Optimal. Leaf size=23 \[ -\frac {2 \left (x^6+x^2\right )^{3/4}}{x \left (x^4+1\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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 = 0.02, size = 14, normalized size = 0.61 \begin {gather*} -\frac {2 x}{\sqrt [4]{x^6+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \left (x^2+x^6\right )^{3/4}}{x \left (1+x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 18, normalized size = 0.78 \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 = 0.08, size = 13, normalized size = 0.57
method | result | size |
gosper | \(-\frac {2 x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(-\frac {2 x}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(15\) |
trager | \(-\frac {2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{4}+1\right )}\) | \(22\) |
meijerg | \(-2 \hypergeom \left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], -x^{4}\right ) \sqrt {x}+\frac {2 \hypergeom \left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], -x^{4}\right ) x^{\frac {9}{2}}}{9}\) | \(34\) |
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 = 0.13, size = 21, normalized size = 0.91 \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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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