3.570 \(\int \frac {1}{\sqrt {-9-4 x^2}} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]

[Out]

1/2*arctan(2*x/(-4*x^2-9)^(1/2))

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {217, 203} \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-9 - 4*x^2],x]

[Out]

ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-9-4 x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {x}{\sqrt {-9-4 x^2}}\right )\\ &=\frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-9-4 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{2} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-9 - 4*x^2],x]

[Out]

ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2

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fricas [C]  time = 0.79, size = 47, normalized size = 2.47 \[ \frac {1}{4} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) - \frac {1}{4} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/4*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) - 1/4*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-4 \, x^{2} - 9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-4*x^2 - 9), x)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \[ \frac {\arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2-9)^(1/2),x)

[Out]

1/2*arctan(2/(-4*x^2-9)^(1/2)*x)

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maxima [C]  time = 2.98, size = 6, normalized size = 0.32 \[ -\frac {1}{2} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

-1/2*I*arcsinh(2/3*x)

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mupad [B]  time = 0.11, size = 15, normalized size = 0.79 \[ \frac {\mathrm {atan}\left (\frac {2\,x}{\sqrt {-4\,x^2-9}}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 4*x^2 - 9)^(1/2),x)

[Out]

atan((2*x)/(- 4*x^2 - 9)^(1/2))/2

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sympy [A]  time = 0.32, size = 17, normalized size = 0.89 \[ \frac {\operatorname {atan}{\left (\frac {2 x}{\sqrt {- 4 x^{2} - 9}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2-9)**(1/2),x)

[Out]

atan(2*x/sqrt(-4*x**2 - 9))/2

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