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

Optimal. Leaf size=241 \[ -\frac {\log \left (d^{2/3} x^4-2 d^{2/3} x^2+d^{2/3}+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+\left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}\right )}{4 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [3]{d}+\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}\right )}{2 \sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}{-2 \sqrt [3]{d} x^2+2 \sqrt [3]{d}+\sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{2 \sqrt [3]{d}} \]

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Rubi [F]  time = 0.82, 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 {\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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 11.80, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

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]

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IntegrateAlgebraic [A]  time = 5.55, size = 241, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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]

[Out]

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

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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(((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")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (k^{2}-2\right ) x +k^{2} x^{3}}{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )^{\frac {1}{3}} \left (-1+d +\left (k^{2}-2 d \right ) x^{2}+d \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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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(((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)

[Out]

Timed out

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