Optimal. Leaf size=19 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
[Out]
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Rubi [A] time = 0.0123318, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-9 + 4*x^2],x]
[Out]
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Rubi in Sympy [A] time = 1.21891, size = 15, normalized size = 0.79 \[ \frac{\operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4*x**2-9)**(1/2),x)
[Out]
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Mathematica [B] time = 0.00505957, size = 43, normalized size = 2.26 \[ \frac{1}{4} \log \left (\frac{2 x}{\sqrt{4 x^2-9}}+1\right )-\frac{1}{4} \log \left (1-\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-9 + 4*x^2],x]
[Out]
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Maple [A] time = 0.003, size = 22, normalized size = 1.2 \[{\frac{\sqrt{4}}{4}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4*x^2-9)^(1/2),x)
[Out]
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Maxima [A] time = 1.49784, size = 24, normalized size = 1.26 \[ \frac{1}{2} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^2 - 9),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211873, size = 22, normalized size = 1.16 \[ -\frac{1}{2} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^2 - 9),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.31438, size = 7, normalized size = 0.37 \[ \frac{\operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4*x**2-9)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215207, size = 23, normalized size = 1.21 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^2 - 9),x, algorithm="giac")
[Out]