Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
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Rubi [A] time = 0.0200329, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In] Int[(a^2 - b^2*x^2)^(-1),x]
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Rubi in Sympy [A] time = 2.83151, size = 8, normalized size = 0.57 \[ \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b**2*x**2+a**2),x)
[Out]
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Mathematica [A] time = 0.00544131, size = 14, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 - b^2*x^2)^(-1),x]
[Out]
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Maple [B] time = 0.009, size = 32, normalized size = 2.3 \[{\frac{\ln \left ( bx+a \right ) }{2\,ab}}-{\frac{\ln \left ( bx-a \right ) }{2\,ab}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b^2*x^2+a^2),x)
[Out]
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Maxima [A] time = 1.34838, size = 42, normalized size = 3. \[ \frac{\log \left (b x + a\right )}{2 \, a b} - \frac{\log \left (b x - a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b^2*x^2 - a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22663, size = 34, normalized size = 2.43 \[ \frac{\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b^2*x^2 - a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.158839, size = 20, normalized size = 1.43 \[ - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{2} - \frac{\log{\left (\frac{a}{b} + x \right )}}{2}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b**2*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.21567, size = 45, normalized size = 3.21 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b^2*x^2 - a^2),x, algorithm="giac")
[Out]