Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]
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Rubi [A] time = 0.008832, antiderivative size = 12, 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} \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]
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{15-2 x-x^2}} \, dx\\ &=-\left (\frac{1}{8} \operatorname{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*}
Mathematica [A] time = 0.002709, size = 21, normalized size = 1.75 \[ -2 \sin ^{-1}\left (\frac{\sqrt{3-x}}{2 \sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 7, normalized size = 0.6 \begin{align*} \arcsin \left ({\frac{1}{4}}+{\frac{x}{4}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.10107, size = 11, normalized size = 0.92 \begin{align*} -\arcsin \left (-\frac{1}{4} \, x - \frac{1}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.45716, size = 77, normalized size = 6.42 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 15}{\left (x + 1\right )}}{x^{2} + 2 \, 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 (3 - x\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 = 1.32539, size = 8, normalized size = 0.67 \begin{align*} \arcsin \left (\frac{1}{4} \, x + \frac{1}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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