Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]
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Rubi [A] time = 0.01, antiderivative size = 4, 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} \[ \sin ^{-1}(x+4) \]
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 {-15-8 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*}
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Mathematica [B] time = 0.00, size = 18, normalized size = 4.50 \[ -2 \sin ^{-1}\left (\frac {\sqrt {-x-3}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 29, normalized size = 7.25 \[ -\arctan \left (\frac {\sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 4, normalized size = 1.00 \[ \arcsin \left (x + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 5, normalized size = 1.25 \[ \arcsin \left (x +4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 8, normalized size = 2.00 \[ -\arcsin \left (-x - 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 4, normalized size = 1.00 \[ \mathrm {asin}\left (x+4\right ) \]
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 \[ \int \frac {1}{\sqrt {\left (- x - 3\right ) \left (x + 5\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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