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

Optimal. Leaf size=34 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]

[Out]

ArcTanh[(Sqrt[b - a*c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b - a*c])

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Rubi [A]  time = 0.0308608, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]

Antiderivative was successfully verified.

[In]  Int[(a - (b - a*c)*x^2)^(-1),x]

[Out]

ArcTanh[(Sqrt[b - a*c]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b - a*c])

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Rubi in Sympy [A]  time = 4.85595, size = 29, normalized size = 0.85 \[ \frac{\operatorname{atanh}{\left (\frac{x \sqrt{- a c + b}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{- a c + b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a-(-a*c+b)*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.0176621, size = 36, normalized size = 1.06 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a c-b}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a c-b}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - (b - a*c)*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.005, size = 34, normalized size = 1. \[{1\arctan \left ({x \left ( ac-b \right ){\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a-(-a*c+b)*x^2),x)

[Out]

1/(a*(a*c-b))^(1/2)*arctan((a*c-b)*x/(a*(a*c-b))^(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/((a*c - b)*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222538, 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]

[1/2*log((2*(a^2*c - a*b)*x + sqrt(-a^2*c + a*b)*((a*c - b)*x^2 - a))/((a*c - b)
*x^2 + a))/sqrt(-a^2*c + a*b), arctan(sqrt(a^2*c - a*b)*x/a)/sqrt(a^2*c - a*b)]

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Sympy [A]  time = 0.620811, size = 66, normalized size = 1.94 \[ - \frac{\sqrt{- \frac{1}{a \left (a c - b\right )}} \log{\left (- a \sqrt{- \frac{1}{a \left (a c - b\right )}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a \left (a c - b\right )}} \log{\left (a \sqrt{- \frac{1}{a \left (a c - b\right )}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a-(-a*c+b)*x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209454, size = 49, normalized size = 1.44 \[ \frac{\arctan \left (\frac{a c x - b x}{\sqrt{a^{2} c - a b}}\right )}{\sqrt{a^{2} c - a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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