Optimal. Leaf size=9 \[ \frac {1}{\sqrt {1+\frac {1}{x^2}}} \]
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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, 267}
\begin {gather*} \frac {1}{\sqrt {\frac {1}{x^2}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 267
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] Leaf count is larger than twice the leaf count of optimal. \(20\) vs. \(2(9)=18\).
time = 0.02, size = 20, normalized size = 2.22 \begin {gather*} \frac {\sqrt {1+\frac {1}{x^2}} x^2}{1+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(22\) vs.
\(2(7)=14\).
time = 0.34, size = 23, normalized size = 2.56
method | result | size |
gosper | \(\frac {x^{2} \sqrt {\frac {x^{2}+1}{x^{2}}}}{x^{2}+1}\) | \(23\) |
default | \(\frac {x^{2} \sqrt {\frac {x^{2}+1}{x^{2}}}}{x^{2}+1}\) | \(23\) |
risch | \(\frac {x^{2} \sqrt {\frac {x^{2}+1}{x^{2}}}}{x^{2}+1}\) | \(23\) |
trager | \(\frac {x^{2} \sqrt {-\frac {-x^{2}-1}{x^{2}}}}{x^{2}+1}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 11, normalized size = 1.22 \begin {gather*} \frac {1}{\sqrt {\frac {x^{2} + 1}{x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (7) = 14\).
time = 0.37, size = 28, normalized size = 3.11 \begin {gather*} \frac {x^{2} \sqrt {\frac {x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.56, size = 8, normalized size = 0.89 \begin {gather*} \frac {x}{\sqrt {x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.12, size = 11, normalized size = 1.22 \begin {gather*} \frac {x \mathrm {sgn}\left (x\right )}{\sqrt {x^{2} + 1}} \end {gather*}
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 \begin {gather*} \frac {x^2\,\sqrt {\frac {1}{x^2}+1}}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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