Optimal. Leaf size=45 \[ \frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, 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, 1165, 628} \begin {gather*} \frac {\log \left (x+\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}}-\frac {\log \left (x-\sqrt {2} \sqrt {x}+1\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 827
Rule 1165
Rubi steps
\begin {align*} \int \frac {1-x}{\sqrt {x} \left (1+x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {2}}\\ &=-\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{\sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 99, normalized size = 2.20 \begin {gather*} \frac {1}{12} \left (3 \sqrt {2} \left (-\log \left (x-\sqrt {2} \sqrt {x}+1\right )+\log \left (x+\sqrt {2} \sqrt {x}+1\right )-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )+2 \tan ^{-1}\left (\sqrt {2} \sqrt {x}+1\right )\right )-8 x^{3/2} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 23, normalized size = 0.51 \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 0.73 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} {\left (x + 1\right )} \sqrt {x} + x^{2} + 4 \, x + 1}{x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{4}+\frac {\sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 34, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} \sqrt {x} + x + 1\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {2} \sqrt {x} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 20, normalized size = 0.44 \begin {gather*} \sqrt {2}\,\mathrm {atanh}\left (\frac {8\,\sqrt {2}\,\sqrt {x}}{8\,x+8}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.69, size = 49, normalized size = 1.09 \begin {gather*} - \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{2} + \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} + 4 x + 4 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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