Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]
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Rubi [A] time = 0.0070078, antiderivative size = 4, normalized size of antiderivative = 1., 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 = {1981, 619, 216} \[ \sin ^{-1}(x+4) \]
Antiderivative was successfully verified.
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Rule 1981
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{(-3-x) (5+x)}} \, dx &=\int \frac{1}{\sqrt{-15-8 x-x^2}} \, dx\\ &=-\left (\frac{1}{2} \operatorname{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] time = 0.0027375, size = 18, normalized size = 4.5 \[ -2 \sin ^{-1}\left (\frac{\sqrt{-x-3}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 5, normalized size = 1.3 \begin{align*} \arcsin \left ( 4+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.18335, size = 11, normalized size = 2.75 \begin{align*} -\arcsin \left (-x - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48181, size = 77, normalized size = 19.25 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 8 \, x - 15}{\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (- x - 3\right ) \left (x + 5\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13548, size = 5, normalized size = 1.25 \begin{align*} \arcsin \left (x + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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