Optimal. Leaf size=22 \[ \sqrt {\frac {1}{x}+1} x+\tanh ^{-1}\left (\sqrt {\frac {1}{x}+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {1972, 242, 47, 63, 207} \[ \sqrt {\frac {1}{x}+1} x+\tanh ^{-1}\left (\sqrt {\frac {1}{x}+1}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
Rule 242
Rule 1972
Rubi steps
\begin {align*} \int \sqrt {\frac {1+x}{x}} \, dx &=\int \sqrt {1+\frac {1}{x}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {1+\frac {1}{x}} x-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )\\ &=\sqrt {1+\frac {1}{x}} x-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {1}{x}}\right )\\ &=\sqrt {1+\frac {1}{x}} x+\tanh ^{-1}\left (\sqrt {1+\frac {1}{x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \sqrt {\frac {1}{x}+1} x+\tanh ^{-1}\left (\sqrt {\frac {1}{x}+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 40, normalized size = 1.82 \[ x \sqrt {\frac {x + 1}{x}} + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 31, normalized size = 1.41 \[ -\frac {1}{2} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \mathrm {sgn}\relax (x) + \sqrt {x^{2} + x} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 41, normalized size = 1.86 \[ \frac {\sqrt {\frac {x +1}{x}}\, \left (\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )+2 \sqrt {x^{2}+x}\right ) x}{2 \sqrt {\left (x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 50, normalized size = 2.27 \[ \frac {\sqrt {\frac {x + 1}{x}}}{\frac {x + 1}{x} - 1} + \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {x + 1}{x}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 18, normalized size = 0.82 \[ \mathrm {atanh}\left (\sqrt {\frac {1}{x}+1}\right )+x\,\sqrt {\frac {1}{x}+1} \]
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 \[ \int \sqrt {\frac {x + 1}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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