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 = 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 216
Rule 619
Rule 1981
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*}
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Mathematica [A] time = 0.00, 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 \[ -\arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 15} {\left (x + 1\right )}}{x^{2} + 2 \, x - 15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 6, normalized size = 0.50 \[ \arcsin \left (\frac {1}{4} \, x + \frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 7, normalized size = 0.58 \[ \arcsin \left (\frac {x}{4}+\frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, 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 = 3.37, size = 6, normalized size = 0.50 \[ \mathrm {asin}\left (\frac {x}{4}+\frac {1}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (3 - x\right ) \left (x + 5\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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