Optimal. Leaf size=11 \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]
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Rubi [A] time = 0.0034446, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {25, 261} \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 261
Rubi steps
\begin{align*} \int \frac{\sqrt{1+\frac{1}{x}}}{(1+x)^2} \, dx &=\int \frac{1}{\left (1+\frac{1}{x}\right )^{3/2} x^2} \, dx\\ &=\frac{2}{\sqrt{1+\frac{1}{x}}}\\ \end{align*}
Mathematica [A] time = 0.0052393, size = 11, normalized size = 1. \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 1.6 \begin{align*} 2\,{\frac{x}{1+x}\sqrt{{\frac{1+x}{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{\sqrt{\frac{1}{x} + 1}}{{\left (x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62278, size = 39, normalized size = 3.55 \begin{align*} \frac{2 \, x \sqrt{\frac{x + 1}{x}}}{x + 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{\sqrt{1 + \frac{1}{x}}}{\left (x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10869, size = 31, normalized size = 2.82 \begin{align*} \frac{2 \, \mathrm{sgn}\left (x\right )}{x - \sqrt{x^{2} + x} + 1} - 2 \, \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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