Optimal. Leaf size=26 \[ \frac {\left (-3 e^x+x\right ) \left (\log (x)+\frac {x}{x+\log (x)}\right )}{x+x^2} \]
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Rubi [F]
time = 19.51, 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 {-x^2+e^x \left (3 x+3 x^2-3 x^4\right )+\left (3 x^2+2 x^3-x^4+e^x \left (-6 x-3 x^2-3 x^4\right )\right ) \log (x)+\left (x+x^2-2 x^3+e^x \left (-3+3 x+6 x^2-6 x^3\right )\right ) \log ^2(x)+\left (-x^2+e^x \left (3+3 x-3 x^2\right )\right ) \log ^3(x)}{x^4+2 x^5+x^6+\left (2 x^3+4 x^4+2 x^5\right ) \log (x)+\left (x^2+2 x^3+x^4\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 {-x \left (x+3 e^x \left (-1-x+x^3\right )\right )-x (1+x) \left ((-3+x) x+3 e^x \left (2-x+x^2\right )\right ) \log (x)-(-1+x) \left (x (1+2 x)+e^x \left (-3+6 x^2\right )\right ) \log ^2(x)-\left (x^2+3 e^x \left (-1-x+x^2\right )\right ) \log ^3(x)}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\\ &=\int \left (-\frac {1}{(1+x)^2 (x+\log (x))^2}-\frac {(-3+x) \log (x)}{(1+x) (x+\log (x))^2}-\frac {(-1+x) (1+2 x) \log ^2(x)}{x (1+x)^2 (x+\log (x))^2}-\frac {\log ^3(x)}{(1+x)^2 (x+\log (x))^2}-\frac {3 e^x \left (-x-x^2+x^4+2 x \log (x)+x^2 \log (x)+x^4 \log (x)+\log ^2(x)-x \log ^2(x)-2 x^2 \log ^2(x)+2 x^3 \log ^2(x)-\log ^3(x)-x \log ^3(x)+x^2 \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^x \left (-x-x^2+x^4+2 x \log (x)+x^2 \log (x)+x^4 \log (x)+\log ^2(x)-x \log ^2(x)-2 x^2 \log ^2(x)+2 x^3 \log ^2(x)-\log ^3(x)-x \log ^3(x)+x^2 \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\right )-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {(-3+x) \log (x)}{(1+x) (x+\log (x))^2} \, dx-\int \frac {(-1+x) (1+2 x) \log ^2(x)}{x (1+x)^2 (x+\log (x))^2} \, dx-\int \frac {\log ^3(x)}{(1+x)^2 (x+\log (x))^2} \, dx\\ &=-\left (3 \int \frac {e^x \left (x \left (-1-x+x^3\right )+x \left (2+x+x^3\right ) \log (x)+\left (1-x-2 x^2+2 x^3\right ) \log ^2(x)+\left (-1-x+x^2\right ) \log ^3(x)\right )}{x^2 (1+x)^2 (x+\log (x))^2} \, dx\right )-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \left (-\frac {2 x}{(1+x)^2}+\frac {\log (x)}{(1+x)^2}-\frac {x^3}{(1+x)^2 (x+\log (x))^2}+\frac {3 x^2}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \left (-\frac {(-3+x) x}{(1+x) (x+\log (x))^2}+\frac {-3+x}{(1+x) (x+\log (x))}\right ) \, dx-\int \left (\frac {(-1+x) (1+2 x)}{x (1+x)^2}+\frac {x \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx\\ &=2 \int \frac {x}{(1+x)^2} \, dx+2 \int \frac {-1-x+2 x^2}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {x^2}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (\frac {e^x}{x^2 (1+x)}+\frac {e^x \left (-1-x+x^2\right ) \log (x)}{x^2 (1+x)^2}-\frac {e^x}{x (x+\log (x))^2}+\frac {e^x x}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \frac {(-1+x) (1+2 x)}{x (1+x)^2} \, dx-\int \frac {\log (x)}{(1+x)^2} \, dx-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+\int \frac {x^3}{(1+x)^2 (x+\log (x))^2} \, dx+\int \frac {(-3+x) x}{(1+x) (x+\log (x))^2} \, dx-\int \frac {x \left (-1-x+2 x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {-3+x}{(1+x) (x+\log (x))} \, dx\\ &=-\frac {x \log (x)}{1+x}+2 \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx+2 \int \left (\frac {2}{x+\log (x)}+\frac {2}{(1+x)^2 (x+\log (x))}-\frac {5}{(1+x) (x+\log (x))}\right ) \, dx-3 \int \frac {e^x}{x^2 (1+x)} \, dx-3 \int \frac {e^x \left (-1-x+x^2\right ) \log (x)}{x^2 (1+x)^2} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx-3 \int \frac {e^x x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (\frac {1}{x+\log (x)}+\frac {1}{(1+x)^2 (x+\log (x))}-\frac {2}{(1+x) (x+\log (x))}\right ) \, dx+\int \frac {1}{1+x} \, dx-\int \left (-\frac {1}{x}-\frac {2}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx-\int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+\int \left (-\frac {2}{(x+\log (x))^2}+\frac {x}{(x+\log (x))^2}-\frac {1}{(1+x)^2 (x+\log (x))^2}+\frac {3}{(1+x) (x+\log (x))^2}\right ) \, dx+\int \left (-\frac {4}{(x+\log (x))^2}+\frac {x}{(x+\log (x))^2}+\frac {4}{(1+x) (x+\log (x))^2}\right ) \, dx-\int \left (-\frac {5}{(x+\log (x))^2}+\frac {2 x}{(x+\log (x))^2}-\frac {2}{(1+x)^2 (x+\log (x))^2}+\frac {7}{(1+x) (x+\log (x))^2}\right ) \, dx-\int \left (\frac {1}{x+\log (x)}-\frac {4}{(1+x) (x+\log (x))}\right ) \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x^2 (1+x)} \, dx-3 \int \left (\frac {e^x}{x^2}-\frac {e^x}{x}+\frac {e^x}{1+x}\right ) \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx-3 \int \left (-\frac {e^x}{(1+x)^2 (x+\log (x))}+\frac {e^x}{(1+x) (x+\log (x))}\right ) \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^x}{x^2} \, dx+3 \int \frac {e^x}{x} \, dx-3 \int \frac {e^x}{1+x} \, dx+3 \int \left (\frac {e^x}{x^2}-\frac {e^x}{x}+\frac {e^x}{1+x}\right ) \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\frac {3 e^x}{x}+3 \text {Ei}(x)-\frac {3 \text {Ei}(1+x)}{e}+\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x^2} \, dx-2 \left (3 \int \frac {e^x}{x} \, dx\right )+3 \int \frac {e^x}{1+x} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=-3 \text {Ei}(x)+\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ &=\log (x)-\frac {3 e^x \log (x)}{x}+\frac {3 e^x \log (x)}{1+x}-\frac {x \log (x)}{1+x}-2 \int \frac {1}{(x+\log (x))^2} \, dx-2 \int \frac {x}{(x+\log (x))^2} \, dx+2 \int \frac {1}{(1+x)^2 (x+\log (x))^2} \, dx+3 \int \frac {e^x}{x (x+\log (x))^2} \, dx+3 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-3 \int \frac {1}{x+\log (x)} \, dx-3 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+3 \int \frac {e^x}{(1+x)^2 (x+\log (x))} \, dx-3 \int \frac {e^x}{(1+x) (x+\log (x))} \, dx-4 \int \frac {1}{(x+\log (x))^2} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx+4 \int \frac {1}{x+\log (x)} \, dx+4 \int \frac {1}{(1+x)^2 (x+\log (x))} \, dx+4 \int \frac {1}{(1+x) (x+\log (x))} \, dx+5 \int \frac {1}{(x+\log (x))^2} \, dx+6 \int \frac {1}{(1+x) (x+\log (x))} \, dx-7 \int \frac {1}{(1+x) (x+\log (x))^2} \, dx-10 \int \frac {1}{(1+x) (x+\log (x))} \, dx+2 \int \frac {x}{(x+\log (x))^2} \, dx-2 \int \frac {1}{(1+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 = 0.10, size = 32, normalized size = 1.23 \begin {gather*} \frac {\left (-3 e^x+x\right ) \left (x+x \log (x)+\log ^2(x)\right )}{x (1+x) (x+\log (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 9.62, size = 37, normalized size = 1.42
method | result | size |
risch | \(\frac {\left (x -3 \,{\mathrm e}^{x}\right ) \ln \left (x \right )}{\left (x +1\right ) x}+\frac {x -3 \,{\mathrm e}^{x}}{\left (x +1\right ) \left (x +\ln \left (x \right )\right )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 48, normalized size = 1.85 \begin {gather*} \frac {x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2} - 3 \, {\left (x \log \left (x\right ) + \log \left (x\right )^{2} + x\right )} e^{x}}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 50, normalized size = 1.92 \begin {gather*} \frac {{\left (x - 3 \, e^{x}\right )} \log \left (x\right )^{2} + x^{2} - 3 \, x e^{x} + {\left (x^{2} - 3 \, x e^{x}\right )} \log \left (x\right )}{x^{3} + x^{2} + {\left (x^{2} + x\right )} \log \left (x\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. 60 vs.
\(2 (20) = 40\).
time = 0.24, size = 60, normalized size = 2.31 \begin {gather*} \frac {x}{x^{2} + x + \left (x + 1\right ) \log {\left (x \right )}} + \frac {\left (- 3 x \log {\left (x \right )} - 3 x - 3 \log {\left (x \right )}^{2}\right ) e^{x}}{x^{3} + x^{2} \log {\left (x \right )} + x^{2} + x \log {\left (x \right )}} + \frac {\log {\left (x \right )}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (25) = 50\).
time = 0.42, size = 56, normalized size = 2.15 \begin {gather*} \frac {x^{2} \log \left (x\right ) - 3 \, x e^{x} \log \left (x\right ) + x \log \left (x\right )^{2} - 3 \, e^{x} \log \left (x\right )^{2} + x^{2} - 3 \, x e^{x}}{x^{3} + x^{2} \log \left (x\right ) + x^{2} + x \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\ln \left (x\right )\,\left (3\,x^2+2\,x^3-x^4-{\mathrm {e}}^x\,\left (3\,x^4+3\,x^2+6\,x\right )\right )+{\ln \left (x\right )}^2\,\left (x+x^2-2\,x^3+{\mathrm {e}}^x\,\left (-6\,x^3+6\,x^2+3\,x-3\right )\right )-x^2+{\mathrm {e}}^x\,\left (-3\,x^4+3\,x^2+3\,x\right )+{\ln \left (x\right )}^3\,\left ({\mathrm {e}}^x\,\left (-3\,x^2+3\,x+3\right )-x^2\right )}{\ln \left (x\right )\,\left (2\,x^5+4\,x^4+2\,x^3\right )+x^4+2\,x^5+x^6+{\ln \left (x\right )}^2\,\left (x^4+2\,x^3+x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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