Optimal. Leaf size=79 \[ \frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e} \]
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Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {641, 44, 53, 65,
212} \begin {gather*} \frac {1}{16 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{12 \sqrt {3} e \sqrt {2-e x} (e x+2)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 212
Rule 641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2} (2+e x)^2} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)}+\frac {1}{4} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\sqrt {2-e x}}{16 \sqrt {3} e (2+e x)}+\frac {1}{32} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\sqrt {2-e x}}{16 \sqrt {3} e (2+e x)}-\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{48 e}\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {\sqrt {2-e x}}{16 \sqrt {3} e (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{32 \sqrt {3} e}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 87, normalized size = 1.10 \begin {gather*} \frac {4+6 e x-3 \sqrt {2+e x} \sqrt {4-e^2 x^2} \tanh ^{-1}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{96 e \sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 93, normalized size = 1.18
method | result | size |
default | \(\frac {\sqrt {-3 e^{2} x^{2}+12}\, \left (\sqrt {3}\, \sqrt {-3 e x +6}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}-6 e x -4\right )}{288 \left (e x +2\right )^{\frac {3}{2}} \left (e x -2\right ) e}\) | \(93\) |
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. 137 vs.
\(2 (60) = 120\).
time = 2.87, size = 137, normalized size = 1.73 \begin {gather*} \frac {3 \, \sqrt {3} {\left (x^{3} e^{3} + 2 \, x^{2} e^{2} - 4 \, x e - 8\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} {\left (3 \, x e + 2\right )} \sqrt {x e + 2}}{576 \, {\left (x^{3} e^{4} + 2 \, x^{2} e^{3} - 4 \, x e^{2} - 8 \, 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*} \frac {\sqrt {3} \int \frac {1}{- e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 78, normalized size = 0.99 \begin {gather*} -\frac {1}{192} \, \sqrt {3} e^{\left (-1\right )} \log \left (\sqrt {-x e + 2} + 2\right ) + \frac {1}{192} \, \sqrt {3} e^{\left (-1\right )} \log \left (-\sqrt {-x e + 2} + 2\right ) - \frac {\sqrt {3} {\left (3 \, x e + 2\right )} e^{\left (-1\right )}}{144 \, {\left ({\left (-x e + 2\right )}^{\frac {3}{2}} - 4 \, \sqrt {-x e + 2}\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 = -0.01 \begin {gather*} \int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{3/2}\,\sqrt {e\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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