Optimal. Leaf size=33 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {3 x^2+4 x-2}}\right )}{\sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, 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 = {724, 204} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {3 x^2+4 x-2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 724
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-2+4 x+3 x^2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4+4 x}{\sqrt {-2+4 x+3 x^2}}\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-2+4 x+3 x^2}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 0.82 \[ \frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {\frac {3 x^2}{2}+2 x-1}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 25, normalized size = 0.76 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{\sqrt {3 \, x^{2} + 4 \, x - 2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 30, normalized size = 0.91 \[ \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x - 2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 29, normalized size = 0.88 \[ \frac {\sqrt {2}\, \arctan \left (\frac {\left (4 x -4\right ) \sqrt {2}}{4 \sqrt {3 x^{2}+4 x -2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.99, size = 26, normalized size = 0.79 \[ \frac {1}{2} \, \sqrt {2} \arcsin \left (\frac {\sqrt {10} x}{5 \, {\left | x \right |}} - \frac {\sqrt {10}}{5 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.37, size = 34, normalized size = 1.03 \[ \frac {\sqrt {2}\,\ln \left (\frac {2\,x-2+\sqrt {2}\,\sqrt {3\,x^2+4\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {3 x^{2} + 4 x - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________