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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ 2 \sqrt {x+4}-4 \tanh ^{-1}\left (\frac {\sqrt {x+4}}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 1.17 \[ 2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left (\sqrt {x + 4} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 29, normalized size = 1.21 \[ 2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left ({\left | \sqrt {x + 4} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 1.21 \[ 2 \ln \left (\sqrt {x +4}-2\right )-2 \ln \left (\sqrt {x +4}+2\right )+2 \sqrt {x +4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 28, normalized size = 1.17 \[ 2 \, \sqrt {x + 4} - 2 \, \log \left (\sqrt {x + 4} + 2\right ) + 2 \, \log \left (\sqrt {x + 4} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.75 \[ 2\,\sqrt {x+4}-4\,\mathrm {atanh}\left (\frac {\sqrt {x+4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 44, normalized size = 1.83 \[ \begin {cases} 2 \sqrt {x + 4} - 4 \operatorname {acoth}{\left (\frac {\sqrt {x + 4}}{2} \right )} & \text {for}\: \frac {\left |{x + 4}\right |}{4} > 1 \\2 \sqrt {x + 4} - 4 \operatorname {atanh}{\left (\frac {\sqrt {x + 4}}{2} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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