Optimal. Leaf size=16 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \]
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Rubi [A] time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {620, 206} \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {4 x+x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {4 x+x^2}}\right )\\ &=2 \tanh ^{-1}\left (\frac {x}{\sqrt {4 x+x^2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 33, normalized size = 2.06 \[ \frac {2 \sqrt {x} \sqrt {x+4} \sinh ^{-1}\left (\frac {\sqrt {x}}{2}\right )}{\sqrt {x (x+4)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 17, normalized size = 1.06 \[ -\log \left (-x + \sqrt {x^{2} + 4 \, x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 18, normalized size = 1.12 \[ -\log \left ({\left | -x + \sqrt {x^{2} + 4 \, x} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.88 \[ \ln \left (x +2+\sqrt {x^{2}+4 x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 17, normalized size = 1.06 \[ \log \left (2 \, x + 2 \, \sqrt {x^{2} + 4 \, x} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 11, normalized size = 0.69 \[ \ln \left (x+\sqrt {x\,\left (x+4\right )}+2\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 4 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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