Optimal. Leaf size=12 \[ -\frac{2 x}{\sqrt{x^3+x}} \]
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Rubi [A] time = 0.0701814, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2056, 449} \[ -\frac{2 x}{\sqrt{x^3+x}} \]
Antiderivative was successfully verified.
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Rule 2056
Rule 449
Rubi steps
\begin{align*} \int \frac{-1+x^2}{\left (1+x^2\right ) \sqrt{x+x^3}} \, 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+x^3}}\\ &=-\frac{2 x}{\sqrt{x+x^3}}\\ \end{align*}
Mathematica [A] time = 0.0103464, size = 12, normalized size = 1. \[ -\frac{2 x}{\sqrt{x^3+x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 11, normalized size = 0.9 \begin{align*} -2\,{\frac{x}{\sqrt{{x}^{3}+x}}} \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{x^{3} + 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 = 1.50459, size = 38, normalized size = 3.17 \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{x^{3} + x}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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