Optimal. Leaf size=9 \[ \frac {1}{\sqrt {\frac {1}{x^2}+1}} \]
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Rubi [A] time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {25, 261} \[ \frac {1}{\sqrt {\frac {1}{x^2}+1}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 261
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {1}{x^2}} x}{\left (1+x^2\right )^2} \, dx &=\int \frac {1}{\left (1+\frac {1}{x^2}\right )^{3/2} x^3} \, dx\\ &=\frac {1}{\sqrt {1+\frac {1}{x^2}}}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 20, normalized size = 2.22 \[ \frac {\sqrt {\frac {1}{x^2}+1} x^2}{x^2+1} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 28, normalized size = 3.11 \[ \frac {x^{2} \sqrt {\frac {x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 11, normalized size = 1.22 \[ \frac {x \mathrm {sgn}\relax (x)}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 23, normalized size = 2.56 \[ \frac {\sqrt {\frac {x^{2}+1}{x^{2}}}\, x^{2}}{x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 11, normalized size = 1.22 \[ \frac {1}{\sqrt {\frac {x^{2} + 1}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 18, normalized size = 2.00 \[ \frac {x^2\,\sqrt {\frac {1}{x^2}+1}}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.83, size = 8, normalized size = 0.89 \[ \frac {x}{\sqrt {x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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