Optimal. Leaf size=14 \[ -\frac {2 x}{\sqrt {x \left (1+x^2\right )}} \]
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Rubi [A]
time = 0.10, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {6851, 460}
\begin {gather*} -\frac {2 x}{\sqrt {x \left (x^2+1\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rule 6851
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x \left (1+x^2\right )}} \, 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 \left (1+x^2\right )}}\\ &=-\frac {2 x}{\sqrt {x \left (1+x^2\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2 x}{\sqrt {x+x^3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 13, normalized size = 0.93
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {x \left (x^{2}+1\right )}}\) | \(13\) |
default | \(-\frac {2 x}{\sqrt {x \left (x^{2}+1\right )}}\) | \(13\) |
risch | \(-\frac {2 x}{\sqrt {x \left (x^{2}+1\right )}}\) | \(13\) |
elliptic | \(-\frac {2 x}{\sqrt {x \left (x^{2}+1\right )}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{3}+x}}{x^{2}+1}\) | \(17\) |
meijerg | \(\frac {2 x^{\frac {5}{2}} \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )}{5}-2 \sqrt {x}\, \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -x^{2}\right )\) | \(34\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 16, normalized size = 1.14 \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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.38, size = 138, normalized size = 9.86 \begin {gather*} -\frac {2\,x}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}+\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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