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 = 0.0111079, 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, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]

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])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{a-(b-a c) x^2} \, dx &=\frac{\tanh ^{-1}\left (\frac{\sqrt{b-a c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}}\\ \end{align*}

Mathematica [A]  time = 0.0097602, 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 = 0.003, size = 34, normalized size = 1. \begin{align*}{\arctan \left ({ \left ( ac-b \right ) x{\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( ac-b \right ) }}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46587, size = 219, normalized size = 6.44 \begin{align*} \left [-\frac{\sqrt{-a^{2} c + a b} \log \left (\frac{{\left (a c - b\right )} x^{2} - 2 \, \sqrt{-a^{2} c + a b} x - a}{{\left (a c - b\right )} x^{2} + a}\right )}{2 \,{\left (a^{2} c - a b\right )}}, \frac{\arctan \left (\frac{\sqrt{a^{2} c - a b} x}{a}\right )}{\sqrt{a^{2} c - a b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a^2*c + a*b)*log(((a*c - b)*x^2 - 2*sqrt(-a^2*c + a*b)*x - a)/((a*c - b)*x^2 + a))/(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 [B]  time = 0.256464, size = 66, normalized size = 1.94 \begin{align*} - \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} \end{align*}

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 [A]  time = 2.48133, size = 49, normalized size = 1.44 \begin{align*} \frac{\arctan \left (\frac{a c x - b x}{\sqrt{a^{2} c - a b}}\right )}{\sqrt{a^{2} c - a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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