Optimal. Leaf size=15 \[ 2 \tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
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Rubi [A] time = 0.012346, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1960, 204} \[ 2 \tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 1960
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{-\frac{x}{1+x}}}{x} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-\frac{x}{1+x}}\right )\right )\\ &=2 \tan ^{-1}\left (\sqrt{-\frac{x}{1+x}}\right )\\ \end{align*}
Mathematica [B] time = 0.013448, size = 32, normalized size = 2.13 \[ \frac{2 \sqrt{-\frac{x}{x+1}} \sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 33, normalized size = 2.2 \begin{align*}{(1+x)\sqrt{-{\frac{x}{1+x}}}\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47703, size = 18, normalized size = 1.2 \begin{align*} 2 \, \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68155, size = 38, normalized size = 2.53 \begin{align*} 2 \, \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \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{- \frac{x}{x + 1}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.138, size = 27, normalized size = 1.8 \begin{align*} -\frac{1}{2} \, \pi \mathrm{sgn}\left (x + 1\right ) - \arcsin \left (2 \, x + 1\right ) \mathrm{sgn}\left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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