Optimal. Leaf size=27 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2-a}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0192454, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2-a}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.62496, size = 22, normalized size = 0.81 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{- a + b x^{2}}} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2-a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0119709, size = 27, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2-a}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-a + b*x^2],x]
[Out]
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Maple [A] time = 0.003, size = 23, normalized size = 0.9 \[{1\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}-a} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2-a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^2 - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227841, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (2 \, \sqrt{b x^{2} - a} b x +{\left (2 \, b x^{2} - a\right )} \sqrt{b}\right )}{2 \, \sqrt{b}}, \frac{\arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} - a}}\right )}{\sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^2 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.78788, size = 46, normalized size = 1.7 \[ \begin{cases} \frac{\operatorname{acosh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{for}\: \left |{\frac{b x^{2}}{a}}\right | > 1 \\- \frac{i \operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2-a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220033, size = 34, normalized size = 1.26 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} - a} \right |}\right )}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^2 - a),x, algorithm="giac")
[Out]