Optimal. Leaf size=27 \[ \frac {1}{5} e^{x/3} \left (-\frac {x^3}{3}+\log (2-x)\right )^2 \]
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Rubi [F] time = 2.67, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{x/3} \left (-18 x^3-36 x^5+16 x^6+x^7\right )+e^{x/3} \left (54+108 x^2-42 x^3-6 x^4\right ) \log (2-x)+e^{x/3} (-18+9 x) \log ^2(2-x)}{-270+135 x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 e^{x/3} x^3}{15 (-2+x)}-\frac {4 e^{x/3} x^5}{15 (-2+x)}+\frac {16 e^{x/3} x^6}{135 (-2+x)}+\frac {e^{x/3} x^7}{135 (-2+x)}+\frac {2 e^{x/3} \log (2-x)}{5 (-2+x)}+\frac {4 e^{x/3} x^2 \log (2-x)}{5 (-2+x)}-\frac {14 e^{x/3} x^3 \log (2-x)}{45 (-2+x)}-\frac {2 e^{x/3} x^4 \log (2-x)}{45 (-2+x)}-\frac {2 e^{x/3} \log ^2(2-x)}{15 (-2+x)}+\frac {e^{x/3} x \log ^2(2-x)}{15 (-2+x)}\right ) \, dx\\ &=\frac {1}{135} \int \frac {e^{x/3} x^7}{-2+x} \, dx-\frac {2}{45} \int \frac {e^{x/3} x^4 \log (2-x)}{-2+x} \, dx+\frac {1}{15} \int \frac {e^{x/3} x \log ^2(2-x)}{-2+x} \, dx+\frac {16}{135} \int \frac {e^{x/3} x^6}{-2+x} \, dx-\frac {2}{15} \int \frac {e^{x/3} x^3}{-2+x} \, dx-\frac {2}{15} \int \frac {e^{x/3} \log ^2(2-x)}{-2+x} \, dx-\frac {4}{15} \int \frac {e^{x/3} x^5}{-2+x} \, dx-\frac {14}{45} \int \frac {e^{x/3} x^3 \log (2-x)}{-2+x} \, dx+\frac {2}{5} \int \frac {e^{x/3} \log (2-x)}{-2+x} \, dx+\frac {4}{5} \int \frac {e^{x/3} x^2 \log (2-x)}{-2+x} \, dx\\ &=-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{135} \int \left (64 e^{x/3}+\frac {128 e^{x/3}}{-2+x}+32 e^{x/3} x+16 e^{x/3} x^2+8 e^{x/3} x^3+4 e^{x/3} x^4+2 e^{x/3} x^5+e^{x/3} x^6\right ) \, dx+\frac {2}{45} \int \frac {-3 e^{x/3} \left (-130+46 x-7 x^2+x^3\right )-16 e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx+\frac {1}{15} \int \left (e^{x/3} \log ^2(2-x)+\frac {2 e^{x/3} \log ^2(2-x)}{-2+x}\right ) \, dx+\frac {16}{135} \int \left (32 e^{x/3}+\frac {64 e^{x/3}}{-2+x}+16 e^{x/3} x+8 e^{x/3} x^2+4 e^{x/3} x^3+2 e^{x/3} x^4+e^{x/3} x^5\right ) \, dx-\frac {2}{15} \int \left (4 e^{x/3}+\frac {8 e^{x/3}}{-2+x}+2 e^{x/3} x+e^{x/3} x^2\right ) \, dx-\frac {2}{15} \int \frac {e^{x/3} \log ^2(2-x)}{-2+x} \, dx-\frac {4}{15} \int \left (16 e^{x/3}+\frac {32 e^{x/3}}{-2+x}+8 e^{x/3} x+4 e^{x/3} x^2+2 e^{x/3} x^3+e^{x/3} x^4\right ) \, dx+\frac {14}{45} \int \frac {-3 e^{x/3} \left (16-4 x+x^2\right )-8 e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx-\frac {2}{5} \int \frac {e^{2/3} \text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {4}{5} \int \frac {-3 e^{x/3} (-1+x)-4 e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )}{2-x} \, dx\\ &=-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{135} \int e^{x/3} x^6 \, dx+\frac {2}{135} \int e^{x/3} x^5 \, dx+\frac {4}{135} \int e^{x/3} x^4 \, dx+\frac {2}{45} \int \left (\frac {3 e^{x/3} \left (-130+46 x-7 x^2+x^3\right )}{-2+x}+\frac {16 e^{2/3} \text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {8}{135} \int e^{x/3} x^3 \, dx+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {16}{135} \int e^{x/3} x^2 \, dx+\frac {16}{135} \int e^{x/3} x^5 \, dx-\frac {2}{15} \int e^{x/3} x^2 \, dx+\frac {32}{135} \int e^{x/3} x \, dx+\frac {32}{135} \int e^{x/3} x^4 \, dx-\frac {4}{15} \int e^{x/3} x \, dx-\frac {4}{15} \int e^{x/3} x^4 \, dx+\frac {14}{45} \int \left (\frac {3 e^{x/3} \left (16-4 x+x^2\right )}{-2+x}+\frac {8 e^{2/3} \text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {64}{135} \int e^{x/3} \, dx+\frac {64}{135} \int e^{x/3} x^3 \, dx-\frac {8}{15} \int e^{x/3} \, dx-\frac {8}{15} \int e^{x/3} x^3 \, dx-\frac {4}{5} \int \left (\frac {3 e^{x/3} (-1+x)}{-2+x}+\frac {4 e^{2/3} \text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x}\right ) \, dx+\frac {128}{135} \int \frac {e^{x/3}}{-2+x} \, dx+\frac {128}{135} \int e^{x/3} x^2 \, dx-\frac {16}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {16}{15} \int e^{x/3} x^2 \, dx+\frac {256}{135} \int e^{x/3} x \, dx-\frac {32}{15} \int e^{x/3} x \, dx+\frac {512}{135} \int e^{x/3} \, dx-\frac {64}{15} \int e^{x/3} \, dx+\frac {1024}{135} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {128}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=-\frac {8 e^{x/3}}{5}-\frac {4}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2+\frac {2}{5} e^{x/3} x^5+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {2}{15} \int e^{x/3} x^5 \, dx+\frac {2}{15} \int \frac {e^{x/3} \left (-130+46 x-7 x^2+x^3\right )}{-2+x} \, dx-\frac {2}{9} \int e^{x/3} x^4 \, dx-\frac {16}{45} \int e^{x/3} x^3 \, dx-\frac {8}{15} \int e^{x/3} x^2 \, dx-\frac {32}{45} \int e^{x/3} \, dx-\frac {32}{45} \int e^{x/3} x \, dx+\frac {4}{5} \int e^{x/3} \, dx+\frac {4}{5} \int e^{x/3} x \, dx+\frac {14}{15} \int \frac {e^{x/3} \left (16-4 x+x^2\right )}{-2+x} \, dx-\frac {16}{9} \int e^{x/3} x^4 \, dx-\frac {12}{5} \int \frac {e^{x/3} (-1+x)}{-2+x} \, dx-\frac {128}{45} \int e^{x/3} x^3 \, dx+\frac {16}{5} \int e^{x/3} x^3 \, dx-\frac {64}{15} \int e^{x/3} x^2 \, dx+\frac {24}{5} \int e^{x/3} x^2 \, dx-\frac {256}{45} \int e^{x/3} \, dx-\frac {256}{45} \int e^{x/3} x \, dx+\frac {32}{5} \int e^{x/3} \, dx+\frac {32}{5} \int e^{x/3} x \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=\frac {4 e^{x/3}}{5}+\frac {8}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2-6 e^{x/3} x^4+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {2}{15} \int \left (36 e^{x/3}-\frac {58 e^{x/3}}{-2+x}-5 e^{x/3} x+e^{x/3} x^2\right ) \, dx+\frac {14}{15} \int \left (-2 e^{x/3}+\frac {12 e^{x/3}}{-2+x}+e^{x/3} x\right ) \, dx+2 \int e^{x/3} x^4 \, dx+\frac {32}{15} \int e^{x/3} \, dx-\frac {12}{5} \int e^{x/3} \, dx-\frac {12}{5} \int \left (e^{x/3}+\frac {e^{x/3}}{-2+x}\right ) \, dx+\frac {8}{3} \int e^{x/3} x^3 \, dx+\frac {16}{5} \int e^{x/3} x \, dx+\frac {16}{5} \int e^{x/3} x^2 \, dx+\frac {256}{15} \int e^{x/3} \, dx-\frac {96}{5} \int e^{x/3} \, dx+\frac {64}{3} \int e^{x/3} x^3 \, dx+\frac {128}{5} \int e^{x/3} x \, dx+\frac {128}{5} \int e^{x/3} x^2 \, dx-\frac {144}{5} \int e^{x/3} x \, dx-\frac {144}{5} \int e^{x/3} x^2 \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=-\frac {32 e^{x/3}}{5}+\frac {8}{5} e^{x/3} x-\frac {2}{5} e^{x/3} x^2+72 e^{x/3} x^3+\frac {1}{45} e^{x/3} x^6-\frac {16}{15} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right )-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {2}{15} \int e^{x/3} x^2 \, dx-\frac {2}{3} \int e^{x/3} x \, dx+\frac {14}{15} \int e^{x/3} x \, dx-\frac {28}{15} \int e^{x/3} \, dx-\frac {12}{5} \int e^{x/3} \, dx-\frac {12}{5} \int \frac {e^{x/3}}{-2+x} \, dx+\frac {24}{5} \int e^{x/3} \, dx-\frac {116}{15} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {48}{5} \int e^{x/3} \, dx+\frac {56}{5} \int \frac {e^{x/3}}{-2+x} \, dx-\frac {96}{5} \int e^{x/3} x \, dx-24 \int e^{x/3} x^2 \, dx-24 \int e^{x/3} x^3 \, dx-\frac {384}{5} \int e^{x/3} \, dx+\frac {432}{5} \int e^{x/3} \, dx-\frac {768}{5} \int e^{x/3} x \, dx+\frac {864}{5} \int e^{x/3} x \, dx-192 \int e^{x/3} x^2 \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=-\frac {24 e^{x/3}}{5}+\frac {12}{5} e^{x/3} x-648 e^{x/3} x^2+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {4}{5} \int e^{x/3} x \, dx+2 \int e^{x/3} \, dx-\frac {14}{5} \int e^{x/3} \, dx+\frac {288}{5} \int e^{x/3} \, dx+144 \int e^{x/3} x \, dx+216 \int e^{x/3} x^2 \, dx+\frac {2304}{5} \int e^{x/3} \, dx-\frac {2592}{5} \int e^{x/3} \, dx+1152 \int e^{x/3} x \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=-\frac {36 e^{x/3}}{5}+3888 e^{x/3} x+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+\frac {12}{5} \int e^{x/3} \, dx-432 \int e^{x/3} \, dx-1296 \int e^{x/3} x \, dx-3456 \int e^{x/3} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=-11664 e^{x/3}+\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx+3888 \int e^{x/3} \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ &=\frac {1}{45} e^{x/3} x^6-\frac {2}{15} e^{x/3} x^3 \log (2-x)+\frac {2}{5} e^{2/3} \text {Ei}\left (\frac {1}{3} (-2+x)\right ) \log (2-x)+\frac {1}{15} \int e^{x/3} \log ^2(2-x) \, dx-\frac {1}{5} \left (2 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (32 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx+\frac {1}{45} \left (112 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx-\frac {1}{5} \left (16 e^{2/3}\right ) \int \frac {\text {Ei}\left (-\frac {2}{3}+\frac {x}{3}\right )}{-2+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{45} e^{x/3} \left (x^3-3 \log (2-x)\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 39, normalized size = 1.44 \begin {gather*} \frac {1}{45} \, x^{6} e^{\left (\frac {1}{3} \, x\right )} - \frac {2}{15} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right ) + \frac {1}{5} \, e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 39, normalized size = 1.44 \begin {gather*} \frac {1}{45} \, x^{6} e^{\left (\frac {1}{3} \, x\right )} - \frac {2}{15} \, x^{3} e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right ) + \frac {1}{5} \, e^{\left (\frac {1}{3} \, x\right )} \log \left (-x + 2\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 40, normalized size = 1.48
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right )^{2}}{5}-\frac {2 x^{3} {\mathrm e}^{\frac {x}{3}} \ln \left (2-x \right )}{15}+\frac {x^{6} {\mathrm e}^{\frac {x}{3}}}{45}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{45} \, {\left (x^{6} - 6 \, x^{3} \log \left (-x + 2\right ) + 9 \, \log \left (-x + 2\right )^{2}\right )} e^{\left (\frac {1}{3} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 22, normalized size = 0.81 \begin {gather*} \frac {{\mathrm {e}}^{x/3}\,{\left (3\,\ln \left (2-x\right )-x^3\right )}^2}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 27, normalized size = 1.00 \begin {gather*} \frac {\left (x^{6} - 6 x^{3} \log {\left (2 - x \right )} + 9 \log {\left (2 - x \right )}^{2}\right ) e^{\frac {x}{3}}}{45} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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