Integrand size = 100, antiderivative size = 32 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=-e^{e^x+x}+\frac {(-2+x)^4 x^4}{25 \log ^2\left (5 e^{-3+x}\right )} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(103\) vs. \(2(32)=64\).
Time = 1.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.22, number of steps used = 85, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.110, Rules used = {12, 6820, 2320, 2207, 2225, 6874, 2199, 2190, 2189, 2188, 29} \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\frac {x^8}{25 \log ^2\left (5 e^{x-3}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{x-3}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{x-3}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{x-3}\right )}+\frac {16 x^4}{25 \log ^2\left (5 e^{x-3}\right )}+e^{e^x}-e^{e^x} \left (e^x+1\right ) \]
[In]
[Out]
Rule 12
Rule 29
Rule 2188
Rule 2189
Rule 2190
Rule 2199
Rule 2207
Rule 2225
Rule 2320
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{25} \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{\log ^3\left (5 e^{-3+x}\right )} \, dx \\ & = \frac {1}{25} \int \left (-25 e^{e^x+x} \left (1+e^x\right )-\frac {2 (-2+x)^4 x^4}{\log ^3\left (5 e^{-3+x}\right )}+\frac {8 (-2+x)^3 (-1+x) x^3}{\log ^2\left (5 e^{-3+x}\right )}\right ) \, dx \\ & = -\left (\frac {2}{25} \int \frac {(-2+x)^4 x^4}{\log ^3\left (5 e^{-3+x}\right )} \, dx\right )+\frac {8}{25} \int \frac {(-2+x)^3 (-1+x) x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\int e^{e^x+x} \left (1+e^x\right ) \, dx \\ & = -\left (\frac {2}{25} \int \left (\frac {16 x^4}{\log ^3\left (5 e^{-3+x}\right )}-\frac {32 x^5}{\log ^3\left (5 e^{-3+x}\right )}+\frac {24 x^6}{\log ^3\left (5 e^{-3+x}\right )}-\frac {8 x^7}{\log ^3\left (5 e^{-3+x}\right )}+\frac {x^8}{\log ^3\left (5 e^{-3+x}\right )}\right ) \, dx\right )+\frac {8}{25} \int \left (\frac {8 x^3}{\log ^2\left (5 e^{-3+x}\right )}-\frac {20 x^4}{\log ^2\left (5 e^{-3+x}\right )}+\frac {18 x^5}{\log ^2\left (5 e^{-3+x}\right )}-\frac {7 x^6}{\log ^2\left (5 e^{-3+x}\right )}+\frac {x^7}{\log ^2\left (5 e^{-3+x}\right )}\right ) \, dx-\text {Subst}\left (\int e^x (1+x) \, dx,x,e^x\right ) \\ & = -e^{e^x} \left (1+e^x\right )-\frac {2}{25} \int \frac {x^8}{\log ^3\left (5 e^{-3+x}\right )} \, dx+\frac {8}{25} \int \frac {x^7}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {16}{25} \int \frac {x^7}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {32}{25} \int \frac {x^4}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {48}{25} \int \frac {x^6}{\log ^3\left (5 e^{-3+x}\right )} \, dx-\frac {56}{25} \int \frac {x^6}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {64}{25} \int \frac {x^5}{\log ^3\left (5 e^{-3+x}\right )} \, dx+\frac {64}{25} \int \frac {x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {144}{25} \int \frac {x^5}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\frac {32}{5} \int \frac {x^4}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\text {Subst}\left (\int e^x \, dx,x,e^x\right ) \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {64 x^3}{25 \log \left (5 e^{-3+x}\right )}+\frac {32 x^4}{5 \log \left (5 e^{-3+x}\right )}-\frac {144 x^5}{25 \log \left (5 e^{-3+x}\right )}+\frac {56 x^6}{25 \log \left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log \left (5 e^{-3+x}\right )}-\frac {8}{25} \int \frac {x^7}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {56}{25} \int \frac {x^6}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {56}{25} \int \frac {x^6}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {64}{25} \int \frac {x^3}{\log ^2\left (5 e^{-3+x}\right )} \, dx-\frac {144}{25} \int \frac {x^5}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {32}{5} \int \frac {x^4}{\log ^2\left (5 e^{-3+x}\right )} \, dx+\frac {192}{25} \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {336}{25} \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {128}{5} \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {144}{5} \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {96 x^2}{25}-\frac {128 x^3}{15}+\frac {36 x^4}{5}-\frac {336 x^5}{125}+\frac {28 x^6}{75}+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {56}{25} \int \frac {x^6}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {192}{25} \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {336}{25} \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {128}{5} \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {144}{5} \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {192}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )-\frac {64}{5} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {48}{5} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )-\frac {84}{25} x^4 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {56}{125} x^5 \left (x-\log \left (5 e^{-3+x}\right )\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^5}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )-\frac {128}{5} x \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {72}{5} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^2-\frac {112}{25} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {14}{25} x^4 \left (x-\log \left (5 e^{-3+x}\right )\right )^2+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^4}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {144}{5} x \left (x-\log \left (5 e^{-3+x}\right )\right )^3-\frac {168}{25} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^3+\frac {56}{75} x^3 \left (x-\log \left (5 e^{-3+x}\right )\right )^3+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {192}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^2 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (192 \left (-x+\log \left (5 e^{-3+x}\right )\right )^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^3}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )-\frac {336}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )^4+\frac {28}{25} x^2 \left (x-\log \left (5 e^{-3+x}\right )\right )^4+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {128}{5} \left (x-\log \left (5 e^{-3+x}\right )\right )^3 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{5} \left (128 \left (-x+\log \left (5 e^{-3+x}\right )\right )^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x^2}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {56}{25} x \left (x-\log \left (5 e^{-3+x}\right )\right )^5+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {144}{5} \left (x-\log \left (5 e^{-3+x}\right )\right )^4 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{5} \left (144 \left (-x+\log \left (5 e^{-3+x}\right )\right )^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {x}{\log \left (5 e^{-3+x}\right )} \, dx-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {336}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^5 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (336 \left (-x+\log \left (5 e^{-3+x}\right )\right )^5\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \int \frac {1}{\log \left (5 e^{-3+x}\right )} \, dx+\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right ) \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {56}{25} \left (x-\log \left (5 e^{-3+x}\right )\right )^6 \log \left (\log \left (5 e^{-3+x}\right )\right )-\frac {1}{25} \left (56 \left (-x+\log \left (5 e^{-3+x}\right )\right )^6\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (5 e^{-3+x}\right )\right ) \\ & = e^{e^x}-e^{e^x} \left (1+e^x\right )+\frac {16 x^4}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {32 x^5}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {24 x^6}{25 \log ^2\left (5 e^{-3+x}\right )}-\frac {8 x^7}{25 \log ^2\left (5 e^{-3+x}\right )}+\frac {x^8}{25 \log ^2\left (5 e^{-3+x}\right )} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(32)=64\).
Time = 0.60 (sec) , antiderivative size = 198, normalized size of antiderivative = 6.19 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\frac {(-2+x)^4 x^4-\left (25 e^{e^x+x}+(-2+x)^3 x^2 (-6+7 x)\right ) \log ^2\left (5 e^{-3+x}\right )+6 (-2+x)^2 x \left (4-12 x+7 x^2\right ) \log ^3\left (5 e^{-3+x}\right )-3 \left (16-128 x+240 x^2-160 x^3+35 x^4\right ) \log ^4\left (5 e^{-3+x}\right )+4 \left (-32+120 x-120 x^2+35 x^3\right ) \log ^5\left (5 e^{-3+x}\right )-15 \left (8-16 x+7 x^2\right ) \log ^6\left (5 e^{-3+x}\right )+6 (-8+7 x) \log ^7\left (5 e^{-3+x}\right )-7 \log ^8\left (5 e^{-3+x}\right )}{25 \log ^2\left (5 e^{-3+x}\right )} \]
[In]
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Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59
method | result | size |
risch | \(-{\mathrm e}^{{\mathrm e}^{x}+x}-\frac {4 \left (x^{8}-8 x^{7}+24 x^{6}-32 x^{5}+16 x^{4}\right )}{25 {\left (-2 i \ln \left (5\right )-2 i \ln \left ({\mathrm e}^{x}\right )+6 i\right )}^{2}}\) | \(51\) |
parallelrisch | \(\frac {-3360 x^{7}+420 x^{8}+6720 x^{4}+10080 x^{6}-13440 x^{5}-10500 \ln \left (5 \,{\mathrm e}^{-3+x}\right )^{2} {\mathrm e}^{{\mathrm e}^{x}+x}}{10500 \ln \left (5 \,{\mathrm e}^{-3+x}\right )^{2}}\) | \(54\) |
default | \(\text {Expression too large to display}\) | \(889\) |
parts | \(\text {Expression too large to display}\) | \(889\) |
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (27) = 54\).
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 6.03 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\frac {x^{8} - 2 \, {\left (7 \, x - 60\right )} \log \left (5\right )^{7} - 7 \, \log \left (5\right )^{8} - 8 \, x^{7} - {\left (7 \, x^{2} - 198 \, x + 876\right )} \log \left (5\right )^{6} + 24 \, x^{6} + 2 \, {\left (39 \, x^{2} - 579 \, x + 1772\right )} \log \left (5\right )^{5} - 32 \, x^{5} - {\left (345 \, x^{2} - 3614 \, x + 8658\right )} \log \left (5\right )^{4} + 16 \, x^{4} + 2 \, {\left (386 \, x^{2} - 3237 \, x + 6516\right )} \log \left (5\right )^{3} - 3 \, {\left (307 \, x^{2} - 2214 \, x + 3924\right )} \log \left (5\right )^{2} - 135 \, x^{2} - 25 \, {\left (x^{2} + 2 \, {\left (x - 3\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 6 \, x + 9\right )} e^{\left (x + e^{x}\right )} + 18 \, {\left (31 \, x^{2} - 201 \, x + 324\right )} \log \left (5\right ) + 810 \, x - 1215}{25 \, {\left (x^{2} + 2 \, {\left (x - 3\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 6 \, x + 9\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (26) = 52\).
Time = 1.40 (sec) , antiderivative size = 291, normalized size of antiderivative = 9.09 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\frac {x^{6}}{25} + x^{5} \left (- \frac {2 \log {\left (5 \right )}}{25} - \frac {2}{25}\right ) + x^{4} \left (- \frac {2 \log {\left (5 \right )}}{25} + \frac {3}{25} + \frac {3 \log {\left (5 \right )}^{2}}{25}\right ) + x^{3} \left (- \frac {12 \log {\left (5 \right )}}{25} - \frac {4 \log {\left (5 \right )}^{3}}{25} + \frac {4}{25} + \frac {12 \log {\left (5 \right )}^{2}}{25}\right ) + x^{2} \left (- \frac {28 \log {\left (5 \right )}^{3}}{25} - \frac {44 \log {\left (5 \right )}}{25} + \frac {13}{25} + \frac {\log {\left (5 \right )}^{4}}{5} + \frac {54 \log {\left (5 \right )}^{2}}{25}\right ) + x \left (- \frac {156 \log {\left (5 \right )}^{3}}{25} - \frac {158 \log {\left (5 \right )}}{25} - \frac {6 \log {\left (5 \right )}^{5}}{25} + \frac {42}{25} + 2 \log {\left (5 \right )}^{4} + \frac {228 \log {\left (5 \right )}^{2}}{25}\right ) - e^{x + e^{x}} + \frac {x \left (- 3544 \log {\left (5 \right )}^{3} - 648 \log {\left (5 \right )}^{5} - 1944 \log {\left (5 \right )} - 8 \log {\left (5 \right )}^{7} + 432 + 112 \log {\left (5 \right )}^{6} + 3600 \log {\left (5 \right )}^{2} + 2000 \log {\left (5 \right )}^{4}\right ) - 8658 \log {\left (5 \right )}^{4} - 11772 \log {\left (5 \right )}^{2} - 876 \log {\left (5 \right )}^{6} - 1215 - 7 \log {\left (5 \right )}^{8} + 120 \log {\left (5 \right )}^{7} + 5832 \log {\left (5 \right )} + 3544 \log {\left (5 \right )}^{5} + 13032 \log {\left (5 \right )}^{3}}{25 x^{2} + x \left (-150 + 50 \log {\left (5 \right )}\right ) - 150 \log {\left (5 \right )} + 25 \log {\left (5 \right )}^{2} + 225} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (27) = 54\).
Time = 0.33 (sec) , antiderivative size = 1477, normalized size of antiderivative = 46.16 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 415, normalized size of antiderivative = 12.97 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\frac {x^{8} e^{\left (2 \, x\right )} - 7 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{6} - 14 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{7} - 7 \, e^{\left (2 \, x\right )} \log \left (5\right )^{8} - 8 \, x^{7} e^{\left (2 \, x\right )} + 78 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{5} + 198 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{6} + 120 \, e^{\left (2 \, x\right )} \log \left (5\right )^{7} + 24 \, x^{6} e^{\left (2 \, x\right )} - 345 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{4} - 1158 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{5} - 876 \, e^{\left (2 \, x\right )} \log \left (5\right )^{6} - 32 \, x^{5} e^{\left (2 \, x\right )} + 772 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{3} + 3614 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{4} + 3544 \, e^{\left (2 \, x\right )} \log \left (5\right )^{5} + 16 \, x^{4} e^{\left (2 \, x\right )} - 921 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right )^{2} - 6474 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{3} - 8658 \, e^{\left (2 \, x\right )} \log \left (5\right )^{4} + 558 \, x^{2} e^{\left (2 \, x\right )} \log \left (5\right ) + 6642 \, x e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 13032 \, e^{\left (2 \, x\right )} \log \left (5\right )^{3} - 135 \, x^{2} e^{\left (2 \, x\right )} - 25 \, x^{2} e^{\left (3 \, x + e^{x}\right )} - 3618 \, x e^{\left (2 \, x\right )} \log \left (5\right ) - 50 \, x e^{\left (3 \, x + e^{x}\right )} \log \left (5\right ) - 11772 \, e^{\left (2 \, x\right )} \log \left (5\right )^{2} - 25 \, e^{\left (3 \, x + e^{x}\right )} \log \left (5\right )^{2} + 810 \, x e^{\left (2 \, x\right )} + 150 \, x e^{\left (3 \, x + e^{x}\right )} + 5832 \, e^{\left (2 \, x\right )} \log \left (5\right ) + 150 \, e^{\left (3 \, x + e^{x}\right )} \log \left (5\right ) - 1215 \, e^{\left (2 \, x\right )} - 225 \, e^{\left (3 \, x + e^{x}\right )}}{25 \, {\left (x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} \log \left (5\right ) + e^{\left (2 \, x\right )} \log \left (5\right )^{2} - 6 \, x e^{\left (2 \, x\right )} - 6 \, e^{\left (2 \, x\right )} \log \left (5\right ) + 9 \, e^{\left (2 \, x\right )}\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 586, normalized size of antiderivative = 18.31 \[ \int \frac {-32 x^4+64 x^5-48 x^6+16 x^7-2 x^8+\left (64 x^3-160 x^4+144 x^5-56 x^6+8 x^7\right ) \log \left (5 e^{-3+x}\right )+e^{e^x+x} \left (-25-25 e^x\right ) \log ^3\left (5 e^{-3+x}\right )}{25 \log ^3\left (5 e^{-3+x}\right )} \, dx=\text {Too large to display} \]
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