3.1849 \(\int \frac{1}{a+\frac{b}{x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

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Rubi [A]  time = 0.0105006, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {193, 321, 205} \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(-1),x]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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+\frac{b}{x^2}} \, dx &=\int \frac{x^2}{b+a x^2} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^2} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0080479, size = 31, normalized size = 1. \[ \frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(-1),x]

[Out]

x/a - (Sqrt[b]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/a^(3/2)

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Maple [A]  time = 0.002, size = 27, normalized size = 0.9 \begin{align*}{\frac{x}{a}}-{\frac{b}{a}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77539, size = 165, normalized size = 5.32 \begin{align*} \left [\frac{\sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right ) + 2 \, x}{2 \, a}, -\frac{\sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right ) - x}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.287401, size = 56, normalized size = 1.81 \begin{align*} \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (a \sqrt{- \frac{b}{a^{3}}} + x \right )}}{2} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-b/a**3)*log(-a*sqrt(-b/a**3) + x)/2 - sqrt(-b/a**3)*log(a*sqrt(-b/a**3) + x)/2 + x/a

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Giac [A]  time = 1.21775, size = 35, normalized size = 1.13 \begin{align*} -\frac{b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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