3.9.40 \(\int \frac {1}{\sqrt {2-3 x} \sqrt {2+3 x}} \, dx\) [840]

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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {41, 222} \begin {gather*} \frac {1}{3} \text {ArcSin}\left (\frac {3 x}{2}\right ) \end {gather*}

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 222

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.00, size = 24, normalized size = 2.40 \begin {gather*} \frac {1}{3} i \log \left (-3 i x+\sqrt {4-9 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I/3)*Log[(-3*I)*x + Sqrt[4 - 9*x^2]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(6)=12\).
time = 0.52, size = 34, normalized size = 3.40

method	result	size
default	\(\frac {\sqrt {\left (2-3 x \right ) \left (2+3 x \right )}\, \arcsin \left (\frac {3 x}{2}\right )}{3 \sqrt {2-3 x}\, \sqrt {2+3 x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.55, size = 6, normalized size = 0.60 \begin {gather*} \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (6) = 12\).
time = 0.35, size = 25, normalized size = 2.50 \begin {gather*} -\frac {2}{3} \, \arctan \left (\frac {\sqrt {3 \, x + 2} \sqrt {-3 \, x + 2} - 2}{3 \, x}\right ) \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.55, size = 49, normalized size = 4.90 \begin {gather*} \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {3} \sqrt {x + \frac {2}{3}}}{2} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | > \frac {4}{3} \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {3} \sqrt {x + \frac {2}{3}}}{2} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}

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, Abs(x + 2/3) > 4/3), (2*asin(sqrt(3)*sqrt(x + 2/3)/2)/3, Tru
e))

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Giac [A]
time = 3.25, size = 12, normalized size = 1.20 \begin {gather*} \frac {2}{3} \, \arcsin \left (\frac {1}{2} \, \sqrt {3 \, x + 2}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.15, size = 32, normalized size = 3.20 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {2}-\sqrt {2-3\,x}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*atan((2^(1/2) - (2 - 3*x)^(1/2))/(2^(1/2) - (3*x + 2)^(1/2))))/3

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