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

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 = 0.002131, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {41, 216} \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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} \sqrt{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 = 0.0043825, 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]*Sqrt[2 + 3*x]),x]

[Out]

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

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Maple [B]  time = 0.007, size = 34, normalized size = 3.4 \begin{align*}{\frac{1}{3}\sqrt{ \left ( 2-3\,x \right ) \left ( 2+3\,x \right ) }\arcsin \left ({\frac{3\,x}{2}} \right ){\frac{1}{\sqrt{2-3\,x}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((2-3*x)*(2+3*x))^(1/2)/(2-3*x)^(1/2)/(2+3*x)^(1/2)*arcsin(3/2*x)

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Maxima [A]  time = 1.62172, 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)^(1/2)/(2+3*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.50983, size = 74, normalized size = 7.4 \begin{align*} -\frac{2}{3} \, \arctan \left (\frac{\sqrt{3 \, x + 2} \sqrt{-3 \, x + 2} - 2}{3 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.04167, size = 51, normalized size = 5.1 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{3} \sqrt{x + \frac{2}{3}}}{2} \right )}}{3} & \text{for}\: \frac{3 \left |{x + \frac{2}{3}}\right |}{4} > 1 \\\frac{2 \operatorname{asin}{\left (\frac{\sqrt{3} \sqrt{x + \frac{2}{3}}}{2} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(3)*sqrt(x + 2/3)/2)/3, 3*Abs(x + 2/3)/4 > 1), (2*asin(sqrt(3)*sqrt(x + 2/3)/2)/3, T
rue))

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Giac [A]  time = 1.1356, size = 16, normalized size = 1.6 \begin{align*} \frac{2}{3} \, \arcsin \left (\frac{1}{2} \, \sqrt{3 \, x + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*arcsin(1/2*sqrt(3*x + 2))