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