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 = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222}
\begin {gather*} \text {ArcSin}(x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rubi steps
\begin {align*} \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. \(23\) vs. \(2(4)=8\).
time = 0.05, size = 23, normalized size = 5.75 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {-15-8 x-x^2}}{5+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 5, normalized size = 1.25
method | result | size |
default | \(\arcsin \left (x +4\right )\) | \(5\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-x \RootOf \left (\textit {\_Z}^{2}+1\right )-4 \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-8 x -15}\right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, 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 = 0.34, size = 29, normalized size = 7.25 \begin {gather*} -\arctan \left (\frac {\sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- x^{2} - 8 x - 15}}\, dx \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. 24 vs.
\(2 (4) = 8\).
time = 2.49, size = 24, normalized size = 6.00 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} - 8 \, x - 15} {\left (x + 4\right )} + \frac {1}{2} \, \arcsin \left (x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.18, size = 4, normalized size = 1.00 \begin {gather*} \mathrm {asin}\left (x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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