Optimal. Leaf size=75 \[ \frac {1}{4} \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} x+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}+4}{2 \sqrt {2} \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2}}\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1966, 1357, 720, 724, 206} \[ \frac {1}{4} \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} x+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}+4}{2 \sqrt {2} \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2}}\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 1357
Rule 1966
Rubi steps
\begin {align*} \int \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {x}+x}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {2+x+x^2}}{x^3} \, dx,x,\sqrt {\frac {1}{x}}\right )\right )\\ &=\frac {1}{4} \left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} x-\frac {7}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {2+x+x^2}} \, dx,x,\sqrt {\frac {1}{x}}\right )\\ &=\frac {1}{4} \left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} x+\frac {7}{4} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+\sqrt {\frac {1}{x}}}{\sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}}}\right )\\ &=\frac {1}{4} \left (4+\sqrt {\frac {1}{x}}\right ) \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}} x+\frac {7 \tanh ^{-1}\left (\frac {4+\sqrt {\frac {1}{x}}}{2 \sqrt {2} \sqrt {2+\sqrt {\frac {1}{x}}+\frac {1}{x}}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 75, normalized size = 1.00 \[ \frac {1}{16} \left (4 \left (\sqrt {\frac {1}{x}}+4\right ) \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} x+7 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{x}}+4}{2 \sqrt {2} \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 3.69, size = 94, normalized size = 1.25 \[ \frac {1}{4} \, {\left (4 \, x + \sqrt {x}\right )} \sqrt {\frac {2 \, x + \sqrt {x} + 1}{x}} + \frac {7}{64} \, \sqrt {2} \log \left (-2048 \, x^{2} - 64 \, {\left (32 \, x + 9\right )} \sqrt {x} - 8 \, {\left (3 \, \sqrt {2} {\left (32 \, x + 3\right )} \sqrt {x} + 4 \, \sqrt {2} {\left (32 \, x^{2} + 13 \, x\right )}\right )} \sqrt {\frac {2 \, x + \sqrt {x} + 1}{x}} - 1664 \, x - 113\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 74, normalized size = 0.99 \[ -\frac {1}{16} \, \sqrt {2} {\left (2 \, \sqrt {2} - 7 \, \log \left (2 \, \sqrt {2} - 1\right )\right )} + \frac {1}{4} \, \sqrt {2 \, x + \sqrt {x} + 1} {\left (4 \, \sqrt {x} + 1\right )} - \frac {7}{16} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} \sqrt {x} - \sqrt {2 \, x + \sqrt {x} + 1}\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 123, normalized size = 1.64 \[ \frac {\sqrt {\frac {\sqrt {\frac {1}{x}}\, x +2 x +1}{x}}\, \left (7 \sqrt {2}\, \ln \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{x}}\, \sqrt {x}}{4}+\sqrt {2}\, \sqrt {x}+\sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\right )+16 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\, \sqrt {x}+4 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}\, \sqrt {\frac {1}{x}}\, \sqrt {x}\right ) \sqrt {x}}{16 \sqrt {\sqrt {\frac {1}{x}}\, x +2 x +1}} \]
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 \[ \int \sqrt {\frac {1}{\sqrt {x}} + \frac {1}{x} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {\sqrt {\frac {1}{x}}+\frac {1}{x}+2} \,d x \]
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 \[ \int \sqrt {\sqrt {\frac {1}{x}} + 2 + \frac {1}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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