Optimal. Leaf size=24 \[ 2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right ) \]
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Rubi [A] time = 0.0054734, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {50, 63, 207} \[ 2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{4+x}}{x} \, dx &=2 \sqrt{4+x}+4 \int \frac{1}{x \sqrt{4+x}} \, dx\\ &=2 \sqrt{4+x}+8 \operatorname{Subst}\left (\int \frac{1}{-4+x^2} \, dx,x,\sqrt{4+x}\right )\\ &=2 \sqrt{4+x}-4 \tanh ^{-1}\left (\frac{\sqrt{4+x}}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0045924, size = 24, normalized size = 1. \[ 2 \sqrt{x+4}-4 \tanh ^{-1}\left (\frac{\sqrt{x+4}}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 29, normalized size = 1.2 \begin{align*} 2\,\sqrt{4+x}-2\,\ln \left ( \sqrt{4+x}+2 \right ) +2\,\ln \left ( \sqrt{4+x}-2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940056, size = 38, normalized size = 1.58 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left (\sqrt{x + 4} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79127, size = 88, normalized size = 3.67 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left (\sqrt{x + 4} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.716193, size = 44, normalized size = 1.83 \begin{align*} \begin{cases} 2 \sqrt{x + 4} - 4 \operatorname{acoth}{\left (\frac{\sqrt{x + 4}}{2} \right )} & \text{for}\: \frac{\left |{x + 4}\right |}{4} > 1 \\2 \sqrt{x + 4} - 4 \operatorname{atanh}{\left (\frac{\sqrt{x + 4}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05978, size = 39, normalized size = 1.62 \begin{align*} 2 \, \sqrt{x + 4} - 2 \, \log \left (\sqrt{x + 4} + 2\right ) + 2 \, \log \left ({\left | \sqrt{x + 4} - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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