3.2423 \(\int \frac{1}{x \sqrt{-2+4 x+3 x^2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{3 x^2+4 x-2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTan[(Sqrt[2]*(1 - x))/Sqrt[-2 + 4*x + 3*x^2]]/Sqrt[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0363673, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{3 x^2+4 x-2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-2 + 4*x + 3*x^2]),x]

[Out]

-(ArcTan[(Sqrt[2]*(1 - x))/Sqrt[-2 + 4*x + 3*x^2]]/Sqrt[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.95507, size = 32, normalized size = 0.97 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (4 x - 4\right )}{4 \sqrt{3 x^{2} + 4 x - 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(3*x**2+4*x-2)**(1/2),x)

[Out]

sqrt(2)*atan(sqrt(2)*(4*x - 4)/(4*sqrt(3*x**2 + 4*x - 2)))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0273038, 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]  Integrate[1/(x*Sqrt[-2 + 4*x + 3*x^2]),x]

[Out]

ArcTan[(-1 + x)/Sqrt[-1 + 2*x + (3*x^2)/2]]/Sqrt[2]

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 29, normalized size = 0.9 \[{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( -4+4\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+4\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^2+4*x-2)^(1/2),x)

[Out]

1/2*2^(1/2)*arctan(1/4*(-4+4*x)*2^(1/2)/(3*x^2+4*x-2)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.750031, size = 35, normalized size = 1.06 \[ \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{\sqrt{10} x}{5 \,{\left | x \right |}} - \frac{\sqrt{10}}{5 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 4*x - 2)*x),x, algorithm="maxima")

[Out]

1/2*sqrt(2)*arcsin(1/5*sqrt(10)*x/abs(x) - 1/5*sqrt(10)/abs(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.227348, size = 34, normalized size = 1.03 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )}}{\sqrt{3 \, x^{2} + 4 \, x - 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 4*x - 2)*x),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(sqrt(2)*(x - 1)/sqrt(3*x^2 + 4*x - 2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{3 x^{2} + 4 x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(3*x**2+4*x-2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(3*x**2 + 4*x - 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229034, size = 41, normalized size = 1.24 \[ \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(3*x^2 + 4*x - 2)*x),x, algorithm="giac")

[Out]

sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(3)*x - sqrt(3*x^2 + 4*x - 2)))