Optimal. Leaf size=33 \[ \frac {\frac {1}{9} \left (1+e^x\right )^2-\log (2 x)}{\left (\frac {2}{3}-x\right ) (3+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(33)=66\).
time = 1.65, antiderivative size = 113, normalized size of antiderivative = 3.42, number of steps
used = 32, number of rules used = 11, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.124, Rules used = {6873, 12,
6874, 46, 90, 78, 2208, 2209, 2404, 2351, 31} \begin {gather*} \frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{11 (2-3 x)}+\frac {2 e^x}{33 (x+3)}+\frac {e^{2 x}}{33 (x+3)}+\frac {1}{33 (x+3)}-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (x+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 46
Rule 78
Rule 90
Rule 2208
Rule 2209
Rule 2351
Rule 2404
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{3 x \left (6-7 x-3 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-54+70 x+33 x^2+e^{2 x} \left (19 x-8 x^2-6 x^3\right )+e^x \left (26 x-2 x^2-6 x^3\right )+\left (-63 x-54 x^2\right ) \log (2 x)}{x \left (6-7 x-3 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {70}{(3+x)^2 (-2+3 x)^2}-\frac {54}{x (3+x)^2 (-2+3 x)^2}+\frac {33 x}{(3+x)^2 (-2+3 x)^2}-\frac {2 e^x \left (-13+x+3 x^2\right )}{(3+x)^2 (-2+3 x)^2}-\frac {e^{2 x} \left (-19+8 x+6 x^2\right )}{(3+x)^2 (-2+3 x)^2}-\frac {9 (7+6 x) \log (2 x)}{(3+x)^2 (-2+3 x)^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {e^{2 x} \left (-19+8 x+6 x^2\right )}{(3+x)^2 (-2+3 x)^2} \, dx\right )-\frac {2}{3} \int \frac {e^x \left (-13+x+3 x^2\right )}{(3+x)^2 (-2+3 x)^2} \, dx-3 \int \frac {(7+6 x) \log (2 x)}{(3+x)^2 (-2+3 x)^2} \, dx+11 \int \frac {x}{(3+x)^2 (-2+3 x)^2} \, dx-18 \int \frac {1}{x (3+x)^2 (-2+3 x)^2} \, dx+\frac {70}{3} \int \frac {1}{(3+x)^2 (-2+3 x)^2} \, dx\\ &=-\left (\frac {1}{3} \int \left (\frac {e^{2 x}}{11 (3+x)^2}-\frac {2 e^{2 x}}{11 (3+x)}-\frac {9 e^{2 x}}{11 (-2+3 x)^2}+\frac {6 e^{2 x}}{11 (-2+3 x)}\right ) \, dx\right )-\frac {2}{3} \int \left (\frac {e^x}{11 (3+x)^2}-\frac {e^x}{11 (3+x)}-\frac {9 e^x}{11 (-2+3 x)^2}+\frac {3 e^x}{11 (-2+3 x)}\right ) \, dx-3 \int \left (-\frac {\log (2 x)}{11 (3+x)^2}+\frac {9 \log (2 x)}{11 (-2+3 x)^2}\right ) \, dx+11 \int \left (-\frac {3}{121 (3+x)^2}-\frac {7}{1331 (3+x)}+\frac {6}{121 (-2+3 x)^2}+\frac {21}{1331 (-2+3 x)}\right ) \, dx-18 \int \left (\frac {1}{36 x}-\frac {1}{363 (3+x)^2}-\frac {29}{11979 (3+x)}+\frac {27}{242 (-2+3 x)^2}-\frac {405}{5324 (-2+3 x)}\right ) \, dx+\frac {70}{3} \int \left (\frac {1}{121 (3+x)^2}+\frac {6}{1331 (3+x)}+\frac {9}{121 (-2+3 x)^2}-\frac {18}{1331 (-2+3 x)}\right ) \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {9}{22} \log (2-3 x)-\frac {\log (x)}{2}+\frac {1}{11} \log (3+x)-\frac {1}{33} \int \frac {e^{2 x}}{(3+x)^2} \, dx-\frac {2}{33} \int \frac {e^x}{(3+x)^2} \, dx+\frac {2}{33} \int \frac {e^x}{3+x} \, dx+\frac {2}{33} \int \frac {e^{2 x}}{3+x} \, dx-\frac {2}{11} \int \frac {e^x}{-2+3 x} \, dx-\frac {2}{11} \int \frac {e^{2 x}}{-2+3 x} \, dx+\frac {3}{11} \int \frac {e^{2 x}}{(-2+3 x)^2} \, dx+\frac {3}{11} \int \frac {\log (2 x)}{(3+x)^2} \, dx+\frac {6}{11} \int \frac {e^x}{(-2+3 x)^2} \, dx-\frac {27}{11} \int \frac {\log (2 x)}{(-2+3 x)^2} \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {2 e^x}{33 (3+x)}+\frac {e^{2 x}}{33 (3+x)}-\frac {2}{33} e^{4/3} \text {Ei}\left (-\frac {2}{3} (2-3 x)\right )+\frac {2 \text {Ei}(3+x)}{33 e^3}+\frac {2 \text {Ei}(2 (3+x))}{33 e^6}-\frac {2}{33} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+3 x)\right )+\frac {9}{22} \log (2-3 x)-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (3+x)}+\frac {1}{11} \log (3+x)-\frac {2}{33} \int \frac {e^x}{3+x} \, dx-\frac {2}{33} \int \frac {e^{2 x}}{3+x} \, dx-\frac {1}{11} \int \frac {1}{3+x} \, dx+\frac {2}{11} \int \frac {e^x}{-2+3 x} \, dx+\frac {2}{11} \int \frac {e^{2 x}}{-2+3 x} \, dx-\frac {27}{22} \int \frac {1}{-2+3 x} \, dx\\ &=\frac {1}{11 (2-3 x)}+\frac {2 e^x}{11 (2-3 x)}+\frac {e^{2 x}}{11 (2-3 x)}+\frac {1}{33 (3+x)}+\frac {2 e^x}{33 (3+x)}+\frac {e^{2 x}}{33 (3+x)}-\frac {\log (x)}{2}-\frac {27 x \log (2 x)}{22 (2-3 x)}+\frac {x \log (2 x)}{11 (3+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.59, size = 32, normalized size = 0.97 \begin {gather*} \frac {-\left (1+e^x\right )^2+9 \log (2 x)}{3 \left (-6+7 x+3 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs.
\(2(32)=64\).
time = 0.53, size = 120, normalized size = 3.64
method | result | size |
risch | \(\frac {3 \ln \left (2 x \right )}{3 x^{2}+7 x -6}-\frac {2 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+1}{3 \left (3 x^{2}+7 x -6\right )}\) | \(44\) |
default | \(-\frac {1}{11 \left (3 x -2\right )}+\frac {9 \ln \left (3 x -2\right )}{22}-\frac {\ln \left (x \right )}{2}+\frac {1}{99+33 x}+\frac {\ln \left (3+x \right )}{11}-\frac {9 \ln \left (6 x -4\right )}{22}+\frac {27 \ln \left (2 x \right ) x}{11 \left (6 x -4\right )}-\frac {\ln \left (2 x +6\right )}{11}+\frac {2 \ln \left (2 x \right ) x}{11 \left (2 x +6\right )}+\frac {2 \,{\mathrm e}^{x}}{33 \left (3+x \right )}-\frac {2 \,{\mathrm e}^{x}}{33 \left (x -\frac {2}{3}\right )}+\frac {{\mathrm e}^{2 x}}{99+33 x}-\frac {{\mathrm e}^{2 x}}{33 \left (x -\frac {2}{3}\right )}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (27) = 54\).
time = 0.54, size = 104, normalized size = 3.15 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} + 7 \, x\right )} \log \left (x\right ) - 2 \, e^{\left (2 \, x\right )} - 4 \, e^{x} + 18 \, \log \left (2\right )}{6 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {3 \, {\left (21 \, x + 85\right )}}{121 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} + \frac {7 \, x - 12}{11 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {70 \, {\left (6 \, x + 7\right )}}{363 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} - \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 30, normalized size = 0.91 \begin {gather*} -\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \left (2 \, x\right ) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (22) = 44\).
time = 0.19, size = 76, normalized size = 2.30 \begin {gather*} \frac {\left (- 18 x^{2} - 42 x + 36\right ) e^{x} + \left (- 9 x^{2} - 21 x + 18\right ) e^{2 x}}{81 x^{4} + 378 x^{3} + 117 x^{2} - 756 x + 324} - \frac {1}{9 x^{2} + 21 x - 18} + \frac {3 \log {\left (2 x \right )}}{3 x^{2} + 7 x - 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 30, normalized size = 0.91 \begin {gather*} -\frac {e^{\left (2 \, x\right )} + 2 \, e^{x} - 9 \, \log \left (2 \, x\right ) + 1}{3 \, {\left (3 \, x^{2} + 7 \, x - 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 64, normalized size = 1.94 \begin {gather*} \frac {\ln \left (2\,x\right )}{x^2+\frac {7\,x}{3}-2}-\frac {{\mathrm {e}}^{2\,x}}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {1}{9\,\left (x^2+\frac {7\,x}{3}-2\right )}-\frac {2\,{\mathrm {e}}^x}{9\,\left (x^2+\frac {7\,x}{3}-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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