Optimal. Leaf size=26 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0180275, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a - b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.32923, size = 22, normalized size = 0.85 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{\sqrt{b}} \]
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 = 0.0104618, size = 26, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a-b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a - b*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 21, normalized size = 0.8 \[{1\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{b}}}} \]
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 = 0.233812, 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 = 3.68542, size = 46, normalized size = 1.77 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{for}\: \left |{\frac{b x^{2}}{a}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
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 = 0.217663, size = 38, normalized size = 1.46 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x + \sqrt{-b x^{2} + a} \right |}\right )}{\sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-b*x^2 + a),x, algorithm="giac")
[Out]