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

Optimal. Leaf size=226 \[ \frac {\log \left (\sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+x^2-1\right )}{2 d^{2/3}}-\frac {\log \left (d^{2/3} \left (k^2 x^4+\left (-k^2-1\right ) x^2+1\right )^{2/3}+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}+x^4-2 x^2+1\right )}{4 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}}{\sqrt [3]{d} \sqrt [3]{k^2 x^4+\left (-k^2-1\right ) x^2+1}-2 x^2+2}\right )}{2 d^{2/3}} \]

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Rubi [F]  time = 0.80, 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+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 + x^4)),x]

[Out]

Defer[Subst][Defer[Int][(-2 + k^2 + k^2*x)/((1 - d + (-2 + d*k^2)*x + 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+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+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+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+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 = 12.42, 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+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 + 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 + x^4)), x]

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IntegrateAlgebraic [A]  time = 3.74, size = 226, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2-2 x^2+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 d^{2/3}}+\frac {\log \left (-1+x^2+\sqrt [3]{d} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{2 d^{2/3}}-\frac {\log \left (1-2 x^2+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}+d^{2/3} \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{4 d^{2/3}} \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 + x^4))
,x]

[Out]

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

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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+(d*k^2-2)*x^2+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 (x^{4} + {\left (d k^{2} - 2\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+(d*k^2-2)*x^2+x^4),x, algorithm="giac")

[Out]

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

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+(d*k^2-2)*x^2+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+(d*k^2-2)*x^2+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 (x^{4} + {\left (d k^{2} - 2\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+(d*k^2-2)*x^2+x^4),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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