Optimal. Leaf size=21 \[ -\frac {x^3}{3}-\frac {1}{3} \left (x^2+1\right )^{3/2} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2106, 30, 261} \[ -\frac {x^3}{3}-\frac {1}{3} \left (x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 261
Rule 2106
Rubi steps
\begin {align*} \int \frac {x}{x-\sqrt {1+x^2}} \, dx &=-\int x^2 \, dx-\int x \sqrt {1+x^2} \, dx\\ &=-\frac {x^3}{3}-\frac {1}{3} \left (1+x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 1.00 \[ -\frac {x^3}{3}-\frac {1}{3} \left (x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 15, normalized size = 0.71 \[ -\frac {1}{3} \, x^{3} - \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 15, normalized size = 0.71 \[ -\frac {1}{3} \, x^{3} - \frac {1}{3} \, {\left (x^{2} + 1\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.76 \[ -\frac {x^{3}}{3}-\frac {\left (x^{2}+1\right )^{\frac {3}{2}}}{3} \]
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 \[ \int \frac {x}{x - \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 22, normalized size = 1.05 \[ -\sqrt {x^2+1}\,\left (\frac {x^2}{3}+\frac {1}{3}\right )-\frac {x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.37, size = 56, normalized size = 2.67 \[ \frac {2 x^{2}}{3 x - 3 \sqrt {x^{2} + 1}} - \frac {x \sqrt {x^{2} + 1}}{3 x - 3 \sqrt {x^{2} + 1}} + \frac {1}{3 x - 3 \sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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