Optimal. Leaf size=25 \[ -\frac {2 \left (x^6-x^2\right )^{3/4}}{x \left (x^4-1\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 16, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2056, 449} \begin {gather*} -\frac {2 x}{\sqrt [4]{x^6-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-1+x^4}\right ) \int \frac {1+x^4}{\sqrt {x} \left (-1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{-x^2+x^6}}\\ &=-\frac {2 x}{\sqrt [4]{-x^2+x^6}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 0.64 \begin {gather*} -\frac {2 x}{\sqrt [4]{x^2 \left (x^4-1\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 25, normalized size = 1.00 \begin {gather*} -\frac {2 \left (x^6-x^2\right )^{3/4}}{x \left (x^4-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 22, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (x^{6} - x^{2}\right )}^{\frac {3}{4}}}{x^{5} - x} \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 {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 15, normalized size = 0.60 \begin {gather*} -\frac {2 x}{\left (x^{6}-x^{2}\right )^{\frac {1}{4}}} \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 {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 23, normalized size = 0.92 \begin {gather*} -\frac {2\,{\left (x^6-x^2\right )}^{3/4}}{x\,\left (x^4-1\right )} \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 {x^{4} + 1}{\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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