3.1.21 \(\int \frac {1}{\sqrt {3 x-4 x^2}} \, dx\) [21]

Optimal. Leaf size=12 \[ -\frac {1}{2} \sin ^{-1}\left (1-\frac {8 x}{3}\right ) \]

[Out]

1/2*arcsin(-1+8/3*x)

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {633, 222} \begin {gather*} -\frac {1}{2} \text {ArcSin}\left (1-\frac {8 x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3*x - 4*x^2],x]

[Out]

-1/2*ArcSin[1 - (8*x)/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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {3 x-4 x^2}} \, dx &=-\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,3-8 x\right )\right )\\ &=-\frac {1}{2} \sin ^{-1}\left (1-\frac {8 x}{3}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(12)=24\).
time = 0.03, size = 46, normalized size = 3.83 \begin {gather*} -\frac {\sqrt {x} \sqrt {-3+4 x} \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )}{\sqrt {-x (-3+4 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3*x - 4*x^2],x]

[Out]

-((Sqrt[x]*Sqrt[-3 + 4*x]*Log[-2*Sqrt[x] + Sqrt[-3 + 4*x]])/Sqrt[-(x*(-3 + 4*x))])

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Maple [A]
time = 0.39, size = 9, normalized size = 0.75

method	result	size
default	\(\frac {\arcsin \left (-1+\frac {8 x}{3}\right )}{2}\)	\(9\)
meijerg	\(\arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )\)	\(10\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}-3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+3*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*arcsin(-1+8/3*x)

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Maxima [A]
time = 0.54, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{2} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="maxima")

[Out]

-1/2*arcsin(-8/3*x + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (8) = 16\).
time = 1.33, size = 19, normalized size = 1.58 \begin {gather*} -\arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="fricas")

[Out]

-arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 4 x^{2} + 3 x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2+3*x)**(1/2),x)

[Out]

Integral(1/sqrt(-4*x**2 + 3*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (8) = 16\).
time = 3.70, size = 27, normalized size = 2.25 \begin {gather*} \frac {1}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )} + \frac {9}{64} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+3*x)^(1/2),x, algorithm="giac")

[Out]

1/16*sqrt(-4*x^2 + 3*x)*(8*x - 3) + 9/64*arcsin(8/3*x - 1)

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Mupad [B]
time = 0.11, size = 8, normalized size = 0.67 \begin {gather*} \frac {\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x - 4*x^2)^(1/2),x)

[Out]

asin((8*x)/3 - 1)/2

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