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

Optimal. Leaf size=19 \[ -\frac{\sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{3}} \]

[Out]

-(ArcSin[(2 - 3*x)/Sqrt[10]]/Sqrt[3])

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Rubi [A]  time = 0.0263589, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-(ArcSin[(2 - 3*x)/Sqrt[10]]/Sqrt[3])

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Rubi in Sympy [A]  time = 1.53222, size = 34, normalized size = 1.79 \[ - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x + 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*atan(sqrt(3)*(-6*x + 4)/(6*sqrt(-3*x**2 + 4*x + 2)))/3

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Mathematica [A]  time = 0.0162651, size = 19, normalized size = 1. \[ -\frac{\sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-(ArcSin[(2 - 3*x)/Sqrt[10]]/Sqrt[3])

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[{\frac{\sqrt{3}}{3}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+4*x+2)^(1/2),x)

[Out]

1/3*3^(1/2)*arcsin(3/10*10^(1/2)*(x-2/3))

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Maxima [A]  time = 0.883958, size = 22, normalized size = 1.16 \[ -\frac{1}{3} \, \sqrt{3} \arcsin \left (-\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*arcsin(-1/10*sqrt(10)*(3*x - 2))

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Fricas [A]  time = 0.232517, size = 38, normalized size = 2. \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (3 \, x - 2\right )}}{3 \, \sqrt{-3 \, x^{2} + 4 \, x + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215309, size = 22, normalized size = 1.16 \[ \frac{1}{3} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arcsin(1/10*sqrt(10)*(3*x - 2))