3.115 \(\int \frac{1}{\sqrt{3-4 x-4 x^2}} \, dx\)

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

[Out]

ArcSin[1/2 + x]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0067516, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {619, 216} \[ \frac{1}{2} \sin ^{-1}\left (x+\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

ArcSin[1/2 + x]/2

Rule 619

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]

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{3-4 x-4 x^2}} \, dx &=-\left (\frac{1}{16} \operatorname{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*}

Mathematica [A]  time = 0.0060364, size = 14, normalized size = 1.4 \[ -\frac{1}{2} \sin ^{-1}\left (\frac{1}{2} (-2 x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

-ArcSin[(-1 - 2*x)/2]/2

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 7, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arcsin \left ( x+{\frac{1}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.6966, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2} \, \arcsin \left (-x - \frac{1}{2}\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 1.8221, size = 88, normalized size = 8.8 \begin{align*} -\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{align*}

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))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]  time = 1.15997, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \arcsin \left (x + \frac{1}{2}\right ) \end{align*}

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]

1/2*arcsin(x + 1/2)