Optimal. Leaf size=34 \[ \frac {5 e^{1-e^{2/x}-\frac {4 \log (15)}{x}}}{x \left (-\frac {1}{x}+x\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(34)=68\).
time = 0.66, antiderivative size = 126, normalized size of antiderivative = 3.71, number of steps
used = 3, number of rules used = 3, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {1608, 28, 2326}
\begin {gather*} -\frac {2\ 3^{-4/x} 5^{1-\frac {4}{x}} e^{\frac {x-e^{2/x} x}{x}} \left (e^{2/x} \left (1-x^2\right )+2 \left (1-x^2\right ) \log (15)\right )}{x^2 \left (1-x^2\right )^2 \left (\frac {-e^{2/x}+\frac {2 e^{2/x}}{x}+1}{x}-\frac {-e^{2/x} x+x-4 \log (15)}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 1608
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {x-e^{2/x} x-4 \log (15)}{x}} \left (-10 x^3+e^{2/x} \left (-10+10 x^2\right )+\left (-20+20 x^2\right ) \log (15)\right )}{x^2 \left (1-2 x^2+x^4\right )} \, dx\\ &=\int \frac {e^{\frac {x-e^{2/x} x-4 \log (15)}{x}} \left (-10 x^3+e^{2/x} \left (-10+10 x^2\right )+\left (-20+20 x^2\right ) \log (15)\right )}{x^2 \left (-1+x^2\right )^2} \, dx\\ &=-\frac {2\ 3^{-4/x} 5^{1-\frac {4}{x}} e^{\frac {x-e^{2/x} x}{x}} \left (e^{2/x} \left (1-x^2\right )+2 \left (1-x^2\right ) \log (15)\right )}{x^2 \left (1-x^2\right )^2 \left (\frac {1-e^{2/x}+\frac {2 e^{2/x}}{x}}{x}-\frac {x-e^{2/x} x-4 \log (15)}{x^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.64, size = 37, normalized size = 1.09 \begin {gather*} \frac {5^{\frac {-4+x}{x}} 81^{-1/x} e^{1-e^{2/x}}}{-1+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.53, size = 30, normalized size = 0.88
method | result | size |
norman | \(\frac {5 \,{\mathrm e}^{\frac {-x \,{\mathrm e}^{\frac {2}{x}}-4 \ln \left (15\right )+x}{x}}}{x^{2}-1}\) | \(30\) |
risch | \(\frac {5 \left (\frac {1}{625}\right )^{\frac {1}{x}} \left (\frac {1}{81}\right )^{\frac {1}{x}} {\mathrm e}^{-{\mathrm e}^{\frac {2}{x}}+1}}{x^{2}-1}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 34, normalized size = 1.00 \begin {gather*} \frac {5 \, e^{\left (-\frac {4 \, \log \left (5\right )}{x} - \frac {4 \, \log \left (3\right )}{x} - e^{\frac {2}{x}} + 1\right )}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 31, normalized size = 0.91 \begin {gather*} \frac {5 \, e^{\left (-\frac {x e^{\frac {2}{x}} - x + 4 \, \log \left (15\right )}{x}\right )}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 22, normalized size = 0.65 \begin {gather*} \frac {5 e^{\frac {- x e^{\frac {2}{x}} + x - 4 \log {\left (15 \right )}}{x}}}{x^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.27, size = 29, normalized size = 0.85 \begin {gather*} \frac {5\,{\mathrm {e}}^{-{\mathrm {e}}^{2/x}}\,\mathrm {e}}{{15}^{4/x}\,\left (x^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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