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.
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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.
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IntegrateAlgebraic [A] time = 0.33, size = 143, normalized size = 1.00 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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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.
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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.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2}-4\right ) \left (-2 x^{4}-x^{2}+2\right )^{\frac {1}{4}}}{x^{2} \left (x^{2}-2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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.
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