3.103 \(\int \frac{1}{r \sqrt{-a^2+2 H r^2}} \, dx\)

Optimal. Leaf size=21 \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2])

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Rubi [A]  time = 0.0161468, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {8} \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(r*Sqrt[-a^2 + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{r \sqrt{-a^2+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2+2 H r^2}}\\ \end{align*}

Mathematica [A]  time = 0.0000386, size = 21, normalized size = 1. \[ \frac{x}{r \sqrt{2 H r^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(r*Sqrt[-a^2 + 2*H*r^2]),x]

[Out]

x/(r*Sqrt[-a^2 + 2*H*r^2])

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Maple [A]  time = 0.002, size = 20, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-{a}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/r/(2*H*r^2-a^2)^(1/2),x)

[Out]

x/r/(2*H*r^2-a^2)^(1/2)

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Maxima [A]  time = 0.931835, size = 26, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r^2-a^2)^(1/2),x, algorithm="maxima")

[Out]

x/(sqrt(2*H*r^2 - a^2)*r)

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Fricas [A]  time = 1.73989, size = 55, normalized size = 2.62 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2}} x}{2 \, H r^{3} - a^{2} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2*H*r^2 - a^2)*x/(2*H*r^3 - a^2*r)

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Sympy [A]  time = 0.052351, size = 15, normalized size = 0.71 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - a^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/r/(2*H*r**2-a**2)**(1/2),x)

[Out]

x/(r*sqrt(2*H*r**2 - a**2))

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Giac [A]  time = 1.04571, size = 26, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2}} r} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(sqrt(2*H*r^2 - a^2)*r)