Optimal. Leaf size=34 \[ 2 \sqrt {x+\sqrt {x}}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt {x}}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2010, 2013, 620, 206} \[ 2 \sqrt {x+\sqrt {x}}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt {x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 2010
Rule 2013
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sqrt {x}+x}} \, dx &=2 \sqrt {\sqrt {x}+x}-\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {\sqrt {x}+x}} \, dx\\ &=2 \sqrt {\sqrt {x}+x}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=2 \sqrt {\sqrt {x}+x}-2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ &=2 \sqrt {\sqrt {x}+x}-2 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {\sqrt {x}+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.15 \[ 2 \sqrt {x+\sqrt {x}} \left (1-\frac {\sinh ^{-1}\left (\sqrt [4]{x}\right )}{\sqrt {\sqrt {x}+1} \sqrt [4]{x}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 39, normalized size = 1.15 \[ 2 \, \sqrt {x + \sqrt {x}} + \frac {1}{2} \, \log \left (4 \, \sqrt {x + \sqrt {x}} {\left (2 \, \sqrt {x} + 1\right )} - 8 \, x - 8 \, \sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 27, normalized size = 0.79 \[ 2 \, \sqrt {x + \sqrt {x}} + \log \left (-2 \, \sqrt {x + \sqrt {x}} + 2 \, \sqrt {x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 1.29 \[ -\frac {\sqrt {x +\sqrt {x}}\, \left (\ln \left (\sqrt {x}+\frac {1}{2}+\sqrt {x +\sqrt {x}}\right )-2 \sqrt {x +\sqrt {x}}\right )}{\sqrt {\left (\sqrt {x}+1\right ) \sqrt {x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x + \sqrt {x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.33, size = 39, normalized size = 1.15 \[ \frac {2\,\sqrt {x}\,\left (\sqrt {x}+1\right )+x^{1/4}\,\mathrm {asin}\left (x^{1/4}\,1{}\mathrm {i}\right )\,\sqrt {\sqrt {x}+1}\,2{}\mathrm {i}}{\sqrt {x+\sqrt {x}}} \]
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 \[ \int \frac {1}{\sqrt {\sqrt {x} + x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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