3.21.14 \(\int \frac {(-4+x^2) \sqrt [4]{2-x^2-2 x^4}}{x^2 (-2+x^2)} \, dx\)

Optimal. Leaf size=143 \[ -\frac {2 \sqrt [4]{-2 x^4-x^2+2}}{x}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{-2 x^4-x^2+2}}{\sqrt {2} x^2-\sqrt {-2 x^4-x^2+2}}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-2 x^4-x^2+2}}{2 x^2+\sqrt {2} \sqrt {-2 x^4-x^2+2}}\right )}{\sqrt [4]{2}} \]

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Rubi [F]  time = 0.43, 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 (-4+x^2\right ) \sqrt [4]{2-x^2-2 x^4}}{x^2 \left (-2+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(2 - x^2 - 2*x^4)^(1/4)*AppellF1[-1/2, -1/4, -1/4, 1/2, (-4*x^2)/(1 + Sqrt[17]), (-4*x^2)/(1 - Sqrt[17])])
/(x*(1 + (4*x^2)/(1 - Sqrt[17]))^(1/4)*(1 + (4*x^2)/(1 + Sqrt[17]))^(1/4)) + Defer[Int][(2 - x^2 - 2*x^4)^(1/4
)/(2 - x^2), x]

Rubi steps

\begin {align*} \int \frac {\left (-4+x^2\right ) \sqrt [4]{2-x^2-2 x^4}}{x^2 \left (-2+x^2\right )} \, dx &=\int \left (\frac {2 \sqrt [4]{2-x^2-2 x^4}}{x^2}+\frac {\sqrt [4]{2-x^2-2 x^4}}{2-x^2}\right ) \, dx\\ &=2 \int \frac {\sqrt [4]{2-x^2-2 x^4}}{x^2} \, dx+\int \frac {\sqrt [4]{2-x^2-2 x^4}}{2-x^2} \, dx\\ &=\frac {\left (2 \sqrt [4]{2-x^2-2 x^4}\right ) \int \frac {\sqrt [4]{1-\frac {4 x^2}{-1-\sqrt {17}}} \sqrt [4]{1-\frac {4 x^2}{-1+\sqrt {17}}}}{x^2} \, dx}{\sqrt [4]{1-\frac {4 x^2}{-1-\sqrt {17}}} \sqrt [4]{1-\frac {4 x^2}{-1+\sqrt {17}}}}+\int \frac {\sqrt [4]{2-x^2-2 x^4}}{2-x^2} \, dx\\ &=-\frac {2 \sqrt [4]{2-x^2-2 x^4} F_1\left (-\frac {1}{2};-\frac {1}{4},-\frac {1}{4};\frac {1}{2};-\frac {4 x^2}{1+\sqrt {17}},-\frac {4 x^2}{1-\sqrt {17}}\right )}{x \sqrt [4]{1+\frac {4 x^2}{1-\sqrt {17}}} \sqrt [4]{1+\frac {4 x^2}{1+\sqrt {17}}}}+\int \frac {\sqrt [4]{2-x^2-2 x^4}}{2-x^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4+x^2\right ) \sqrt [4]{2-x^2-2 x^4}}{x^2 \left (-2+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.30, size = 143, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{2-x^2-2 x^4}}{x}+\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{2-x^2-2 x^4}}{\sqrt {2} x^2-\sqrt {2-x^2-2 x^4}}\right )}{\sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{2-x^2-2 x^4}}{2 x^2+\sqrt {2} \sqrt {2-x^2-2 x^4}}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((x^2-4)*(-2*x^4-x^2+2)^(1/4)/x^2/(x^2-2),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 {{\left (-2 \, x^{4} - x^{2} + 2\right )}^{\frac {1}{4}} {\left (x^{2} - 4\right )}}{{\left (x^{2} - 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 246.14, size = 321, normalized size = 2.24

method	result	size
trager	\(-\frac {2 \left (-2 x^{4}-x^{2}+2\right )^{\frac {1}{4}}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-4 \left (-2 x^{4}-x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {-2 x^{4}-x^{2}+2}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}+4 \left (-2 x^{4}-x^{2}+2\right )^{\frac {3}{4}} x -2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{x^{2}-2}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}+4 \left (-2 x^{4}-x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{3}-4 \sqrt {-2 x^{4}-x^{2}+2}\, \RootOf \left (\textit {\_Z}^{4}+2\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{2}+4 \left (-2 x^{4}-x^{2}+2\right )^{\frac {3}{4}} x +2 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{x^{2}-2}\right )}{2}\)	\(321\)
risch	\(\frac {4 x^{4}+2 x^{2}-4}{\left (-2 x^{4}-x^{2}+2\right )^{\frac {3}{4}} x}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (\frac {-16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{12}+8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{9}-20 x^{10} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{7}+32 x^{8} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-14 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{5}+31 x^{6} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+8 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{6}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {3}{4}} x^{3}-26 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{4}+4 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{4}+8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x -12 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-8 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}}{\left (x^{2}-2\right ) \left (2 x^{4}+x^{2}-2\right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {16 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{12}+8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{9}+20 x^{10} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{7}-32 x^{8} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}-14 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{5}-31 x^{6} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}+8 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{6}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {3}{4}} x^{3}+26 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{4}+4 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{4}+8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8\right )^{\frac {1}{4}} x +12 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-8 \sqrt {-8 x^{12}-12 x^{10}+18 x^{8}+23 x^{6}-18 x^{4}-12 x^{2}+8}\, x^{2}-8 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2}}{\left (x^{2}-2\right ) \left (2 x^{4}+x^{2}-2\right )^{2}}\right )}{2}\right ) \left (-\left (2 x^{4}+x^{2}-2\right )^{3}\right )^{\frac {1}{4}}}{\left (-2 x^{4}-x^{2}+2\right )^{\frac {3}{4}}}\)	\(1168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-4)*(-2*x^4-x^2+2)^(1/4)/x^2/(x^2-2),x,method=_RETURNVERBOSE)

[Out]

-2*(-2*x^4-x^2+2)^(1/4)/x-1/2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*ln(-(4*RootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)
^2)*x^4-4*(-2*x^4-x^2+2)^(1/4)*RootOf(_Z^4+2)^2*x^3-4*(-2*x^4-x^2+2)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+R
ootOf(_Z^4+2)^2*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^2+4*(-2*x^4-x^2+2)^(3/4)*x-2*RootOf(_Z^4+2)^2*RootOf(_Z^2+Root
Of(_Z^4+2)^2))/(x^2-2))-1/2*RootOf(_Z^4+2)*ln(-(-4*RootOf(_Z^4+2)^3*x^4+4*(-2*x^4-x^2+2)^(1/4)*RootOf(_Z^4+2)^
2*x^3-4*(-2*x^4-x^2+2)^(1/2)*RootOf(_Z^4+2)*x^2-RootOf(_Z^4+2)^3*x^2+4*(-2*x^4-x^2+2)^(3/4)*x+2*RootOf(_Z^4+2)
^3)/(x^2-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 4)*(2 - 2*x^4 - x^2)^(1/4))/(x^2*(x^2 - 2)),x)

[Out]

int(((x^2 - 4)*(2 - 2*x^4 - x^2)^(1/4))/(x^2*(x^2 - 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x - 2)*(x + 2)*(-2*x**4 - x**2 + 2)**(1/4)/(x**2*(x**2 - 2)), x)

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