Optimal. Leaf size=18 \[ -\frac {2 \sqrt {x^3+x}}{x^2+1} \]
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Rubi [A] time = 0.07, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 449} \begin {gather*} -\frac {2 x}{\sqrt {x^3+x}} \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 {x+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^2}\right ) \int \frac {-1+x^2}{\sqrt {x} \left (1+x^2\right )^{3/2}} \, dx}{\sqrt {x+x^3}}\\ &=-\frac {2 x}{\sqrt {x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 0.67 \begin {gather*} -\frac {2 x}{\sqrt {x^3+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^3}}{1+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 16, normalized size = 0.89 \begin {gather*} -\frac {2 \, \sqrt {x^{3} + x}}{x^{2} + 1} \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}{\sqrt {x^{3} + x} {\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 = 0.08, size = 11, normalized size = 0.61
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {x^{3}+x}}\) | \(11\) |
default | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
risch | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
elliptic | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{3}+x}}{x^{2}+1}\) | \(17\) |
meijerg | \(\frac {2 \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right ) x^{\frac {5}{2}}}{5}-2 \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -x^{2}\right ) \sqrt {x}\) | \(34\) |
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^{2} - 1}{\sqrt {x^{3} + x} {\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 = 0.05, size = 10, normalized size = 0.56 \begin {gather*} -\frac {2\,x}{\sqrt {x^3+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 - 1\right ) \left (x + 1\right )}{\sqrt {x \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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