Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac {1}{4} (-x-1)\right ) \]
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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 = 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 216
Rule 619
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*}
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Mathematica [A] time = 0.02, 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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fricas [B] time = 0.41, size = 29, normalized size = 2.42 \[ -\arctan \left (\frac {\sqrt {x + 5} {\left (x + 1\right )} \sqrt {-x + 3}}{x^{2} + 2 \, x - 15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 13, normalized size = 1.08 \[ 2 \, \arcsin \left (\frac {1}{4} \, \sqrt {2} \sqrt {x + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 31, normalized size = 2.58 \[ \frac {\sqrt {\left (-x +3\right ) \left (x +5\right )}\, \arcsin \left (\frac {x}{4}+\frac {1}{4}\right )}{\sqrt {-x +3}\, \sqrt {x +5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.96, size = 8, normalized size = 0.67 \[ -\arcsin \left (-\frac {1}{4} \, x - \frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 30, normalized size = 2.50 \[ 4\,\mathrm {atan}\left (\frac {\sqrt {3}-\sqrt {3-x}}{\sqrt {x+5}-\sqrt {5}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.02, size = 41, normalized size = 3.42 \[ \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{4} \right )} & \text {for}\: \frac {\left |{x + 5}\right |}{8} > 1 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{4} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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