Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]
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Rubi [A] time = 0.0074626, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {53, 619, 216} \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]
Antiderivative was successfully verified.
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Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-x} \sqrt{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 = 0.011606, 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 [B] time = 0.006, size = 31, normalized size = 2.6 \begin{align*}{\sqrt{ \left ( 3-x \right ) \left ( 5+x \right ) }\arcsin \left ({\frac{1}{4}}+{\frac{x}{4}} \right ){\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{5+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60581, 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 = 1.44129, size = 81, normalized size = 6.75 \begin{align*} -\arctan \left (\frac{\sqrt{x + 5}{\left (x + 1\right )} \sqrt{-x + 3}}{x^{2} + 2 \, x - 15}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.01498, size = 41, normalized size = 3.42 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{4} \right )} & \text{for}\: \frac{\left |{x + 5}\right |}{8} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{4} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14132, size = 18, normalized size = 1.5 \begin{align*} 2 \, \arcsin \left (\frac{1}{4} \, \sqrt{2} \sqrt{x + 5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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