Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\left (\sqrt {2}-1\right ) \sqrt {x-3}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\left (1+\sqrt {2}\right ) \sqrt {x-3}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {827, 1163, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {x-3}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x-3}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 827
Rule 1163
Rubi steps
\begin {align*} \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+6 x^2+x^4} \, dx,x,\sqrt {-3+x}\right )\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{3-2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right )+\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{3+2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 69, normalized size = 1.53 \begin {gather*} \frac {\left (2+\sqrt {2}\right ) \left (\tan ^{-1}\left (\sqrt {3-2 \sqrt {2}} \sqrt {x-3}\right )+\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} \sqrt {x-3}\right )\right )}{2 \sqrt {3+2 \sqrt {2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.21, size = 37, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {x-3}{2 \sqrt {2}}-\frac {1}{2 \sqrt {2}}}{\sqrt {x-3}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 0.42 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 23, normalized size = 0.51 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 119, normalized size = 2.64 \begin {gather*} \frac {2 \arctan \left (\frac {2 \sqrt {x -3}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -3}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}+\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -3}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}+\frac {2 \arctan \left (\frac {2 \sqrt {x -3}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}} \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}{{\left (x^{2} - 8\right )} \sqrt {x - 3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 39, normalized size = 0.87 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-3}}{4}\right )+\mathrm {atan}\left (\frac {7\,\sqrt {2}\,\sqrt {x-3}}{4}+\frac {\sqrt {2}\,{\left (x-3\right )}^{3/2}}{4}\right )\right )}{2} \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 {x - 2}{\sqrt {x - 3} \left (x^{2} - 8\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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