Optimal. Leaf size=11 \[ \frac {2}{\sqrt {\frac {1}{x}+1}} \]
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Rubi [A] time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {25, 261} \[ \frac {2}{\sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 261
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx &=\int \frac {1}{\left (1+\frac {1}{x}\right )^{3/2} x^2} \, dx\\ &=\frac {2}{\sqrt {1+\frac {1}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {2}{\sqrt {\frac {1}{x}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 17, normalized size = 1.55 \[ \frac {2 \, x \sqrt {\frac {x + 1}{x}}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 23, normalized size = 2.09 \[ \frac {2 \, \mathrm {sgn}\relax (x)}{x - \sqrt {x^{2} + x} + 1} - 2 \, \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 1.64 \[ \frac {2 \sqrt {\frac {x +1}{x}}\, x}{x +1} \]
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 \[ \int \frac {\sqrt {\frac {1}{x} + 1}}{{\left (x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.13, size = 15, normalized size = 1.36 \[ \frac {2\,x\,\sqrt {\frac {1}{x}+1}}{x+1} \]
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 \[ \int \frac {\sqrt {1 + \frac {1}{x}}}{\left (x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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