Optimal. Leaf size=32 \[ -\frac{\tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]
[Out]
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Rubi [A] time = 0.0514027, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*a*x - b*x^2)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 7.22237, size = 27, normalized size = 0.84 \[ - \frac{\operatorname{atanh}{\left (\frac{a - b x}{\sqrt{a^{2} + b^{2}}} \right )}}{\sqrt{a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x**2+2*a*x+b),x)
[Out]
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Mathematica [A] time = 0.0184163, size = 41, normalized size = 1.28 \[ -\frac{\tan ^{-1}\left (\frac{b x-a}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*a*x - b*x^2)^(-1),x]
[Out]
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Maple [A] time = 0.005, size = 31, normalized size = 1. \[{1{\it Artanh} \left ({\frac{2\,bx-2\,a}{2}{\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x^2+2*a*x+b),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/(b*x^2 - 2*a*x - b),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215828, size = 115, normalized size = 3.59 \[ \frac{\log \left (-\frac{2 \, a^{3} + 2 \, a b^{2} - 2 \,{\left (a^{2} b + b^{3}\right )} x -{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} + b^{2}\right )} \sqrt{a^{2} + b^{2}}}{b x^{2} - 2 \, a x - b}\right )}{2 \, \sqrt{a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b*x^2 - 2*a*x - b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.653943, size = 102, normalized size = 3.19 \[ - \frac{\sqrt{\frac{1}{a^{2} + b^{2}}} \log{\left (x + \frac{- a^{2} \sqrt{\frac{1}{a^{2} + b^{2}}} - a - b^{2} \sqrt{\frac{1}{a^{2} + b^{2}}}}{b} \right )}}{2} + \frac{\sqrt{\frac{1}{a^{2} + b^{2}}} \log{\left (x + \frac{a^{2} \sqrt{\frac{1}{a^{2} + b^{2}}} - a + b^{2} \sqrt{\frac{1}{a^{2} + b^{2}}}}{b} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x**2+2*a*x+b),x)
[Out]
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GIAC/XCAS [A] time = 0.21338, size = 74, normalized size = 2.31 \[ -\frac{{\rm ln}\left (\frac{{\left | 2 \, b x - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b x - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{2 \, \sqrt{a^{2} + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(b*x^2 - 2*a*x - b),x, algorithm="giac")
[Out]