Optimal. Leaf size=11 \[ \frac {x}{\sqrt {1+x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {197}
\begin {gather*} \frac {x}{\sqrt {x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx &=\frac {x}{\sqrt {1+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} \frac {x}{\sqrt {1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.89, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 10, normalized size = 0.91
method | result | size |
gosper | \(\frac {x}{\sqrt {x^{2}+1}}\) | \(10\) |
default | \(\frac {x}{\sqrt {x^{2}+1}}\) | \(10\) |
trager | \(\frac {x}{\sqrt {x^{2}+1}}\) | \(10\) |
meijerg | \(\frac {x}{\sqrt {x^{2}+1}}\) | \(10\) |
risch | \(\frac {x}{\sqrt {x^{2}+1}}\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {x^{2} + 1}} \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. 22 vs.
\(2 (9) = 18\).
time = 0.32, size = 22, normalized size = 2.00 \begin {gather*} \frac {x^{2} + \sqrt {x^{2} + 1} x + 1}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 8, normalized size = 0.73 \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 = 0.00, size = 20, normalized size = 1.82 \begin {gather*} \frac {2 x \sqrt {x^{2}+1}}{2 \left (x^{2}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 9, normalized size = 0.82 \begin {gather*} \frac {x}{\sqrt {x^2+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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