Optimal. Leaf size=4 \[ \sin ^{-1}(4+x) \]
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Rubi [A]
time = 0.00, 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 = {55, 633, 222}
\begin {gather*} \text {ArcSin}(x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 55
Rule 222
Rule 633
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} \text {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] Leaf count is larger than twice the leaf count of optimal. \(44\) vs. \(2(4)=8\).
time = 0.03, size = 44, normalized size = 11.00 \begin {gather*} \frac {2 \sqrt {3+x} \sqrt {5+x} \tanh ^{-1}\left (\frac {\sqrt {3+x}}{\sqrt {5+x}}\right )}{\sqrt {-((3+x) (5+x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs.
\(2(4)=8\).
time = 0.62, size = 29, normalized size = 7.25
method | result | size |
default | \(\frac {\sqrt {\left (-3-x \right ) \left (5+x \right )}\, \arcsin \left (x +4\right )}{\sqrt {-3-x}\, \sqrt {5+x}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (4) = 8\).
time = 0.35, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.49, size = 39, normalized size = 9.75 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 5}}{2} \right )} & \text {for}\: \left |{x + 5}\right | > 2 \\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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs.
\(2 (4) = 8\).
time = 2.86, 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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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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