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]

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

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Rubi [A]  time = 0.0037278, antiderivative size = 10, normalized size of antiderivative = 1., 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 1972

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

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]

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*}

Mathematica [A]  time = 0.0040046, size = 10, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

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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Maple [A]  time = 0.006, size = 7, normalized size = 0.7 \begin{align*}{\frac{1}{3}\arcsin \left ({\frac{3\,x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.67563, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \end{align*}

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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Fricas [B]  time = 1.42755, size = 58, normalized size = 5.8 \begin{align*} -\frac{2}{3} \, \arctan \left (\frac{\sqrt{-9 \, x^{2} + 4} - 2}{3 \, x}\right ) \end{align*}

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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Sympy [A]  time = 1.24983, size = 7, normalized size = 0.7 \begin{align*} \frac{\operatorname{asin}{\left (\frac{3 x}{2} \right )}}{3} \end{align*}

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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Giac [A]  time = 1.18089, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \end{align*}

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)