\(\int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx\) [846]

Optimal result

Integrand size = 17, antiderivative size = 4 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\arcsin (4+x) \]

[Out]

arcsin(4+x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {55, 633, 222} \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\arcsin (x+4) \]

[In]

Int[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]

[Out]

ArcSin[4 + x]

Rule 55

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

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}& = \int \frac {1}{\sqrt {-15-8 x-x^2}} \, dx \\ & = -\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. \(44\) vs. \(2(4)=8\).

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 11.00 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\frac {2 \sqrt {3+x} \sqrt {5+x} \text {arctanh}\left (\frac {\sqrt {5+x}}{\sqrt {3+x}}\right )}{\sqrt {-((3+x) (5+x))}} \]

[In]

Integrate[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]

[Out]

(2*Sqrt[3 + x]*Sqrt[5 + x]*ArcTanh[Sqrt[5 + x]/Sqrt[3 + x]])/Sqrt[-((3 + x)*(5 + x))]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(4)=8\).

Time = 1.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 7.25

method	result	size
default	\(\frac {\sqrt {\left (-3-x \right ) \left (5+x \right )}\, \arcsin \left (x +4\right )}{\sqrt {-3-x}\, \sqrt {5+x}}\)	\(29\)

[In]

int(1/(-3-x)^(1/2)/(5+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((-3-x)*(5+x))^(1/2)/(-3-x)^(1/2)/(5+x)^(1/2)*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 = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 7.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=-\arctan \left (\frac {\sqrt {x + 5} {\left (x + 4\right )} \sqrt {-x - 3}}{x^{2} + 8 \, x + 15}\right ) \]

[In]

integrate(1/(-3-x)^(1/2)/(5+x)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(x + 5)*(x + 4)*sqrt(-x - 3)/(x^2 + 8*x + 15))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 39, normalized size of antiderivative = 9.75 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=\begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{2} \right )} & \text {for}\: \left |{x + 5}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{2} \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(-3-x)**(1/2)/(5+x)**(1/2),x)

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 5)/2), Abs(x + 5) > 2), (2*asin(sqrt(2)*sqrt(x + 5)/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=-\arcsin \left (-x - 4\right ) \]

[In]

integrate(1/(-3-x)^(1/2)/(5+x)^(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. 13 vs. \(2 (4) = 8\).

Time = 0.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 3.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 5}\right ) \]

[In]

integrate(1/(-3-x)^(1/2)/(5+x)^(1/2),x, algorithm="giac")

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(x + 5))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 8.25 \[ \int \frac {1}{\sqrt {-3-x} \sqrt {5+x}} \, dx=4\,\mathrm {atan}\left (\frac {-\sqrt {-x-3}+\sqrt {3}\,1{}\mathrm {i}}{\sqrt {x+5}-\sqrt {5}}\right ) \]

[In]

int(1/((- x - 3)^(1/2)*(x + 5)^(1/2)),x)

[Out]

4*atan((3^(1/2)*1i - (- x - 3)^(1/2))/((x + 5)^(1/2) - 5^(1/2)))