Optimal. Leaf size=19 \[ -\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 19, 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 = {619, 215} \[ -\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {5-4 x+3 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{44}}} \, dx,x,-4+6 x\right )}{2 \sqrt {33}}\\ &=-\frac {\sinh ^{-1}\left (\frac {2-3 x}{\sqrt {11}}\right )}{\sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 0.95 \[ \frac {\sinh ^{-1}\left (\frac {3 x-2}{\sqrt {11}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.08, size = 33, normalized size = 1.74 \[ -\frac {\log \left (\sqrt {3} \sqrt {3 x^2-4 x+5}-3 x+2\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.97, size = 38, normalized size = 2.00 \[ \frac {1}{6} \, \sqrt {3} \log \left (-2 \, \sqrt {3} \sqrt {3 \, x^{2} - 4 \, x + 5} {\left (3 \, x - 2\right )} - 18 \, x^{2} + 24 \, x - 19\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.08, size = 33, normalized size = 1.74 \[ -\frac {1}{3} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - 4 \, x + 5}\right )} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 15, normalized size = 0.79
method | result | size |
default | \(\frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {11}\, \left (x -\frac {2}{3}\right )}{11}\right )}{3}\) | \(15\) |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +2 \RootOf \left (\textit {\_Z}^{2}-3\right )+3 \sqrt {3 x^{2}-4 x +5}\right )}{3}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.00, size = 16, normalized size = 0.84 \[ \frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{11} \, \sqrt {11} {\left (3 \, x - 2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 26, normalized size = 1.37 \[ \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x-\frac {2}{3}\right )+\sqrt {3\,x^2-4\,x+5}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{2} - 4 x + 5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________