\(\int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx\) [845]

Optimal result

Integrand size = 14, antiderivative size = 4 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\arcsin (4+x) \]

[Out]

arcsin(4+x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 4, 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} \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\arcsin (x+4) \]

[In]

Int[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

ArcSin[4 + 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,-8-2 x\right )\right ) \\ & = \sin ^{-1}(4+x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(4)=8\).

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 5.75 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {-15-8 x-x^2}}{5+x}\right ) \]

[In]

Integrate[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

-2*ArcTan[Sqrt[-15 - 8*x - x^2]/(5 + x)]

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25

method	result	size
default	\(\arcsin \left (x +4\right )\)	\(5\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-8 x -15}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\)	\(39\)

[In]

int(1/(-x^2-8*x-15)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arcsin(x+4)

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 = 7.25 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-\arctan \left (\frac {\sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \]

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(-x^2 - 8*x - 15)*(x + 4)/(x^2 + 8*x + 15))

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\operatorname {asin}{\left (x + 4 \right )} \]

[In]

integrate(1/(-x**2-8*x-15)**(1/2),x)

[Out]

asin(x + 4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=-\arcsin \left (-x - 4\right ) \]

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

Giac [B] (verification not implemented)

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

Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )} + \frac {1}{2} \, \arcsin \left (x + 4\right ) \]

[In]

integrate(1/(-x^2-8*x-15)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 - 8*x - 15)*(x + 4) + 1/2*arcsin(x + 4)

Mupad [B] (verification not implemented)

Time = 21.15 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx=\mathrm {asin}\left (x+4\right ) \]

[In]

int(1/(- 8*x - x^2 - 15)^(1/2),x)

[Out]

asin(x + 4)