Integrand size = 116, antiderivative size = 30 \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=e^{\frac {2 \left (x+\frac {\log \left (\frac {5}{\frac {40}{3}+e^x}\right )}{x (4+x)}\right )}{x}} \]
[Out]
\[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=\int \frac {\exp \left (\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}\right ) \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {2\ 15^{\frac {2}{x^2 (4+x)}} e^2 \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} \left (-3 e^x x (4+x)-\left (40+3 e^x\right ) (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )\right )}{x^3 (4+x)^2} \, dx \\ & = \left (2 e^2\right ) \int \frac {15^{\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} \left (-3 e^x x (4+x)-\left (40+3 e^x\right ) (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )\right )}{x^3 (4+x)^2} \, dx \\ & = \left (2 e^2\right ) \int \left (-\frac {8\ 3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )}{x^3 (4+x)^2}-\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} \left (4 x+x^2+8 \log \left (\frac {15}{40+3 e^x}\right )+3 x \log \left (\frac {15}{40+3 e^x}\right )\right )}{x^3 (4+x)^2}\right ) \, dx \\ & = -\left (\left (2 e^2\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} \left (4 x+x^2+8 \log \left (\frac {15}{40+3 e^x}\right )+3 x \log \left (\frac {15}{40+3 e^x}\right )\right )}{x^3 (4+x)^2} \, dx\right )-\left (16 e^2\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )}{x^3 (4+x)^2} \, dx \\ & = -\left (\left (2 e^2\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} \left (x (4+x)+(8+3 x) \log \left (\frac {15}{40+3 e^x}\right )\right )}{x^3 (4+x)^2} \, dx\right )+\left (16 e^2\right ) \int \frac {3 e^x \left (-8 \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx\right )}{16 \left (40+3 e^x\right )} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = -\left (\left (2 e^2\right ) \int \left (\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2 (4+x)}+\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )}{x^3 (4+x)^2}\right ) \, dx\right )+\left (3 e^2\right ) \int \frac {e^x \left (-8 \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx\right )}{40+3 e^x} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = -\left (\left (2 e^2\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2 (4+x)} \, dx\right )-\left (2 e^2\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}} (8+3 x) \log \left (\frac {15}{40+3 e^x}\right )}{x^3 (4+x)^2} \, dx+\left (3 e^2\right ) \int \left (-\frac {8 e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x}+\frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x}-\frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x}\right ) \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = -\left (\left (2 e^2\right ) \int \left (\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{4 x^2}-\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{16 x}+\frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{16 (4+x)}\right ) \, dx\right )+\left (2 e^2\right ) \int \frac {3 e^x \left (-8 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx\right )}{16 \left (40+3 e^x\right )} \, dx+\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x} \, dx-\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x} \, dx-\left (24 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x} \, dx+\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = \frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x} \, dx-\frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{4+x} \, dx+\frac {1}{8} \left (3 e^2\right ) \int \frac {e^x \left (-8 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx\right )}{40+3 e^x} \, dx-\frac {1}{2} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx+\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x} \, dx-\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x} \, dx-\left (24 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x} \, dx+\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = \frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x} \, dx-\frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{4+x} \, dx+\frac {1}{8} \left (3 e^2\right ) \int \left (-\frac {8 e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x}+\frac {e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x}-\frac {e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x}\right ) \, dx-\frac {1}{2} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx+\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x} \, dx-\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x} \, dx-\left (24 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x} \, dx+\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ & = \frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x} \, dx-\frac {1}{8} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{4+x} \, dx+\frac {1}{8} \left (3 e^2\right ) \int \frac {e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x} \, dx-\frac {1}{8} \left (3 e^2\right ) \int \frac {e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x} \, dx-\frac {1}{2} e^2 \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (3 e^2\right ) \int \frac {e^x \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x} \, dx+\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx}{40+3 e^x} \, dx-\left (3 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx}{40+3 e^x} \, dx-\left (24 e^2\right ) \int \frac {e^x \int \frac {5^{1+\frac {2}{x^2 (4+x)}} 9^{\frac {1}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx}{40+3 e^x} \, dx+\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\frac {1}{8} \left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{1+\frac {2}{x^2 (4+x)}} 5^{\frac {2}{x^2 (4+x)}} e^x \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx+\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^2} \, dx-\left (e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{(4+x)^2} \, dx-\left (8 e^2 \log \left (\frac {15}{40+3 e^x}\right )\right ) \int \frac {3^{\frac {2}{x^2 (4+x)}} 5^{1+\frac {2}{x^2 (4+x)}} \left (\frac {1}{40+3 e^x}\right )^{1+\frac {2}{x^2 (4+x)}}}{x^3} \, dx \\ \end{align*}
\[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=\int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx \]
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Time = 56.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\left ({\mathrm e}^{x}+\frac {40}{3}\right )^{-\frac {2}{x^{2} \left (4+x \right )}} 5^{\frac {2}{x^{2} \left (4+x \right )}} {\mathrm e}^{2}\) | \(34\) |
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=e^{\left (\frac {2 \, {\left (x^{3} + 4 \, x^{2} + \log \left (\frac {15}{3 \, e^{x} + 40}\right )\right )}}{x^{3} + 4 \, x^{2}}\right )} \]
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Timed out. \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (28) = 56\).
Time = 0.56 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=e^{\left (\frac {\log \left (5\right )}{8 \, {\left (x + 4\right )}} - \frac {\log \left (5\right )}{8 \, x} + \frac {\log \left (3\right )}{8 \, {\left (x + 4\right )}} - \frac {\log \left (3\right )}{8 \, x} - \frac {\log \left (3 \, e^{x} + 40\right )}{8 \, {\left (x + 4\right )}} + \frac {\log \left (3 \, e^{x} + 40\right )}{8 \, x} + \frac {\log \left (5\right )}{2 \, x^{2}} + \frac {\log \left (3\right )}{2 \, x^{2}} - \frac {\log \left (3 \, e^{x} + 40\right )}{2 \, x^{2}} + 2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (28) = 56\).
Time = 0.41 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx=e^{\left (\frac {2 \, x^{3}}{x^{3} + 4 \, x^{2}} + \frac {8 \, x^{2}}{x^{3} + 4 \, x^{2}} + \frac {2 \, \log \left (\frac {15}{3 \, e^{x} + 40}\right )}{x^{3} + 4 \, x^{2}}\right )} \]
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Time = 12.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \frac {e^{\frac {2 \left (4 x^2+x^3+\log \left (\frac {15}{40+3 e^x}\right )\right )}{4 x^2+x^3}} \left (e^x \left (-24 x-6 x^2\right )+\left (-640+e^x (-48-18 x)-240 x\right ) \log \left (\frac {15}{40+3 e^x}\right )\right )}{640 x^3+320 x^4+40 x^5+e^x \left (48 x^3+24 x^4+3 x^5\right )} \, dx={\mathrm {e}}^{\frac {2\,x^3}{x^3+4\,x^2}}\,{\mathrm {e}}^{\frac {8\,x^2}{x^3+4\,x^2}}\,{\left (\frac {225}{{\left (3\,{\mathrm {e}}^x+40\right )}^2}\right )}^{\frac {1}{x^3+4\,x^2}} \]
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