Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]
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Rubi [A] time = 0.0059976, antiderivative size = 4, 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} \[ \sin ^{-1}(x+4) \]
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{-15-8 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 = 0.0103522, 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 [B] time = 0.006, size = 29, normalized size = 7.3 \begin{align*}{\arcsin \left ( 4+x \right ) \sqrt{ \left ( -3-x \right ) \left ( 5+x \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 [B] time = 2.03335, 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 = 1.51496, size = 81, normalized size = 20.25 \begin{align*} -\arctan \left (\frac{\sqrt{x + 5}{\left (x + 4\right )} \sqrt{-x - 3}}{x^{2} + 8 \, x + 15}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.01342, size = 41, normalized size = 10.25 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{for}\: \frac{\left |{x + 5}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1544, size = 18, normalized size = 4.5 \begin{align*} 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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