3.54 \(\int \frac {1}{\sqrt {2+x-x^2}} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {619, 216} \[ -\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + x - x^2],x]

[Out]

-ArcSin[(1 - 2*x)/3]

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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ -\sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[2 + x - x^2],x]

[Out]

-ArcSin[(1 - 2*x)/3]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 21, normalized size = 1.75 \[ -2 \tan ^{-1}\left (\frac {\sqrt {-x^2+x+2}}{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[2 + x - x^2],x]

[Out]

-2*ArcTan[Sqrt[2 + x - x^2]/(1 + x)]

________________________________________________________________________________________

fricas [B]  time = 0.91, size = 30, normalized size = 2.50 \[ -\arctan \left (\frac {\sqrt {-x^{2} + x + 2} {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - x - 2\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.08, size = 6, normalized size = 0.50 \[ \arcsin \left (\frac {2}{3} \, x - \frac {1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2/3*x - 1/3)

________________________________________________________________________________________

maple [A]  time = 0.29, size = 7, normalized size = 0.58

method	result	size
default	\(\arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )\)	\(7\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{2}+x +2}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 8, normalized size = 0.67 \[ -\arcsin \left (-\frac {2}{3} \, x + \frac {1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-2/3*x + 1/3)

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 6, normalized size = 0.50 \[ \mathrm {asin}\left (\frac {2\,x}{3}-\frac {1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - x^2 + 2)^(1/2),x)

[Out]

asin((2*x)/3 - 1/3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x^{2} + x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-x**2 + x + 2), x)

________________________________________________________________________________________