3.1851 \(\int \frac{1}{(a+\frac{b}{x^2}) x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0071655, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^2),x]

[Out]

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

Rule 263

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

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

Mathematica [A]  time = 0.0038884, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^2),x]

[Out]

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

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Maple [A]  time = 0.002, size = 16, normalized size = 0.7 \begin{align*}{\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^2,x)

[Out]

1/(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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67696, size = 151, normalized size = 6.29 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{a x^{2} - 2 \, \sqrt{-a b} x - b}{a x^{2} + b}\right )}{2 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{b}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.123505, size = 53, normalized size = 2.21 \begin{align*} - \frac{\sqrt{- \frac{1}{a b}} \log{\left (- b \sqrt{- \frac{1}{a b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b}} \log{\left (b \sqrt{- \frac{1}{a b}} + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18363, size = 20, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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