Optimal. Leaf size=37 \[ \frac {e^{2 x} \left (e^{2 x}-\frac {1+x^2}{x}\right )^2 \left (3-\log ^2(3)\right )^2}{x^2} \]
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Rubi [B] time = 1.44, antiderivative size = 115, normalized size of antiderivative = 3.11, number of steps used = 33, number of rules used = 7, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 6742, 2197, 2199, 2177, 2178, 2194} \begin {gather*} \frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+e^{2 x} \left (3-\log ^2(3)\right )^2-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2177
Rule 2178
Rule 2194
Rule 2197
Rule 2199
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\left (3-\log ^2(3)\right )^2 \int \frac {e^{2 x} \left (-4+2 x-4 x^2+4 x^3+2 x^5+e^{4 x} \left (-2 x^2+6 x^3\right )+e^{2 x} \left (6 x-8 x^2+2 x^3-8 x^4\right )\right )}{x^5} \, dx\\ &=\left (3-\log ^2(3)\right )^2 \int \left (\frac {2 e^{6 x} (-1+3 x)}{x^3}-\frac {2 e^{4 x} \left (-3+4 x-x^2+4 x^3\right )}{x^4}+\frac {2 e^{2 x} \left (-2+x-2 x^2+2 x^3+x^5\right )}{x^5}\right ) \, dx\\ &=\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{6 x} (-1+3 x)}{x^3} \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x} \left (-3+4 x-x^2+4 x^3\right )}{x^4} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x} \left (-2+x-2 x^2+2 x^3+x^5\right )}{x^5} \, dx\\ &=\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \left (e^{2 x}-\frac {2 e^{2 x}}{x^5}+\frac {e^{2 x}}{x^4}-\frac {2 e^{2 x}}{x^3}+\frac {2 e^{2 x}}{x^2}\right ) \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \left (-\frac {3 e^{4 x}}{x^4}+\frac {4 e^{4 x}}{x^3}-\frac {e^{4 x}}{x^2}+\frac {4 e^{4 x}}{x}\right ) \, dx\\ &=\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int e^{2 x} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^4} \, dx+\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^5} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx+\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\left (6 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^4} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^3} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x^3}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {4 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{x}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}-8 \text {Ei}(4 x) \left (3-\log ^2(3)\right )^2+\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx-\left (2 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^4} \, dx-\left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^3} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx-\left (16 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {14 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}+8 \text {Ei}(2 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^3} \, dx+\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx-\left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (16 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x^2} \, dx-\left (64 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {4 e^{2 x} \left (3-\log ^2(3)\right )^2}{3 x}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}-64 \text {Ei}(4 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (4 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x^2} \, dx+\frac {1}{3} \left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx+\left (64 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{4 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}+\frac {8}{3} \text {Ei}(2 x) \left (3-\log ^2(3)\right )^2-\frac {1}{3} \left (8 \left (3-\log ^2(3)\right )^2\right ) \int \frac {e^{2 x}}{x} \, dx\\ &=e^{2 x} \left (3-\log ^2(3)\right )^2+\frac {e^{2 x} \left (3-\log ^2(3)\right )^2}{x^4}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x^3}+\frac {2 e^{2 x} \left (3-\log ^2(3)\right )^2}{x^2}+\frac {e^{6 x} \left (3-\log ^2(3)\right )^2}{x^2}-\frac {2 e^{4 x} \left (3-\log ^2(3)\right )^2}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.88, size = 32, normalized size = 0.86 \begin {gather*} \frac {e^{2 x} \left (1-e^{2 x} x+x^2\right )^2 \left (-3+\log ^2(3)\right )^2}{x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 205, normalized size = 5.54 \begin {gather*} \frac {x^{2} e^{\left (6 \, x + 6 \, \log \left (-\log \relax (3)^{2} + 3\right )\right )} - 2 \, {\left ({\left (x^{3} + x\right )} \log \relax (3)^{4} + 9 \, x^{3} - 6 \, {\left (x^{3} + x\right )} \log \relax (3)^{2} + 9 \, x\right )} e^{\left (4 \, x + 4 \, \log \left (-\log \relax (3)^{2} + 3\right )\right )} + {\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (3)^{8} - 12 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (3)^{6} + 54 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (3)^{4} + 81 \, x^{4} - 108 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \relax (3)^{2} + 162 \, x^{2} + 81\right )} e^{\left (2 \, x + 2 \, \log \left (-\log \relax (3)^{2} + 3\right )\right )}}{x^{4} \log \relax (3)^{8} - 12 \, x^{4} \log \relax (3)^{6} + 54 \, x^{4} \log \relax (3)^{4} - 108 \, x^{4} \log \relax (3)^{2} + 81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 197, normalized size = 5.32 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} \log \relax (3)^{4} - 2 \, x^{3} e^{\left (4 \, x\right )} \log \relax (3)^{4} - 6 \, x^{4} e^{\left (2 \, x\right )} \log \relax (3)^{2} + x^{2} e^{\left (6 \, x\right )} \log \relax (3)^{4} + 2 \, x^{2} e^{\left (2 \, x\right )} \log \relax (3)^{4} + 12 \, x^{3} e^{\left (4 \, x\right )} \log \relax (3)^{2} - 2 \, x e^{\left (4 \, x\right )} \log \relax (3)^{4} + 9 \, x^{4} e^{\left (2 \, x\right )} - 6 \, x^{2} e^{\left (6 \, x\right )} \log \relax (3)^{2} - 12 \, x^{2} e^{\left (2 \, x\right )} \log \relax (3)^{2} + e^{\left (2 \, x\right )} \log \relax (3)^{4} - 18 \, x^{3} e^{\left (4 \, x\right )} + 12 \, x e^{\left (4 \, x\right )} \log \relax (3)^{2} + 9 \, x^{2} e^{\left (6 \, x\right )} + 18 \, x^{2} e^{\left (2 \, x\right )} - 6 \, e^{\left (2 \, x\right )} \log \relax (3)^{2} - 18 \, x e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 72, normalized size = 1.95
method | result | size |
risch | \(\frac {\left (-\ln \relax (3)^{2}+3\right )^{2} {\mathrm e}^{6 x}}{x^{2}}-\frac {2 \left (-\ln \relax (3)^{2}+3\right )^{2} \left (x^{2}+1\right ) {\mathrm e}^{4 x}}{x^{3}}+\frac {\left (-\ln \relax (3)^{2}+3\right )^{2} \left (x^{4}+2 x^{2}+1\right ) {\mathrm e}^{2 x}}{x^{4}}\) | \(72\) |
default | \(-6 \ln \relax (3)^{2} {\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{2 x}}{x^{4}}+\frac {18 \,{\mathrm e}^{2 x}}{x^{2}}+\ln \relax (3)^{4} {\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{6 x}}{x^{2}}-\frac {18 \,{\mathrm e}^{4 x}}{x}-\frac {18 \,{\mathrm e}^{4 x}}{x^{3}}-\frac {6 \ln \relax (3)^{2} {\mathrm e}^{2 x}}{x^{4}}+\frac {{\mathrm e}^{6 x} \ln \relax (3)^{4}}{x^{2}}-\frac {6 \,{\mathrm e}^{6 x} \ln \relax (3)^{2}}{x^{2}}-\frac {2 \,{\mathrm e}^{4 x} \ln \relax (3)^{4}}{x}+\frac {12 \,{\mathrm e}^{4 x} \ln \relax (3)^{2}}{x}-\frac {2 \,{\mathrm e}^{4 x} \ln \relax (3)^{4}}{x^{3}}+\frac {12 \,{\mathrm e}^{4 x} \ln \relax (3)^{2}}{x^{3}}+\frac {2 \ln \relax (3)^{4} {\mathrm e}^{2 x}}{x^{2}}-\frac {12 \ln \relax (3)^{2} {\mathrm e}^{2 x}}{x^{2}}+\frac {\ln \relax (3)^{4} {\mathrm e}^{2 x}}{x^{4}}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.40, size = 86, normalized size = 2.32 \begin {gather*} -{\left (\log \relax (3)^{2} - 3\right )}^{2} {\left (8 \, {\rm Ei}\left (4 \, x\right ) - e^{\left (2 \, x\right )} - 8 \, \Gamma \left (-1, -2 \, x\right ) - 8 \, \Gamma \left (-1, -4 \, x\right ) - 36 \, \Gamma \left (-1, -6 \, x\right ) - 16 \, \Gamma \left (-2, -2 \, x\right ) - 128 \, \Gamma \left (-2, -4 \, x\right ) - 72 \, \Gamma \left (-2, -6 \, x\right ) - 16 \, \Gamma \left (-3, -2 \, x\right ) - 384 \, \Gamma \left (-3, -4 \, x\right ) - 64 \, \Gamma \left (-4, -2 \, x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.53, size = 30, normalized size = 0.81 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,{\left ({\ln \relax (3)}^2-3\right )}^2\,{\left (x^2-x\,{\mathrm {e}}^{2\,x}+1\right )}^2}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.37, size = 165, normalized size = 4.46 \begin {gather*} \frac {\left (- 6 x^{7} \log {\relax (3 )}^{2} + x^{7} \log {\relax (3 )}^{4} + 9 x^{7}\right ) e^{6 x} + \left (- 18 x^{8} - 2 x^{8} \log {\relax (3 )}^{4} + 12 x^{8} \log {\relax (3 )}^{2} - 18 x^{6} - 2 x^{6} \log {\relax (3 )}^{4} + 12 x^{6} \log {\relax (3 )}^{2}\right ) e^{4 x} + \left (- 6 x^{9} \log {\relax (3 )}^{2} + x^{9} \log {\relax (3 )}^{4} + 9 x^{9} - 12 x^{7} \log {\relax (3 )}^{2} + 2 x^{7} \log {\relax (3 )}^{4} + 18 x^{7} - 6 x^{5} \log {\relax (3 )}^{2} + x^{5} \log {\relax (3 )}^{4} + 9 x^{5}\right ) e^{2 x}}{x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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