Optimal. Leaf size=28 \[ -\frac {x^2}{2}-\frac {1}{2} \sqrt {x^2+1} x-\frac {1}{2} \sinh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2106, 30, 195, 215} \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \sqrt {x^2+1} x-\frac {1}{2} \sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 195
Rule 215
Rule 2106
Rubi steps
\begin {align*} \int \frac {1}{x-\sqrt {1+x^2}} \, dx &=-\int x \, dx-\int \sqrt {1+x^2} \, dx\\ &=-\frac {x^2}{2}-\frac {1}{2} x \sqrt {1+x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=-\frac {x^2}{2}-\frac {1}{2} x \sqrt {1+x^2}-\frac {1}{2} \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.36 \begin {gather*} \frac {1}{2} \log \left (x-\sqrt {x^2+1}\right )-\frac {1}{4 \left (x-\sqrt {x^2+1}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 40, normalized size = 1.43 \begin {gather*} -\frac {x^2}{2}-\frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \log \left (\sqrt {x^2+1}-x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 30, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 30, normalized size = 1.07 \begin {gather*} -\frac {1}{2} \, x^{2} - \frac {1}{2} \, \sqrt {x^{2} + 1} x + \frac {1}{2} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.75 \begin {gather*} -\frac {x^{2}}{2}-\frac {\sqrt {x^{2}+1}\, x}{2}-\frac {\arcsinh \relax (x )}{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 {1}{x - \sqrt {x^{2} + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 20, normalized size = 0.71 \begin {gather*} -\frac {\mathrm {asinh}\relax (x)}{2}-\frac {x\,\sqrt {x^2+1}}{2}-\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.35, size = 58, normalized size = 2.07 \begin {gather*} - \frac {x \operatorname {asinh}{\relax (x )}}{2 x - 2 \sqrt {x^{2} + 1}} + \frac {x}{2 x - 2 \sqrt {x^{2} + 1}} + \frac {\sqrt {x^{2} + 1} \operatorname {asinh}{\relax (x )}}{2 x - 2 \sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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