Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {444, 63, 207} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(3-x) \sqrt {5-x}} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {5-x^2}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 24, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 48, normalized size = 2.00 \[ \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} - 4 \, \sqrt {2} {\left (x^{2} - 7\right )} \sqrt {-x^{2} + 5} - 22 \, x^{2} + 89}{x^{4} - 6 \, x^{2} + 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 42, normalized size = 1.75 \[ \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x^{2} + 5}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | -\sqrt {2} + \sqrt {-x^{2} + 5} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 49, normalized size = 2.04
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {-x^{2}+5}}{x^{2}-3}\right )}{4}\) | \(49\) |
default | \(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4+2 \sqrt {3}\, \left (x +\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+2}}\right )}{4}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4-2 \sqrt {3}\, \left (x -\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+2}}\right )}{4}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 112, normalized size = 4.67 \[ \frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} \sqrt {2} \log \left (\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \sqrt {3} \sqrt {2} \log \left (-\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 78, normalized size = 3.25 \[ \frac {\sqrt {2}\,\left (\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x+5\right )\,1{}\mathrm {i}}{2}+\sqrt {5-x^2}\,1{}\mathrm {i}}{x+\sqrt {3}}\right )+\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x-5\right )\,1{}\mathrm {i}}{2}-\sqrt {5-x^2}\,1{}\mathrm {i}}{x-\sqrt {3}}\right )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.72, size = 61, normalized size = 2.54 \[ - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} < \frac {1}{2} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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