Optimal. Leaf size=31 \[ -\sqrt {2 x^2+1}+\tan ^{-1}\left (\sqrt {2 x^2+1}\right )-x+\tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2107, 321, 203, 444, 50, 63} \[ -\sqrt {2 x^2+1}+\tan ^{-1}\left (\sqrt {2 x^2+1}\right )-x+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 321
Rule 444
Rule 2107
Rubi steps
\begin {align*} \int \frac {x}{x-\sqrt {1+2 x^2}} \, dx &=-\int \frac {x^2}{1+x^2} \, dx-\int \frac {x \sqrt {1+2 x^2}}{1+x^2} \, dx\\ &=-x-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x}}{1+x} \, dx,x,x^2\right )+\int \frac {1}{1+x^2} \, dx\\ &=-x-\sqrt {1+2 x^2}+\tan ^{-1}(x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )\\ &=-x-\sqrt {1+2 x^2}+\tan ^{-1}(x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {x^2}{2}} \, dx,x,\sqrt {1+2 x^2}\right )\\ &=-x-\sqrt {1+2 x^2}+\tan ^{-1}(x)+\tan ^{-1}\left (\sqrt {1+2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 1.00 \[ -\sqrt {2 x^2+1}+\tan ^{-1}\left (\sqrt {2 x^2+1}\right )-x+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 41, normalized size = 1.32 \[ -x - \sqrt {2 \, x^{2} + 1} + \arctan \relax (x) - \arctan \left (-\frac {x^{2} - \sqrt {2 \, x^{2} + 1} + 1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 63, normalized size = 2.03 \[ -\frac {1}{2} \, \pi - x - \sqrt {2 \, x^{2} + 1} + \arctan \relax (x) + \arctan \left (-\frac {{\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} + 1}{2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 28, normalized size = 0.90 \[ -x +\arctan \relax (x )+\arctan \left (\sqrt {2 x^{2}+1}\right )-\sqrt {2 x^{2}+1} \]
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 \[ \int \frac {x}{x - \sqrt {2 \, x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 64, normalized size = 2.06 \[ -x-\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}-\ln \left (x-\mathrm {i}\right )\,1{}\mathrm {i}+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
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 \[ \int \frac {x}{x - \sqrt {2 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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