3.90 \(\int \frac{1}{b+2 a x-b x^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac{\tanh ^{-1}\left (\frac{a-b x}{\sqrt{a^2+b^2}}\right )}{\sqrt{a^2+b^2}} \]

[Out]

-(ArcTanh[(a - b*x)/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])

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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]

-(ArcTanh[(a - b*x)/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])

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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]

-atanh((a - b*x)/sqrt(a**2 + b**2))/sqrt(a**2 + b**2)

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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]

-(ArcTan[(-a + b*x)/Sqrt[-a^2 - b^2]]/Sqrt[-a^2 - b^2])

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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]

1/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*x-2*a)/(a^2+b^2)^(1/2))

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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]

Exception raised: ValueError

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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]

1/2*log(-(2*a^3 + 2*a*b^2 - 2*(a^2*b + b^3)*x - (b^2*x^2 - 2*a*b*x + 2*a^2 + b^2
)*sqrt(a^2 + b^2))/(b*x^2 - 2*a*x - b))/sqrt(a^2 + b^2)

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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]

-sqrt(1/(a**2 + b**2))*log(x + (-a**2*sqrt(1/(a**2 + b**2)) - a - b**2*sqrt(1/(a
**2 + b**2)))/b)/2 + sqrt(1/(a**2 + b**2))*log(x + (a**2*sqrt(1/(a**2 + b**2)) -
a + b**2*sqrt(1/(a**2 + b**2)))/b)/2

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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]

-1/2*ln(abs(2*b*x - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*x - 2*a + 2*sqrt(a^2 + b^2)
))/sqrt(a^2 + b^2)