3.9.44 \(\int \frac {1}{\sqrt {(3-x) (5+x)}} \, dx\) [844]

Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac {1}{4} (-1-x)\right ) \]

[Out]

arcsin(1/4+1/4*x)

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1976, 633, 222} \begin {gather*} -\text {ArcSin}\left (\frac {1}{4} (-x-1)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(3 - x)*(5 + x)],x]

[Out]

-ArcSin[(-1 - x)/4]

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]

Rule 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {(3-x) (5+x)}} \, dx &=\int \frac {1}{\sqrt {15-2 x-x^2}} \, dx\\ &=-\left (\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{64}}} \, dx,x,-2-2 x\right )\right )\\ &=-\sin ^{-1}\left (\frac {1}{4} (-1-x)\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(12)=24\).
time = 0.00, size = 44, normalized size = 3.67 \begin {gather*} \frac {2 \sqrt {-3+x} \sqrt {5+x} \tanh ^{-1}\left (\frac {\sqrt {5+x}}{\sqrt {-3+x}}\right )}{\sqrt {-((-3+x) (5+x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(3 - x)*(5 + x)],x]

[Out]

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

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Maple [A]
time = 0.57, size = 7, normalized size = 0.58

method	result	size
default	\(\arcsin \left (\frac {1}{4}+\frac {x}{4}\right )\)	\(7\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-2 x +15}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(1/4+1/4*x)

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Maxima [A]
time = 0.51, size = 8, normalized size = 0.67 \begin {gather*} -\arcsin \left (-\frac {1}{4} \, x - \frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-1/4*x - 1/4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (6) = 12\).
time = 0.37, size = 29, normalized size = 2.42 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 15} {\left (x + 1\right )}}{x^{2} + 2 \, x - 15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(-x^2 - 2*x + 15)*(x + 1)/(x^2 + 2*x - 15))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (3 - x\right ) \left (x + 5\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt((3 - x)*(x + 5)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (6) = 12\).
time = 3.17, size = 26, normalized size = 2.17 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 15} {\left (x + 1\right )} + 8 \, \arcsin \left (\frac {1}{4} \, x + \frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.37, size = 6, normalized size = 0.50 \begin {gather*} \mathrm {asin}\left (\frac {x}{4}+\frac {1}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-(x - 3)*(x + 5))^(1/2),x)

[Out]

asin(x/4 + 1/4)

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