Optimal. Leaf size=30 \[ -3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (-3+(-2+\log (x))^2\right )} \]
[Out]
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Rubi [F]
time = 7.39, 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 {-3 x+e^{2 x} \left (4 e^5+2 x\right )+e^x \left (-x+x^2+e^5 (4+x)\right )+\left (24 x+e^{2 x} \left (-2 e^5+18 x\right )+e^x \left (e^5 (-2-4 x)+42 x-4 x^2\right )\right ) \log (x)+\left (-54 x-53 e^{2 x} x+e^x \left (-107 x+e^5 x+x^2\right )\right ) \log ^2(x)+\left (24 x+48 e^x x+24 e^{2 x} x\right ) \log ^3(x)+\left (-3 x-6 e^x x-3 e^{2 x} x\right ) \log ^4(x)}{x+2 e^x x+e^{2 x} x+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log (x)+\left (18 x+36 e^x x+18 e^{2 x} x\right ) \log ^2(x)+\left (-8 x-16 e^x x-8 e^{2 x} x\right ) \log ^3(x)+\left (x+2 e^x x+e^{2 x} x\right ) \log ^4(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{5+2 x}-3 x+2 e^{2 x} x+e^x (-1+x) x+e^{5+x} (4+x)-2 \left (e^{5+2 x}-12 x-9 e^{2 x} x+e^x x (-21+2 x)+e^{5+x} (1+2 x)\right ) \log (x)+\left (-54-53 e^{2 x}+e^{5+x}+e^x (-107+x)\right ) x \log ^2(x)+24 \left (1+e^x\right )^2 x \log ^3(x)-3 \left (1+e^x\right )^2 x \log ^4(x)}{\left (1+e^x\right )^2 x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \left (-\frac {e^5+x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}+\frac {-4 e^5-5 \left (1-\frac {e^5}{5}\right ) x+x^2+2 e^5 \log (x)+6 \left (1-\frac {2 e^5}{3}\right ) x \log (x)-4 x^2 \log (x)-\left (1-e^5\right ) x \log ^2(x)+x^2 \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {4 e^5+2 x-2 e^5 \log (x)+18 x \log (x)-53 x \log ^2(x)+24 x \log ^3(x)-3 x \log ^4(x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=-\int \frac {e^5+x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \frac {-4 e^5-5 \left (1-\frac {e^5}{5}\right ) x+x^2+2 e^5 \log (x)+6 \left (1-\frac {2 e^5}{3}\right ) x \log (x)-4 x^2 \log (x)-\left (1-e^5\right ) x \log ^2(x)+x^2 \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {4 e^5+2 x-2 e^5 \log (x)+18 x \log (x)-53 x \log ^2(x)+24 x \log ^3(x)-3 x \log ^4(x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx\\ &=\int \frac {e^5 (-4+x)+(-5+x) x+\left (e^5 (2-4 x)+2 (3-2 x) x\right ) \log (x)+x \left (-1+e^5+x\right ) \log ^2(x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \left (-3-\frac {2 \left (e^5+x\right ) (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {1}{1-4 \log (x)+\log ^2(x)}\right ) \, dx-\int \left (\frac {e^5}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}+\frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )}\right ) \, dx\\ &=-3 x-2 \int \frac {\left (e^5+x\right ) (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \frac {1}{1-4 \log (x)+\log ^2(x)} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \left (-\frac {5 \left (1-\frac {e^5}{5}\right )}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {4 e^5}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {6 \left (1-\frac {2 e^5}{3}\right ) \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {2 e^5 \log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {4 x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}-\frac {\left (1-e^5\right ) \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx\\ &=-3 x-2 \int \left (\frac {-2+\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {e^5 (-2+\log (x))}{x \left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\int \left (-\frac {1}{\sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}-\frac {1}{\sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx\\ &=-3 x-2 \int \frac {-2+\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {\int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx}{\sqrt {3}}-\frac {\int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx}{\sqrt {3}}-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx-\left (2 e^5\right ) \int \frac {-2+\log (x)}{x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x-2 \int \left (-\frac {2}{\left (1-4 \log (x)+\log ^2(x)\right )^2}+\frac {\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {\text {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )}{\sqrt {3}}-\frac {\text {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )}{\sqrt {3}}-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (2 e^5\right ) \text {Subst}\left (\int \frac {-2+x}{\left (1-4 x+x^2\right )^2} \, dx,x,\log (x)\right )-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-2 \int \frac {\log (x)}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+4 \int \frac {1}{\left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-2 \int \left (\frac {4+2 \sqrt {3}}{6 \left (4+2 \sqrt {3}-2 \log (x)\right )^2}+\frac {1}{3 \sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}+\frac {4-2 \sqrt {3}}{6 \left (-4+2 \sqrt {3}+2 \log (x)\right )^2}+\frac {1}{3 \sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+4 \int \left (\frac {1}{3 \left (4+2 \sqrt {3}-2 \log (x)\right )^2}+\frac {1}{6 \sqrt {3} \left (4+2 \sqrt {3}-2 \log (x)\right )}+\frac {1}{3 \left (-4+2 \sqrt {3}+2 \log (x)\right )^2}+\frac {1}{6 \sqrt {3} \left (-4+2 \sqrt {3}+2 \log (x)\right )}\right ) \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}+\frac {4}{3} \int \frac {1}{\left (4+2 \sqrt {3}-2 \log (x)\right )^2} \, dx+\frac {4}{3} \int \frac {1}{\left (-4+2 \sqrt {3}+2 \log (x)\right )^2} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\left (-4+2 \sqrt {3}+2 \log (x)\right )^2} \, dx-\frac {1}{3} \left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {3}-2 \log (x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-\frac {2}{3} \int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx+\frac {2}{3} \int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (-2-\sqrt {3}\right ) \int \frac {1}{4+2 \sqrt {3}-2 \log (x)} \, dx-\frac {1}{3} \left (2-\sqrt {3}\right ) \int \frac {1}{-4+2 \sqrt {3}+2 \log (x)} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-\frac {2}{3} \text {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )+\frac {2}{3} \text {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\frac {1}{3} \left (-2-\sqrt {3}\right ) \text {Subst}\left (\int \frac {e^x}{4+2 \sqrt {3}-2 x} \, dx,x,\log (x)\right )-\frac {1}{3} \left (2-\sqrt {3}\right ) \text {Subst}\left (\int \frac {e^x}{-4+2 \sqrt {3}+2 x} \, dx,x,\log (x)\right )-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ &=-3 x+\frac {1}{3} e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )+\frac {e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {1}{6} \left (2+\sqrt {3}\right ) e^{2+\sqrt {3}} \text {Ei}\left (-2-\sqrt {3}+\log (x)\right )+\frac {1}{3} e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )-\frac {e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )}{2 \sqrt {3}}-\frac {1}{6} \left (2-\sqrt {3}\right ) e^{2-\sqrt {3}} \text {Ei}\left (-2+\sqrt {3}+\log (x)\right )+\frac {x}{3 \left (2-\sqrt {3}-\log (x)\right )}-\frac {\left (2-\sqrt {3}\right ) x}{6 \left (2-\sqrt {3}-\log (x)\right )}+\frac {x}{3 \left (2+\sqrt {3}-\log (x)\right )}-\frac {\left (2+\sqrt {3}\right ) x}{6 \left (2+\sqrt {3}-\log (x)\right )}+\frac {e^5}{1-4 \log (x)+\log ^2(x)}-4 \int \frac {x \log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-e^5 \int \frac {1}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx+\left (2 e^5\right ) \int \frac {\log (x)}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (4 e^5\right ) \int \frac {1}{\left (1+e^x\right ) x \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (2 \left (3-2 e^5\right )\right ) \int \frac {\log (x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\left (5-e^5\right ) \int \frac {1}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\left (-1+e^5\right ) \int \frac {\log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx+\int \frac {x \log ^2(x)}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )^2} \, dx-\int \frac {x}{\left (1+e^x\right )^2 \left (1-4 \log (x)+\log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.18, size = 32, normalized size = 1.07 \begin {gather*} -3 x+\frac {e^x \left (e^5+x\right )}{\left (1+e^x\right ) \left (1-4 \log (x)+\log ^2(x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 30, normalized size = 1.00
method | result | size |
risch | \(-3 x +\frac {\left ({\mathrm e}^{5}+x \right ) {\mathrm e}^{x}}{\left ({\mathrm e}^{x}+1\right ) \left (\ln \left (x \right )^{2}-4 \ln \left (x \right )+1\right )}\) | \(30\) |
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. 66 vs.
\(2 (27) = 54\).
time = 0.41, size = 66, normalized size = 2.20 \begin {gather*} -\frac {3 \, x \log \left (x\right )^{2} + {\left (3 \, x \log \left (x\right )^{2} - 12 \, x \log \left (x\right ) + 2 \, x - e^{5}\right )} e^{x} - 12 \, x \log \left (x\right ) + 3 \, x}{{\left (\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} e^{x} + \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (27) = 54\).
time = 0.39, size = 62, normalized size = 2.07 \begin {gather*} -\frac {3 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + {\left (2 \, x - e^{5}\right )} e^{x} - 12 \, {\left (x e^{x} + x\right )} \log \left (x\right ) + 3 \, x}{{\left (e^{x} + 1\right )} \log \left (x\right )^{2} - 4 \, {\left (e^{x} + 1\right )} \log \left (x\right ) + e^{x} + 1} \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. 53 vs.
\(2 (24) = 48\).
time = 0.17, size = 53, normalized size = 1.77 \begin {gather*} - 3 x + \frac {- x - e^{5}}{\left (\log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 1\right ) e^{x} + \log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 1} + \frac {x + e^{5}}{\log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 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. 71 vs.
\(2 (27) = 54\).
time = 0.43, size = 71, normalized size = 2.37 \begin {gather*} -\frac {3 \, x e^{x} \log \left (x\right )^{2} - 12 \, x e^{x} \log \left (x\right ) + 3 \, x \log \left (x\right )^{2} + x e^{x} - 12 \, x \log \left (x\right ) + 3 \, x - 2 \, e^{\left (x + 5\right )}}{e^{x} \log \left (x\right )^{2} - 4 \, e^{x} \log \left (x\right ) + \log \left (x\right )^{2} + e^{x} - 4 \, \log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.92, size = 44, normalized size = 1.47 \begin {gather*} \frac {{\mathrm {e}}^{x+5}+{\mathrm {e}}^{2\,x+5}+x\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^x+1\right )}^2\,\left ({\ln \left (x\right )}^2-4\,\ln \left (x\right )+1\right )}-3\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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