Optimal. Leaf size=29 \[ \tan ^{-1}\left (\sqrt{-\frac{x+1}{x}}\right )-x \sqrt{-\frac{x+1}{x}} \]
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Rubi [A] time = 0.0118638, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1972, 242, 51, 63, 204} \[ \tan ^{-1}\left (\sqrt{-\frac{x+1}{x}}\right )-x \sqrt{-\frac{x+1}{x}} \]
Antiderivative was successfully verified.
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Rule 1972
Rule 242
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{-1-x}{x}}} \, dx &=\int \frac{1}{\sqrt{-1-\frac{1}{x}}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^2} \, dx,x,\frac{1}{x}\right )\\ &=-x \sqrt{-\frac{1+x}{x}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,\frac{1}{x}\right )\\ &=-x \sqrt{-\frac{1+x}{x}}-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-\frac{1+x}{x}}\right )\\ &=-x \sqrt{-\frac{1+x}{x}}+\tan ^{-1}\left (\sqrt{-\frac{1+x}{x}}\right )\\ \end{align*}
Mathematica [A] time = 0.0132626, size = 43, normalized size = 1.48 \[ \frac{\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )}{\sqrt{x} \sqrt{-\frac{x+1}{x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 1.5 \begin{align*}{\frac{1+x}{2} \left ( 2\,\sqrt{-{x}^{2}-x}+\arcsin \left ( 1+2\,x \right ) \right ){\frac{1}{\sqrt{-{\frac{1+x}{x}}}}}{\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 = 1.61727, size = 47, normalized size = 1.62 \begin{align*} -\frac{\sqrt{-\frac{x + 1}{x}}}{\frac{x + 1}{x} - 1} + \arctan \left (\sqrt{-\frac{x + 1}{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46477, size = 65, normalized size = 2.24 \begin{align*} -x \sqrt{-\frac{x + 1}{x}} + \arctan \left (\sqrt{-\frac{x + 1}{x}}\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{1}{\sqrt{\frac{- x - 1}{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19008, size = 47, normalized size = 1.62 \begin{align*} \frac{1}{4} \, \pi \mathrm{sgn}\left (x\right ) - \frac{\arcsin \left (2 \, x + 1\right )}{2 \, \mathrm{sgn}\left (x\right )} - \frac{\sqrt{-x^{2} - x}}{\mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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