Integrand size = 153, antiderivative size = 36 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-2+\log \left (\frac {4-e^{3+x}}{x+\frac {x}{3-e^{e^{\frac {x^2}{2}}}}}\right ) \]
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\[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx \]
[In]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-48-e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )-e^{3+x} (-12+12 x)-e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{\left (12-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}\right ) \left (4-e^{3+x}\right ) x} \, dx \\ & = \int \left (\frac {48}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {12 e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}-\frac {7 e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{\frac {1}{2} \left (2 e^{\frac {x^2}{2}}+x^2\right )} x}{-12+7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}}+\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx \\ & = -\left (7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\right )+12 \int \frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{\frac {1}{2} \left (2 e^{\frac {x^2}{2}}+x^2\right )} x}{-12+7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {e^{\frac {1}{2} \left (2 e^{x/2}+x\right )}}{-12+7 e^{e^{x/2}}-e^{2 e^{x/2}}} \, dx,x,x^2\right )-7 \int \left (\frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (\frac {e^{3+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx-28 \int \left (\frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+48 \int \left (\frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+\int \left (\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx \\ & = -\left (7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\right )-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\text {Subst}\left (\int \frac {e^{e^x+x}}{-12+7 e^{e^x}-e^{2 e^x}} \, dx,x,\frac {x^2}{2}\right ) \\ & = -\left (7 \int \left (\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx\right )-7 \int \left (-\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (\frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (-\frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx+\int \left (-\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx+\text {Subst}\left (\int \frac {e^x}{-12+7 e^x-e^{2 x}} \, dx,x,e^{\frac {x^2}{2}}\right ) \\ & = 4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx+\text {Subst}\left (\int \frac {1}{-12+7 x-x^2} \, dx,x,e^{e^{\frac {x^2}{2}}}\right ) \\ & = 4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (\frac {e^{2 e^{\frac {x^2}{2}}}}{-4+e^{e^{\frac {x^2}{2}}}}-\frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx-\int \left (\frac {e^{2 e^{\frac {x^2}{2}}}}{-3+e^{e^{\frac {x^2}{2}}}}-\frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx-\text {Subst}\left (\int \frac {1}{3-x} \, dx,x,e^{e^{\frac {x^2}{2}}}\right )+\text {Subst}\left (\int \frac {1}{4-x} \, dx,x,e^{e^{\frac {x^2}{2}}}\right ) \\ & = \log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}}}{-4+e^{e^{\frac {x^2}{2}}}} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}}}{-3+e^{e^{\frac {x^2}{2}}}} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx \\ & = \log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-4+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-3+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (4+e^{e^{\frac {x^2}{2}}}+\frac {16}{-4+e^{e^{\frac {x^2}{2}}}}\right ) \, dx-\int \left (3+e^{e^{\frac {x^2}{2}}}+\frac {9}{-3+e^{e^{\frac {x^2}{2}}}}\right ) \, dx \\ & = x+\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-4+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-3+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-9 \int \frac {1}{-3+e^{e^{\frac {x^2}{2}}}} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+16 \int \frac {1}{-4+e^{e^{\frac {x^2}{2}}}} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )+\log \left (4-e^{3+x}\right )-\log (x) \]
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Time = 3.55 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(-\ln \left (x \right )+\ln \left ({\mathrm e}^{3+x}-4\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-4\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-3\right )\) | \(35\) |
risch | \(-\ln \left (x \right )-3+\ln \left ({\mathrm e}^{3+x}-4\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-3\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-4\right )\) | \(36\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-\log \left (x\right ) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \]
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Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=- \log {\left (x \right )} + \log {\left (e^{x + 3} - 4 \right )} - \log {\left (e^{e^{\frac {x^{2}}{2}}} - 4 \right )} + \log {\left (e^{e^{\frac {x^{2}}{2}}} - 3 \right )} \]
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\log \left ({\left (e^{\left (x + 3\right )} - 4\right )} e^{\left (-3\right )}\right ) - \log \left (x\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \]
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Time = 1.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=-\log \left (x\right ) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \]
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Time = 9.62 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.92 \[ \int \frac {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx=\ln \left ({\mathrm {e}}^3\,{\mathrm {e}}^x-4\right )-\ln \left (24\,x\,\sqrt {{\mathrm {e}}^{x^2}}-6\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}\right )+\ln \left (8\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}-24\,x\,\sqrt {{\mathrm {e}}^{x^2}}\right )-\ln \left (x\right ) \]
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