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 = 0.0230596, antiderivative size = 21, normalized size of antiderivative = 1., 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 2106
Rule 30
Rule 261
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*}
Mathematica [A] time = 0.0238108, size = 21, normalized size = 1. \[ -\frac{x^3}{3}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 16, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{3}}-{\frac{1}{3} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{x - \sqrt{x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4724, size = 43, normalized size = 2.05 \begin{align*} -\frac{1}{3} \, x^{3} - \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.359005, size = 56, normalized size = 2.67 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20324, size = 20, normalized size = 0.95 \begin{align*} -\frac{1}{3} \, x^{3} - \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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