\(\int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx\) [236]

Optimal result

Integrand size = 14, antiderivative size = 8 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=-\arcsin (2-x) \]

[Out]

arcsin(-2+x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=-\arcsin (2-x) \]

[In]

Int[1/Sqrt[-3 + 4*x - x^2],x]

[Out]

-ArcSin[2 - x]

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*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,4-2 x\right )\right ) \\ & = -\arcsin (2-x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.88 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {-3+4 x-x^2}}{-1+x}\right ) \]

[In]

Integrate[1/Sqrt[-3 + 4*x - x^2],x]

[Out]

-2*ArcTan[Sqrt[-3 + 4*x - x^2]/(-1 + x)]

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.62

method	result	size
default	\(\arcsin \left (-2+x \right )\)	\(5\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+4 x -3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\)	\(39\)

[In]

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

[Out]

arcsin(-2+x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (4) = 8\).

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.62 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=-\arctan \left (\frac {\sqrt {-x^{2} + 4 \, x - 3} {\left (x - 2\right )}}{x^{2} - 4 \, x + 3}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=\operatorname {asin}{\left (x - 2 \right )} \]

[In]

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

[Out]

asin(x - 2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=-\arcsin \left (-x + 2\right ) \]

[In]

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

[Out]

-arcsin(-x + 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (4) = 8\).

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 3.00 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 4 \, x - 3} {\left (x - 2\right )} + \frac {1}{2} \, \arcsin \left (x - 2\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt {-3+4 x-x^2}} \, dx=\mathrm {asin}\left (x-2\right ) \]

[In]

int(1/(4*x - x^2 - 3)^(1/2),x)

[Out]

asin(x - 2)