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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0253734, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{b-a c}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b-a c}} \]

Antiderivative was successfully verified.

[In]

Int[(a + (b - a*c)*x^2)^(-1),x]

[Out]

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.0150337, size = 36, normalized size = 1.06 \[ \frac{\tanh ^{-1}\left (\frac{x \sqrt{a c-b}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a c-b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + (b - a*c)*x^2)^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 34, normalized size = 1. \begin{align*}{{\it Artanh} \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)*arctanh((a*c-b)*x/(a*(a*c-b))^(1/2))

________________________________________________________________________________________

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

________________________________________________________________________________________

Fricas [A]  time = 1.52992, size = 219, normalized size = 6.44 \begin{align*} \left [\frac{\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 \, \sqrt{a^{2} c - a b}}, -\frac{\sqrt{-a^{2} c + a b} \arctan \left (\frac{\sqrt{-a^{2} c + a b} x}{a}\right )}{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*log(((a*c - b)*x^2 + 2*sqrt(a^2*c - a*b)*x + a)/((a*c - b)*x^2 - a))/sqrt(a^2*c - a*b), -sqrt(-a^2*c + a*
b)*arctan(sqrt(-a^2*c + a*b)*x/a)/(a^2*c - a*b)]

________________________________________________________________________________________

Sympy [B]  time = 0.23586, size = 60, normalized size = 1.76 \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

________________________________________________________________________________________

Giac [A]  time = 2.77855, size = 50, normalized size = 1.47 \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)