Optimal. Leaf size=45 \[ -\frac {\log \left (1-\sqrt {2} \sqrt {x}+x\right )}{\sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {x}+x\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.02, 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 = {841, 1179, 642}
\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 642
Rule 841
Rule 1179
Rubi steps
\begin {align*} \int \frac {1-x}{\sqrt {x} \left (1+x^2\right )} \, dx &=2 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {2}}-\frac {\text {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 [A]
time = 0.03, size = 23, normalized size = 0.51 \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs.
\(2(34)=68\).
time = 0.55, size = 112, normalized size = 2.49
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 x^{\frac {3}{2}}+\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x}}{x^{2}+1}\right )}{2}\) | \(54\) |
derivativedivides | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )\right )}{4}\) | \(112\) |
default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x +\sqrt {2}\, \sqrt {x}}{1+x -\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x -\sqrt {2}\, \sqrt {x}}{1+x +\sqrt {2}\, \sqrt {x}}\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}+1\right )+2 \arctan \left (\sqrt {2}\, \sqrt {x}-1\right )\right )}{4}\) | \(112\) |
meijerg | \(-\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}+\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {3}{4}}}-\frac {x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {x^{\frac {3}{2}} \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {3}{4}}}-\frac {x^{\frac {3}{2}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}\) | \(276\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, 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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Fricas [A]
time = 3.29, 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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Sympy [A]
time = 0.46, 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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Giac [A]
time = 0.57, 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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