Optimal. Leaf size=28 \[ e^{x-\frac {3 x}{2+\frac {2}{x}}}-\frac {621+x}{x+\log (x)} \]
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Rubi [F]
time = 9.71, 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 {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {x^2}{2+2 x}} \left (2 e^{\frac {x^2}{2+2 x}} (1+x)^2 (621+622 x)-e^{\frac {x}{1+x}} x^3 \left (-2+2 x+x^2\right )-2 x \left (e^{\frac {x^2}{2+2 x}} (1+x)^2+e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right )\right ) \log (x)-e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right ) \log ^2(x)\right )}{2 x (1+x)^2 (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-\frac {x^2}{2+2 x}} \left (2 e^{\frac {x^2}{2+2 x}} (1+x)^2 (621+622 x)-e^{\frac {x}{1+x}} x^3 \left (-2+2 x+x^2\right )-2 x \left (e^{\frac {x^2}{2+2 x}} (1+x)^2+e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right )\right ) \log (x)-e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right ) \log ^2(x)\right )}{x (1+x)^2 (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2}-\frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2}-\frac {2 (-621-622 x+x \log (x))}{x (x+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx\right )-\frac {1}{2} \int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {-621-622 x+x \log (x)}{x (x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx\right )-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \left (\frac {-621-622 x-x^2}{x (x+\log (x))^2}+\frac {1}{x+\log (x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx\right )-\frac {1}{2} \int \left (\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right )}{(1+x)^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \frac {-621-622 x-x^2}{x (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx-\int \left (-\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right )}{(1+x)^2} \, dx\right )-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx-\int \left (-\frac {622}{(x+\log (x))^2}-\frac {621}{x (x+\log (x))^2}-\frac {x}{(x+\log (x))^2}\right ) \, dx\\ &=e^{\frac {(2-x) x}{2 (1+x)}}-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+621 \int \frac {1}{x (x+\log (x))^2} \, dx+622 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x}{(x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx+\int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx\\ &=e^{\frac {(2-x) x}{2 (1+x)}}-2 \left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx\right )+2 \left (\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx\right )+2 \left (\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx\right )-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-2 \left (3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx\right )+6 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+621 \int \frac {1}{x (x+\log (x))^2} \, dx+622 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x}{(x+\log (x))^2} \, dx+\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.20, size = 33, normalized size = 1.18 \begin {gather*} \frac {1}{2} \left (2 e^{-\frac {(-2+x) x}{2 (1+x)}}-\frac {2 (621+x)}{x+\log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 20.65, size = 25, normalized size = 0.89
method | result | size |
risch | \({\mathrm e}^{-\frac {\left (x -2\right ) x}{2 \left (x +1\right )}}-\frac {621+x}{x +\ln \left (x \right )}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 41, normalized size = 1.46 \begin {gather*} -\frac {{\left ({\left (x + 621\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x e^{\frac {3}{2}} + e^{\frac {3}{2}} \log \left (x\right )\right )} e^{\left (-\frac {3}{2 \, {\left (x + 1\right )}}\right )}\right )} e^{\left (-\frac {1}{2} \, x\right )}}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 47, normalized size = 1.68 \begin {gather*} \frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 22, normalized size = 0.79 \begin {gather*} \frac {- x - 621}{x + \log {\left (x \right )}} + e^{\frac {- x^{2} + 2 x}{2 x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 47, normalized size = 1.68 \begin {gather*} \frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.55, size = 60, normalized size = 2.14 \begin {gather*} {\mathrm {e}}^{\frac {2\,x}{2\,x+2}-\frac {x^2}{2\,x+2}}+\frac {1}{x+1}-\frac {\frac {622\,x+621}{x+1}-\frac {x\,\ln \left (x\right )}{x+1}}{x+\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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