Optimal. Leaf size=15 \[ 2 \tan ^{-1}\left (\sqrt {-\frac {x}{x+1}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 204
Rule 1960
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*}
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Mathematica [B] time = 0.02, 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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fricas [A] time = 0.44, size = 13, normalized size = 0.87 \[ 2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 20, normalized size = 1.33 \[ -\frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) - \arcsin \left (2 \, x + 1\right ) \mathrm {sgn}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 33, normalized size = 2.20 \[ \frac {\sqrt {-\frac {x}{x +1}}\, \left (x +1\right ) \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{\sqrt {\left (x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 13, normalized size = 0.87 \[ 2 \, \arctan \left (\sqrt {-\frac {x}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 13, normalized size = 0.87 \[ 2\,\mathrm {atan}\left (\sqrt {-\frac {x}{x+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {x}{x + 1}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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