3.841 \(\int \frac {1}{\sqrt {(2-3 x) (2+3 x)}} \, dx\)

Optimal. Leaf size=10 \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]

[Out]

1/3*arcsin(3/2*x)

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1972, 216} \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(2 - 3*x)*(2 + 3*x)],x]

[Out]

ArcSin[(3*x)/2]/3

Rule 216

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

Rule 1972

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[u, x] &&  !BinomialMatchQ[
u, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {(2-3 x) (2+3 x)}} \, dx &=\int \frac {1}{\sqrt {4-9 x^2}} \, dx\\ &=\frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(2 - 3*x)*(2 + 3*x)],x]

[Out]

ArcSin[(3*x)/2]/3

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fricas [B]  time = 0.42, size = 19, normalized size = 1.90 \[ -\frac {2}{3} \, \arctan \left (\frac {\sqrt {-9 \, x^{2} + 4} - 2}{3 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-3*x)*(2+3*x))^(1/2),x, algorithm="fricas")

[Out]

-2/3*arctan(1/3*(sqrt(-9*x^2 + 4) - 2)/x)

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giac [A]  time = 0.39, size = 6, normalized size = 0.60 \[ \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-3*x)*(2+3*x))^(1/2),x, algorithm="giac")

[Out]

1/3*arcsin(3/2*x)

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maple [A]  time = 0.01, size = 7, normalized size = 0.70 \[ \frac {\arcsin \left (\frac {3 x}{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-3*x+2)*(3*x+2))^(1/2),x)

[Out]

1/3*arcsin(3/2*x)

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maxima [A]  time = 1.96, size = 6, normalized size = 0.60 \[ \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-3*x)*(2+3*x))^(1/2),x, algorithm="maxima")

[Out]

1/3*arcsin(3/2*x)

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mupad [B]  time = 0.01, size = 6, normalized size = 0.60 \[ \frac {\mathrm {asin}\left (\frac {3\,x}{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-(3*x - 2)*(3*x + 2))^(1/2),x)

[Out]

asin((3*x)/2)/3

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sympy [A]  time = 1.39, size = 7, normalized size = 0.70 \[ \frac {\operatorname {asin}{\left (\frac {3 x}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((2-3*x)*(2+3*x))**(1/2),x)

[Out]

asin(3*x/2)/3

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