Integrand size = 117, antiderivative size = 23 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=\left (e^x-x\right ) x \left (-x+\log (2)+\log \left (2+e^{7 x}\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(1004\) vs. \(2(23)=46\).
Time = 2.31 (sec) , antiderivative size = 1004, normalized size of antiderivative = 43.65, number of steps used = 100, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {6874, 2286, 2215, 2221, 2317, 2438, 2611, 2320, 6724, 14, 45, 2612, 2207, 2225, 2634, 12, 2280, 327, 207, 648, 632, 210, 642, 31} \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^3-e^x x^2-\log \left (2+e^{7 x}\right ) x^2-\log (2) x^2-\frac {(-1)^{5/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {x^2}{2^{6/7}}+\left (-\frac {1}{2}\right )^{6/7} x^2+2 e^x x-(-1)^{6/7} \sqrt [7]{2} \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right ) x-\sqrt [7]{2} \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right ) x+(-1)^{5/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{4/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right ) x+(-1)^{3/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{2/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right ) x+\sqrt [7]{-2} \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right ) x+e^x \log \left (2+e^{7 x}\right ) x+e^x (5+\log (2)) x-2 e^x+\frac {(-1)^{5/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{4/7} (1-x)^2}{2^{6/7}}+\frac {(-1)^{3/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{2/7} (1-x)^2}{2^{6/7}}+\frac {\sqrt [7]{-1} (1-x)^2}{2^{6/7}}-\frac {(1-x)^2}{2^{6/7}}-\left (-\frac {1}{2}\right )^{6/7} (1-x)^2+7 e^x (1-x)+\sqrt [7]{2} \log \left (\sqrt [7]{2}+e^x\right )-(-1)^{6/7} \sqrt [7]{2} (1-x) \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right )-\sqrt [7]{2} (1-x) \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right )+(-1)^{5/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{4/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right )+(-1)^{3/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{2/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right )+\sqrt [7]{-2} (1-x) \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right )-e^x \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\sqrt [7]{2} \cos \left (\frac {\pi }{7}\right ) \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \cos \left (\frac {\pi }{7}\right )\right )+\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}+2 \sqrt [7]{2} e^x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )+2 \sqrt [7]{2} \arctan \left (\frac {\left (e^x-\sqrt [7]{2} \cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{2}}\right ) \sin \left (\frac {\pi }{7}\right )-\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-e^x (5+\log (2))+2 \sqrt [7]{2} \arctan \left (\frac {\sec \left (\frac {3 \pi }{14}\right ) \left (e^x+\sqrt [7]{2} \sin \left (\frac {3 \pi }{14}\right )\right )}{\sqrt [7]{2}}\right ) \cos \left (\frac {3 \pi }{14}\right )+2 \sqrt [7]{2} \arctan \left (\frac {\sec \left (\frac {\pi }{14}\right ) \left (e^x-\sqrt [7]{2} \sin \left (\frac {\pi }{14}\right )\right )}{\sqrt [7]{2}}\right ) \cos \left (\frac {\pi }{14}\right ) \]
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Rule 12
Rule 14
Rule 31
Rule 45
Rule 207
Rule 210
Rule 327
Rule 632
Rule 642
Rule 648
Rule 2207
Rule 2215
Rule 2221
Rule 2225
Rule 2280
Rule 2286
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 2612
Rule 2634
Rule 6724
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {14 \left (e^x-x\right ) x}{2+e^{7 x}}-2 x \left (2 x+\log (2)+\log \left (2+e^{7 x}\right )\right )+e^x \left (-x^2+5 x \left (1+\frac {\log (2)}{5}\right )+x \log \left (2+e^{7 x}\right )+\log \left (2 \left (2+e^{7 x}\right )\right )\right )\right ) \, dx \\ & = -\left (2 \int x \left (2 x+\log (2)+\log \left (2+e^{7 x}\right )\right ) \, dx\right )-14 \int \frac {\left (e^x-x\right ) x}{2+e^{7 x}} \, dx+\int e^x \left (-x^2+5 x \left (1+\frac {\log (2)}{5}\right )+x \log \left (2+e^{7 x}\right )+\log \left (2 \left (2+e^{7 x}\right )\right )\right ) \, dx \\ & = -\left (2 \int \left (x (2 x+\log (2))+x \log \left (2+e^{7 x}\right )\right ) \, dx\right )-14 \int \left (\frac {e^x x}{2+e^{7 x}}-\frac {x^2}{2+e^{7 x}}\right ) \, dx+\int \left (-e^x x^2+e^x x (5+\log (2))+e^x x \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )\right ) \, dx \\ & = -(2 \int x (2 x+\log (2)) \, dx)-2 \int x \log \left (2+e^{7 x}\right ) \, dx-14 \int \frac {e^x x}{2+e^{7 x}} \, dx+14 \int \frac {x^2}{2+e^{7 x}} \, dx+(5+\log (2)) \int e^x x \, dx-\int e^x x^2 \, dx+\int e^x x \log \left (2+e^{7 x}\right ) \, dx+\int e^x \log \left (2 \left (2+e^{7 x}\right )\right ) \, dx \\ & = -e^x x^2+\frac {7 x^3}{3}+e^x x (5+\log (2))+x^2 \log \left (1+\frac {e^{7 x}}{2}\right )-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )+2 \int e^x x \, dx-2 \int \left (2 x^2+x \log (2)\right ) \, dx-2 \int x \log \left (1+\frac {e^{7 x}}{2}\right ) \, dx-7 \int \frac {e^{7 x} x^2}{2+e^{7 x}} \, dx-14 \int \left (-\frac {x}{7\ 2^{5/7} \left (\sqrt [7]{2}+e^x\right )}-\frac {(-1)^{6/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-\sqrt [7]{-1} e^x\right )}+\frac {\left (-\frac {1}{2}\right )^{5/7} x}{7 \left (\sqrt [7]{2}+(-1)^{2/7} e^x\right )}-\frac {(-1)^{4/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-(-1)^{3/7} e^x\right )}+\frac {(-1)^{3/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}+(-1)^{4/7} e^x\right )}-\frac {(-1)^{2/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-(-1)^{5/7} e^x\right )}+\frac {\sqrt [7]{-1} x}{7\ 2^{5/7} \left (\sqrt [7]{2}+(-1)^{6/7} e^x\right )}\right ) \, dx+(-5-\log (2)) \int e^x \, dx-\int \frac {7 e^{8 x}}{2+e^{7 x}} \, dx-\int \frac {7 e^{8 x} (-1+x)}{2+e^{7 x}} \, dx \\ & = 2 e^x x-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )+\frac {2}{7} x \operatorname {PolyLog}\left (2,-\frac {e^{7 x}}{2}\right )-\frac {2}{7} \int \operatorname {PolyLog}\left (2,-\frac {e^{7 x}}{2}\right ) \, dx-2 \int e^x \, dx+2 \int x \log \left (1+\frac {e^{7 x}}{2}\right ) \, dx-7 \int \frac {e^{8 x}}{2+e^{7 x}} \, dx-7 \int \frac {e^{8 x} (-1+x)}{2+e^{7 x}} \, dx+(-2)^{2/7} \int \frac {x}{\sqrt [7]{2}-(-1)^{5/7} e^x} \, dx+2^{2/7} \int \frac {x}{\sqrt [7]{2}+e^x} \, dx-\left (\sqrt [7]{-1} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{6/7} e^x} \, dx-\left ((-1)^{3/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{4/7} e^x} \, dx+\left ((-1)^{4/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}-(-1)^{3/7} e^x} \, dx-\left ((-1)^{5/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{2/7} e^x} \, dx+\left ((-1)^{6/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}-\sqrt [7]{-1} e^x} \, dx \\ & = -2 e^x+2 e^x x+\left (-\frac {1}{2}\right )^{6/7} x^2+\frac {x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{5/7} x^2}{2^{6/7}}-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\frac {2}{49} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {x}{2}\right )}{x} \, dx,x,e^{7 x}\right )+\frac {2}{7} \int \operatorname {PolyLog}\left (2,-\frac {e^{7 x}}{2}\right ) \, dx-7 \int \left (e^x (-1+x)-\frac {2 e^x (-1+x)}{2+e^{7 x}}\right ) \, dx-7 \text {Subst}\left (\int \frac {x^7}{2+x^7} \, dx,x,e^x\right )-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-\sqrt [7]{-1} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{2/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-(-1)^{3/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{4/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-(-1)^{5/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{6/7} e^x} \, dx \\ & = -9 e^x+2 e^x x+\left (-\frac {1}{2}\right )^{6/7} x^2+\frac {x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{5/7} x^2}{2^{6/7}}-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-(-1)^{6/7} \sqrt [7]{2} x \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right )-\sqrt [7]{2} x \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right )+(-1)^{5/7} \sqrt [7]{2} x \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{4/7} \sqrt [7]{2} x \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right )+(-1)^{3/7} \sqrt [7]{2} x \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{2/7} \sqrt [7]{2} x \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right )+\sqrt [7]{-2} x \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right )-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\frac {2}{49} \operatorname {PolyLog}\left (3,-\frac {e^{7 x}}{2}\right )+\frac {2}{49} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {x}{2}\right )}{x} \, dx,x,e^{7 x}\right )-7 \int e^x (-1+x) \, dx+14 \int \frac {e^x (-1+x)}{2+e^{7 x}} \, dx+14 \text {Subst}\left (\int \frac {1}{2+x^7} \, dx,x,e^x\right )-\sqrt [7]{-2} \int \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\sqrt [7]{2} \int \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{2/7} \sqrt [7]{2}\right ) \int \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right ) \, dx-\left ((-1)^{3/7} \sqrt [7]{2}\right ) \int \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{4/7} \sqrt [7]{2}\right ) \int \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right ) \, dx-\left ((-1)^{5/7} \sqrt [7]{2}\right ) \int \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{6/7} \sqrt [7]{2}\right ) \int \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=-\left (\left (e^x-x\right ) x \left (x-\log \left (4+2 e^{7 x}\right )\right )\right ) \]
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Time = 0.79 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87
method | result | size |
risch | \(\left ({\mathrm e}^{x} x -x^{2}\right ) \ln \left ({\mathrm e}^{7 x}+2\right )-x^{2} \ln \left (2\right )+x \ln \left (2\right ) {\mathrm e}^{x}+x^{3}-{\mathrm e}^{x} x^{2}\) | \(43\) |
parallelrisch | \(x \ln \left (2\right ) {\mathrm e}^{x}+{\mathrm e}^{x} \ln \left ({\mathrm e}^{7 x}+2\right ) x -{\mathrm e}^{x} x^{2}-x^{2} \ln \left (2\right )-x^{2} \ln \left ({\mathrm e}^{7 x}+2\right )+x^{3}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} \log \left (2\right ) - {\left (x^{2} - x \log \left (2\right )\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \]
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Timed out. \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} \log \left (2\right ) - {\left (x^{2} - x \log \left (2\right )\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx=x^{3} - x^{2} e^{x} - x^{2} \log \left (2\right ) + x e^{x} \log \left (2\right ) - x^{2} \log \left (e^{\left (7 \, x\right )} + 2\right ) + x e^{x} \log \left (e^{\left (7 \, x\right )} + 2\right ) \]
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Time = 13.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {6 x^2-4 x \log (2)+e^x \left (-4 x-2 x^2+(2+2 x) \log (2)\right )+e^{7 x} \left (-4 x^2-2 x \log (2)+e^x \left (5 x-x^2+(1+x) \log (2)\right )\right )+\left (-4 x+e^x (2+2 x)+e^{7 x} \left (-2 x+e^x (1+x)\right )\right ) \log \left (2+e^{7 x}\right )}{2+e^{7 x}} \, dx={\mathrm {e}}^x\,\left (x\,\ln \left (2\right )-x^2\right )-\frac {x^2\,\ln \left (4\right )}{2}+\ln \left ({\mathrm {e}}^{7\,x}+2\right )\,\left (x\,{\mathrm {e}}^x-x^2\right )+x^3 \]
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