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

Optimal. Leaf size=35 \[ -\frac{\tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

[Out]

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

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Rubi [A]  time = 0.0653591, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\tanh ^{-1}\left (\frac{a+b x}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}} \]

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 = 7.51656, size = 27, normalized size = 0.77 \[ - \frac{\operatorname{atanh}{\left (\frac{a + b x}{\sqrt{a^{2} - b^{2}}} \right )}}{\sqrt{a^{2} - b^{2}}} \]

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 = 0.0159774, size = 34, normalized size = 0.97 \[ \frac{\tan ^{-1}\left (\frac{a+b x}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}} \]

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 = 0.008, size = 35, normalized size = 1. \[{1\arctan \left ({\frac{2\,bx+2\,a}{2}{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \]

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)*arctan(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 = 0.217598, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-\frac{2 \, a^{3} - 2 \, a b^{2} + 2 \,{\left (a^{2} b - b^{3}\right )} x -{\left (b^{2} x^{2} + 2 \, a b x + 2 \, a^{2} - b^{2}\right )} \sqrt{a^{2} - b^{2}}}{b x^{2} + 2 \, a x + b}\right )}{2 \, \sqrt{a^{2} - b^{2}}}, \frac{\arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b x + a\right )}}{a^{2} - b^{2}}\right )}{\sqrt{-a^{2} + b^{2}}}\right ] \]

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), arctan(-sqrt(-a^2 + b^
2)*(b*x + a)/(a^2 - b^2))/sqrt(-a^2 + b^2)]

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Sympy [A]  time = 0.57804, size = 100, normalized size = 2.86 \[ \frac{\sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} \log{\left (x + \frac{- a^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} + a + b^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} - \frac{\sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} \log{\left (x + \frac{a^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}} + a - b^{2} \sqrt{\frac{1}{\left (a - b\right ) \left (a + b\right )}}}{b} \right )}}{2} \]

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

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GIAC/XCAS [A]  time = 0.207708, size = 41, normalized size = 1.17 \[ \frac{\arctan \left (\frac{b x + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^2 + 2*a*x + b),x, algorithm="giac")

[Out]

arctan((b*x + a)/sqrt(-a^2 + b^2))/sqrt(-a^2 + b^2)