Optimal. Leaf size=33 \[ -x+\frac {e^{\frac {5}{3-e^2}}}{x \left (x^2+\frac {\log (3)}{4}\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 56, normalized size of antiderivative = 1.70, number of steps
used = 7, number of rules used = 6, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {12, 1608, 28,
1819, 1267, 14} \begin {gather*} -\frac {64 e^{\frac {5}{3-e^2}} x}{\log (3) \left (16 x^2+\log (81)\right )}-x+\frac {4 e^{\frac {5}{3-e^2}}}{x \log (3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 28
Rule 1267
Rule 1608
Rule 1819
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^{\frac {5}{3-e^2}} \int \frac {-48 x^2-4 \log (3)+e^{\frac {5}{-3+e^2}} \left (-16 x^6-8 x^4 \log (3)-x^2 \log ^2(3)\right )}{16 x^6+8 x^4 \log (3)+x^2 \log ^2(3)} \, dx\\ &=e^{\frac {5}{3-e^2}} \int \frac {-48 x^2-4 \log (3)+e^{\frac {5}{-3+e^2}} \left (-16 x^6-8 x^4 \log (3)-x^2 \log ^2(3)\right )}{x^2 \left (16 x^4+8 x^2 \log (3)+\log ^2(3)\right )} \, dx\\ &=\left (16 e^{\frac {5}{3-e^2}}\right ) \int \frac {-48 x^2-4 \log (3)+e^{\frac {5}{-3+e^2}} \left (-16 x^6-8 x^4 \log (3)-x^2 \log ^2(3)\right )}{x^2 \left (16 x^2+4 \log (3)\right )^2} \, dx\\ &=-\frac {64 e^{\frac {5}{3-e^2}} x}{\log (3) \left (16 x^2+\log (81)\right )}-\frac {\left (2 e^{\frac {5}{3-e^2}}\right ) \int \frac {8 \log (3)+8 e^{-\frac {5}{3-e^2}} x^4 \log (3)+2 x^2 \left (16+e^{-\frac {5}{3-e^2}} \log ^2(3)\right )}{x^2 \left (16 x^2+4 \log (3)\right )} \, dx}{\log (3)}\\ &=-\frac {64 e^{\frac {5}{3-e^2}} x}{\log (3) \left (16 x^2+\log (81)\right )}-\frac {\left (2 e^{\frac {5}{3-e^2}}\right ) \int \frac {2+\frac {1}{2} e^{-\frac {5}{3-e^2}} x^2 \log (3)}{x^2} \, dx}{\log (3)}\\ &=-\frac {64 e^{\frac {5}{3-e^2}} x}{\log (3) \left (16 x^2+\log (81)\right )}-\frac {\left (2 e^{\frac {5}{3-e^2}}\right ) \int \left (\frac {2}{x^2}+\frac {1}{2} e^{\frac {5}{-3+e^2}} \log (3)\right ) \, dx}{\log (3)}\\ &=-x+\frac {4 e^{\frac {5}{3-e^2}}}{x \log (3)}-\frac {64 e^{\frac {5}{3-e^2}} x}{\log (3) \left (16 x^2+\log (81)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.24 \begin {gather*} e^{-\frac {5}{-3+e^2}} \left (-e^{\frac {5}{-3+e^2}} x+\frac {4}{4 x^3+x \log (3)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 52, normalized size = 1.58
method | result | size |
risch | \(-x +\frac {4 \,{\mathrm e}^{-\frac {5}{{\mathrm e}^{2}-3}}}{x \left (4 x^{2}+\ln \left (3\right )\right )}\) | \(29\) |
default | \({\mathrm e}^{-\frac {5}{{\mathrm e}^{2}-3}} \left (-{\mathrm e}^{\frac {5}{{\mathrm e}^{2}-3}} x -\frac {16 x}{\ln \left (3\right ) \left (4 x^{2}+\ln \left (3\right )\right )}+\frac {4}{x \ln \left (3\right )}\right )\) | \(52\) |
gosper | \(-\frac {\left (4 \,{\mathrm e}^{\frac {5}{{\mathrm e}^{2}-3}} x^{4}-4+\ln \left (3\right ) {\mathrm e}^{\frac {5}{{\mathrm e}^{2}-3}} x^{2}\right ) {\mathrm e}^{-\frac {5}{{\mathrm e}^{2}-3}}}{x \left (4 x^{2}+\ln \left (3\right )\right )}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 37, normalized size = 1.12 \begin {gather*} -{\left (x e^{\left (\frac {5}{e^{2} - 3}\right )} - \frac {4}{4 \, x^{3} + x \log \left (3\right )}\right )} e^{\left (-\frac {5}{e^{2} - 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 47, normalized size = 1.42 \begin {gather*} -\frac {{\left ({\left (4 \, x^{4} + x^{2} \log \left (3\right )\right )} e^{\left (\frac {5}{e^{2} - 3}\right )} - 4\right )} e^{\left (-\frac {5}{e^{2} - 3}\right )}}{4 \, x^{3} + x \log \left (3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 32, normalized size = 0.97 \begin {gather*} - x + \frac {4}{4 x^{3} e^{- \frac {5}{3 - e^{2}}} + x e^{- \frac {5}{3 - e^{2}}} \log {\left (3 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 37, normalized size = 1.12 \begin {gather*} -{\left (x e^{\left (\frac {5}{e^{2} - 3}\right )} - \frac {4}{4 \, x^{3} + x \log \left (3\right )}\right )} e^{\left (-\frac {5}{e^{2} - 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.86, size = 28, normalized size = 0.85 \begin {gather*} \frac {4\,{\mathrm {e}}^{-\frac {5}{{\mathrm {e}}^2-3}}}{x\,\left (4\,x^2+\ln \left (3\right )\right )}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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