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

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {x}{r \sqrt {2 H r^2-a^2}} \]

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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fricas [A]  time = 0.41, size = 31, normalized size = 1.48 \[ \frac {\sqrt {2 \, H r^{2} - a^{2}} x}{2 \, H r^{3} - a^{2} r} \]

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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giac [A]  time = 0.96, size = 19, normalized size = 0.90 \[ \frac {x}{\sqrt {2 \, H r^{2} - a^{2}} r} \]

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.95 \[ \frac {x}{\sqrt {2 H \,r^{2}-a^{2}}\, r} \]

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 = 0.50, size = 19, normalized size = 0.90 \[ \frac {x}{\sqrt {2 \, H r^{2} - a^{2}} r} \]

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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mupad [B]  time = 0.00, size = 19, normalized size = 0.90 \[ \frac {x}{r\,\sqrt {2\,H\,r^2-a^2}} \]

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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sympy [A]  time = 0.06, size = 15, normalized size = 0.71 \[ \frac {x}{r \sqrt {2 H r^{2} - a^{2}}} \]

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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