Optimal. Leaf size=14 \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]
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Rubi [A] time = 0.149441, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6719, 449} \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 449
Rubi steps
\begin{align*} \int \frac{-1+x^2}{\left (1+x^2\right ) \sqrt{x \left (1+x^2\right )}} \, dx &=\frac{\left (\sqrt{x} \sqrt{1+x^2}\right ) \int \frac{-1+x^2}{\sqrt{x} \left (1+x^2\right )^{3/2}} \, dx}{\sqrt{x \left (1+x^2\right )}}\\ &=-\frac{2 x}{\sqrt{x \left (1+x^2\right )}}\\ \end{align*}
Mathematica [A] time = 0.0257639, size = 12, normalized size = 0.86 \[ -\frac{2 x}{\sqrt{x^3+x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 13, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{\sqrt{x \left ({x}^{2}+1 \right ) }}} \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^{2} - 1}{\sqrt{{\left (x^{2} + 1\right )} x}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45797, size = 38, normalized size = 2.71 \begin{align*} -\frac{2 \, \sqrt{x^{3} + x}}{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x - 1\right ) \left (x + 1\right )}{\sqrt{x \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} - 1}{\sqrt{{\left (x^{2} + 1\right )} x}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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