Optimal. Leaf size=55 \[ -\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e} \]
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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 43, 65,
212} \begin {gather*} \frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e}-\frac {\sqrt {3} \sqrt {2-e x}}{e (e x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 212
Rule 641
Rubi steps
\begin {align*} \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx &=\int \frac {\sqrt {6-3 e x}}{(2+e x)^2} \, dx\\ &=-\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}-\frac {3}{2} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=-\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e}\\ &=-\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 64, normalized size = 1.16 \begin {gather*} \frac {\sqrt {3} \left (-\frac {2 \sqrt {4-e^2 x^2}}{(2+e x)^{3/2}}+\tanh ^{-1}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{2 e} \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. \(87\) vs.
\(2(43)=86\).
time = 0.48, size = 88, normalized size = 1.60
method | result | size |
default | \(\frac {\sqrt {-e^{2} x^{2}+4}\, \left (\sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-2 \sqrt {-3 e x +6}\right ) \sqrt {3}}{2 \sqrt {\left (e x +2\right )^{3}}\, \sqrt {-3 e x +6}\, e}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 115 vs.
\(2 (44) = 88\).
time = 1.97, size = 115, normalized size = 2.09 \begin {gather*} \frac {\sqrt {3} {\left (x^{2} e^{2} + 4 \, x e + 4\right )} \log \left (-\frac {3 \, x^{2} e^{2} - 12 \, x e - 4 \, \sqrt {3} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2} - 36}{x^{2} e^{2} + 4 \, x e + 4}\right ) - 4 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{4 \, {\left (x^{2} e^{3} + 4 \, x e^{2} + 4 \, e\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*} \sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{e^{2} x^{2} \sqrt {e x + 2} + 4 e x \sqrt {e x + 2} + 4 \sqrt {e x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 55, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, \sqrt {3} {\left (\frac {4 \, \sqrt {-x e + 2}}{x e + 2} - \log \left (\sqrt {-x e + 2} + 2\right ) + \log \left (-\sqrt {-x e + 2} + 2\right )\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {12-3\,e^2\,x^2}}{{\left (e\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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