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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {53, 619, 216} \begin {gather*} \sin ^{-1}(x+4) \end {gather*}
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 {-15-8 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 = 0.01, size = 18, normalized size = 4.50 \begin {gather*} -2 \sin ^{-1}\left (\frac {\sqrt {-x-3}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.04, size = 20, normalized size = 5.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {-x-3}}{\sqrt {x+5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 29, normalized size = 7.25 \begin {gather*} -\arctan \left (\frac {\sqrt {x + 5} {\left (x + 4\right )} \sqrt {-x - 3}}{x^{2} + 8 \, x + 15}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 13, normalized size = 3.25 \begin {gather*} 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 29, normalized size = 7.25 \begin {gather*} \frac {\sqrt {\left (-x -3\right ) \left (x +5\right )}\, \arcsin \left (x +4\right )}{\sqrt {-x -3}\, \sqrt {x +5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 8, normalized size = 2.00 \begin {gather*} -\arcsin \left (-x - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 33, normalized size = 8.25 \begin {gather*} 4\,\mathrm {atan}\left (\frac {-\sqrt {-x-3}+\sqrt {3}\,1{}\mathrm {i}}{\sqrt {x+5}-\sqrt {5}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.02, size = 41, normalized size = 10.25 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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