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

Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.006682, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(-1),x]

[Out]

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

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

Mathematica [A]  time = 0.0037806, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(-1),x]

[Out]

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

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Maple [A]  time = 0.002, size = 16, normalized size = 0.7 \begin{align*}{{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31384, size = 151, normalized size = 6.29 \begin{align*} \left [\frac{\sqrt{a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{2 \, a b}, -\frac{\sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.130379, size = 46, normalized size = 1.92 \begin{align*} - \frac{\sqrt{\frac{1}{a b}} \log{\left (- a \sqrt{\frac{1}{a b}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{a b}} \log{\left (a \sqrt{\frac{1}{a b}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.55397, size = 24, normalized size = 1. \begin{align*} -\frac{\arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(b*x/sqrt(-a*b))/sqrt(-a*b)