x-\left (3+d k^2\right ) x^2+\left (-1+d k^2\right ) x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 12.92, size = 86, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{1+x}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{1+x}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, k^{2} x^{2} - {\left (k + 1\right )} {\left (k - 1\right )} x - 2}{{\left ({\left (d k^{2} - 1\right )} x^{3} - {\left (d k^{2} + 3\right )} x^{2} - {\left (d + 3\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-2-\left (-1+k \right ) \left (1+k \right ) x +2 k^{2} x^{2}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {1}{4}} \left (-1+d -\left (3+d \right ) x -\left (d \,k^{2}+3\right ) x^{2}+\left (d \,k^{2}-1\right ) x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, k^{2} x^{2} - {\left (k + 1\right )} {\left (k - 1\right )} x - 2}{{\left ({\left (d k^{2} - 1\right )} x^{3} - {\left (d k^{2} + 3\right )} x^{2} - {\left (d + 3\right )} x + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (k-1\right )\,\left (k+1\right )-2\,k^2\,x^2+2}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/4}\,\left (\left (1-d\,k^2\right )\,x^3+\left (d\,k^2+3\right )\,x^2+\left (d+3\right )\,x-d+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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