Optimal. Leaf size=34 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]
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Rubi [A] time = 0.0557801, antiderivative size = 34, 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 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]
Antiderivative was successfully verified.
[In] Int[(a + (b - a*c)*x^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 4.2146, size = 29, normalized size = 0.85 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt{- a c + b}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{- a c + b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+(-a*c+b)*x**2),x)
[Out]
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Mathematica [A] time = 0.0277972, size = 36, normalized size = 1.06 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{a c-b}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a c-b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + (b - a*c)*x^2)^(-1),x]
[Out]
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Maple [A] time = 0.008, size = 34, normalized size = 1. \[{1{\it Artanh} \left ({ \left ( ac-b \right ) x{\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+(-a*c+b)*x^2),x)
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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/((a*c - b)*x^2 - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228245, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \,{\left (a^{2} c - a b\right )} x + \sqrt{a^{2} c - a b}{\left ({\left (a c - b\right )} x^{2} + a\right )}}{{\left (a c - b\right )} x^{2} - a}\right )}{2 \, \sqrt{a^{2} c - a b}}, \frac{\arctan \left (\frac{\sqrt{-a^{2} c + a b} x}{a}\right )}{\sqrt{-a^{2} c + a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((a*c - b)*x^2 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.690462, size = 60, normalized size = 1.76 \[ - \frac{\sqrt{\frac{1}{a \left (a c - b\right )}} \log{\left (- a \sqrt{\frac{1}{a \left (a c - b\right )}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{a \left (a c - b\right )}} \log{\left (a \sqrt{\frac{1}{a \left (a c - b\right )}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+(-a*c+b)*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.207112, size = 50, normalized size = 1.47 \[ -\frac{\arctan \left (\frac{a c x - b x}{\sqrt{-a^{2} c + a b}}\right )}{\sqrt{-a^{2} c + a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((a*c - b)*x^2 - a),x, algorithm="giac")
[Out]