3.2.15 \(\int \frac {1}{\sqrt {3-4 x-4 x^2}} \, dx\) [115]

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

[Out]

1/2*arcsin(1/2+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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \begin {gather*} \frac {1}{2} \text {ArcSin}\left (x+\frac {1}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

ArcSin[1/2 + x]/2

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 {3-4 x-4 x^2}} \, dx &=-\left (\frac {1}{16} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{64}}} \, dx,x,-4-8 x\right )\right )\\ &=\frac {1}{2} \sin ^{-1}\left (\frac {1}{2}+x\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(10)=20\).
time = 0.05, size = 25, normalized size = 2.50 \begin {gather*} -\tan ^{-1}\left (\frac {\sqrt {3-4 x-4 x^2}}{3+2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

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

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Maple [A]
time = 0.57, size = 7, normalized size = 0.70

method	result	size
default	\(\frac {\arcsin \left (x +\frac {1}{2}\right )}{2}\)	\(7\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (2 x \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-4 x^{2}-4 x +3}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 8, normalized size = 0.80 \begin {gather*} -\frac {1}{2} \, \arcsin \left (-x - \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arcsin(-x - 1/2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (6) = 12\).
time = 1.98, size = 33, normalized size = 3.30 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )}}{4 \, x^{2} + 4 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (6) = 12\).
time = 1.27, size = 24, normalized size = 2.40 \begin {gather*} \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} + \arcsin \left (x + \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.13, size = 6, normalized size = 0.60 \begin {gather*} \frac {\mathrm {asin}\left (x+\frac {1}{2}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3 - 4*x^2 - 4*x)^(1/2),x)

[Out]

asin(x + 1/2)/2

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