3.9.41 \(\int \frac {1}{\sqrt {(2-3 x) (2+3 x)}} \, dx\) [841]

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

[Out]

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

________________________________________________________________________________________

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 = {1976, 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)*(2 + 3*x)],x]

[Out]

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

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]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, 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*}

________________________________________________________________________________________

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)*(2 + 3*x)],x]

[Out]

(I/3)*Log[(-3*I)*x + Sqrt[4 - 9*x^2]]

________________________________________________________________________________________

Maple [A]
time = 0.51, size = 7, normalized size = 0.70

method	result	size
default	\(\frac {\arcsin \left (\frac {3 x}{2}\right )}{3}\)	\(7\)
meijerg	\(\frac {\arcsin \left (\frac {3 x}{2}\right )}{3}\)	\(7\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-9 x^{2}+4}+3 x \right )}{3}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.49, 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)*(2+3*x))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (6) = 12\).
time = 0.39, size = 19, normalized size = 1.90 \begin {gather*} -\frac {2}{3} \, \arctan \left (\frac {\sqrt {-9 \, x^{2} + 4} - 2}{3 \, x}\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.65, size = 7, normalized size = 0.70 \begin {gather*} \frac {\operatorname {asin}{\left (\frac {3 x}{2} \right )}}{3} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (6) = 12\).
time = 3.58, size = 19, normalized size = 1.90 \begin {gather*} \frac {1}{2} \, \sqrt {-9 \, x^{2} + 4} x + \frac {2}{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)*(2+3*x))^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________