Optimal. Leaf size=31 \[ \frac {\left (4-\frac {1}{3 x}\right )^2}{4 e^2 \left (-x-\frac {\log (2)}{x}\right )} \]
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Rubi [A]
time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps
used = 6, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 1598, 21,
1819, 30} \begin {gather*} \frac {x (1-144 \log (2))+24 \log (2)}{36 e^2 \log (2) \left (x^2+\log (2)\right )}-\frac {1}{36 e^2 x \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 30
Rule 1598
Rule 1819
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {x \left (-3 x^2+48 x^3-144 x^4+\left (-1+144 x^2\right ) \log (2)\right )}{\left (-x^2-\log (2)\right ) \left (36 x^5+36 x^3 \log (2)\right )} \, dx}{e^2}\\ &=\frac {\int \frac {-3 x^2+48 x^3-144 x^4+\left (-1+144 x^2\right ) \log (2)}{x^2 \left (-x^2-\log (2)\right ) \left (36 x^2+36 \log (2)\right )} \, dx}{e^2}\\ &=-\frac {\int \frac {-3 x^2+48 x^3-144 x^4+\left (-1+144 x^2\right ) \log (2)}{x^2 \left (-x^2-\log (2)\right )^2} \, dx}{36 e^2}\\ &=\frac {x (1-144 \log (2))+24 \log (2)}{36 e^2 \log (2) \left (x^2+\log (2)\right )}-\frac {\int \frac {2 x^2+\log (4)}{x^2 \left (-x^2-\log (2)\right )} \, dx}{72 e^2 \log (2)}\\ &=\frac {x (1-144 \log (2))+24 \log (2)}{36 e^2 \log (2) \left (x^2+\log (2)\right )}+\frac {\int \frac {1}{x^2} \, dx}{36 e^2 \log (2)}\\ &=-\frac {1}{36 e^2 x \log (2)}+\frac {x (1-144 \log (2))+24 \log (2)}{36 e^2 \log (2) \left (x^2+\log (2)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.81 \begin {gather*} -\frac {(1-12 x)^2}{36 e^2 x \left (x^2+\log (2)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 42, normalized size = 1.35
method | result | size |
risch | \(\frac {{\mathrm e}^{-2} \left (-4 x^{2}+\frac {2}{3} x -\frac {1}{36}\right )}{x \left (\ln \left (2\right )+x^{2}\right )}\) | \(25\) |
gosper | \(-\frac {\left (12 x -1\right )^{2} {\mathrm e}^{-2}}{36 x \left (\ln \left (2\right )+x^{2}\right )}\) | \(30\) |
norman | \(\frac {\frac {2 x^{2} {\mathrm e}^{-2}}{3}-\frac {x \,{\mathrm e}^{-2}}{36}-4 \,{\mathrm e}^{-2} x^{3}}{x^{2} \left (\ln \left (2\right )+x^{2}\right )}\) | \(39\) |
default | \(\frac {{\mathrm e}^{-2} \left (-\frac {1}{x \ln \left (2\right )}-\frac {\left (144 \ln \left (2\right )-1\right ) x -24 \ln \left (2\right )}{\ln \left (2\right ) \left (\ln \left (2\right )+x^{2}\right )}\right )}{36}\) | \(42\) |
meijerg | \(-\frac {{\mathrm e}^{-2} \left (-\ln \left (2\right )-x^{2}\right )^{x -1} \left (-\ln \left (2\right )\right )^{-x} \left (\frac {\Gamma \left (3+x \right ) x^{2} \hypergeom \left (\left [1, 1, 3+x \right ], \left [2, 3\right ], -\frac {x^{2}}{\ln \left (2\right )}\right )}{2 \ln \left (2\right )}-\left (\Psi \left (2+x \right )+\gamma -1+2 \ln \left (x \right )-\ln \left (\ln \left (2\right )\right )\right ) \Gamma \left (2+x \right )-\frac {\Gamma \left (x +1\right ) \ln \left (2\right )}{x^{2}}\right )}{72 \ln \left (2\right ) \Gamma \left (x \right )}-\frac {{\mathrm e}^{-2} \left (-\ln \left (2\right )-x^{2}\right )^{x -1} \left (-\ln \left (2\right )\right )^{-x} \left (-\frac {\Gamma \left (2+x \right ) x^{2} \hypergeom \left (\left [1, 1, 2+x \right ], \left [2, 2\right ], -\frac {x^{2}}{\ln \left (2\right )}\right )}{\ln \left (2\right )}+\left (\Psi \left (x +1\right )+\gamma +2 \ln \left (x \right )-\ln \left (\ln \left (2\right )\right )\right ) \Gamma \left (x +1\right )\right )}{24 \ln \left (2\right ) \Gamma \left (x \right )}+\frac {2 \,{\mathrm e}^{-2} \left (-\ln \left (2\right )-x^{2}\right )^{x -1} \left (-\ln \left (2\right )\right )^{-x} \left (-\frac {\Gamma \left (2+x \right ) x^{2} \hypergeom \left (\left [1, 1, 2+x \right ], \left [2, 2\right ], -\frac {x^{2}}{\ln \left (2\right )}\right )}{\ln \left (2\right )}+\left (\Psi \left (x +1\right )+\gamma +2 \ln \left (x \right )-\ln \left (\ln \left (2\right )\right )\right ) \Gamma \left (x +1\right )\right )}{\Gamma \left (x \right )}-\frac {2 \,{\mathrm e}^{-2} \left (-\ln \left (2\right )-x^{2}\right )^{x -1} \left (-\ln \left (2\right )\right )^{-x} x^{3} \hypergeom \left (\left [1, x +1\right ], \left [2\right ], -\frac {x^{2}}{\ln \left (2\right )}\right )}{\ln \left (2\right )}+\frac {4 \,{\mathrm e}^{-2} \left (-\ln \left (2\right )-x^{2}\right )^{x -1} \left (-\ln \left (2\right )\right )^{-x} x^{2} \hypergeom \left (\left [\frac {1}{2}, x +1\right ], \left [\frac {3}{2}\right ], -\frac {x^{2}}{\ln \left (2\right )}\right )}{3 \ln \left (2\right )}\) | \(372\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 24, normalized size = 0.77 \begin {gather*} -\frac {{\left (144 \, x^{2} - 24 \, x + 1\right )} e^{\left (-2\right )}}{36 \, {\left (x^{3} + x \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 27, normalized size = 0.87 \begin {gather*} -\frac {144 \, x^{2} - 24 \, x + 1}{36 \, {\left (x^{3} e^{2} + x e^{2} \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 27, normalized size = 0.87 \begin {gather*} \frac {- 144 x^{2} + 24 x - 1}{36 x^{3} e^{2} + 36 x e^{2} \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.37, size = 24, normalized size = 0.77 \begin {gather*} -\frac {{\left (144 \, x^{2} - 24 \, x + 1\right )} e^{\left (-2\right )}}{36 \, {\left (x^{3} + x \log \left (2\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 34, normalized size = 1.10 \begin {gather*} -\frac {{\mathrm {e}}^{-2}\,\left (24\,x^3+144\,\ln \left (2\right )\,x^2+\ln \left (2\right )\right )}{36\,x\,\ln \left (2\right )\,\left (x^2+\ln \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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