Optimal. Leaf size=14 \[ \frac {1}{3} \sinh ^{-1}\left (\frac {1}{2} (-1+3 x)\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 221}
\begin {gather*} \frac {1}{3} \sinh ^{-1}\left (\frac {1}{2} (3 x-1)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 633
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {5-6 x+9 x^2}} \, dx &=\frac {1}{36} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{144}}} \, dx,x,-6+18 x\right )\\ &=\frac {1}{3} \sinh ^{-1}\left (\frac {1}{2} (-1+3 x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 24, normalized size = 1.71 \begin {gather*} -\frac {1}{3} \log \left (1-3 x+\sqrt {5-6 x+9 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 9, normalized size = 0.64
method | result | size |
default | \(\frac {\arcsinh \left (\frac {3 x}{2}-\frac {1}{2}\right )}{3}\) | \(9\) |
trager | \(-\frac {\ln \left (\sqrt {9 x^{2}-6 x +5}+1-3 x \right )}{3}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{3} \, \operatorname {arsinh}\left (\frac {3}{2} \, x - \frac {1}{2}\right ) \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. 20 vs.
\(2 (8) = 16\).
time = 2.70, size = 20, normalized size = 1.43 \begin {gather*} -\frac {1}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {9 x^{2} - 6 x + 5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (8) = 16\).
time = 1.51, size = 40, normalized size = 2.86 \begin {gather*} \frac {1}{6} \, \sqrt {9 \, x^{2} - 6 \, x + 5} {\left (3 \, x - 1\right )} - \frac {2}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 20, normalized size = 1.43 \begin {gather*} \frac {\ln \left (3\,x+\sqrt {9\,x^2-6\,x+5}-1\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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