Optimal. Leaf size=23 \[ -\frac {3 \left (x^4+x^2\right )^{2/3}}{x \left (x^2+1\right )} \]
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Rubi [A] time = 0.08, 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 {3 x}{\sqrt [3]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {-1+x^2}{x^{2/3} \left (1+x^2\right )^{4/3}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {3 x}{\sqrt [3]{x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 14, normalized size = 0.61 \begin {gather*} -\frac {3 x}{\sqrt [3]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 23, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^4+x^2\right )^{2/3}}{x \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 18, normalized size = 0.78 \begin {gather*} -\frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + 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^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 0.57 \begin {gather*} -\frac {3 x}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}}} \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^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {3\,{\left (x^4+x^2\right )}^{2/3}}{x\,\left (x^2+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 )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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