Integrand size = 157, antiderivative size = 31 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=1+\frac {1}{x}+4 \left (4+\frac {4}{x-\left (-\frac {e^{-1+x}}{x}+x\right )^2}\right ) \]
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\[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-e^{4 x}+4 e^{1+3 x} x^2-4 e^{3+x} x^5 (1+7 x)+2 e^{2+2 x} x^3 (-15+13 x)-e^4 x^6 \left (17-34 x+x^2\right )}{x^2 \left (e^{2 x}-2 e^{1+x} x^2+e^2 (-1+x) x^3\right )^2} \, dx \\ & = \int \left (-\frac {1}{x^2}-\frac {16 e^3 x^3 \left (4 e^x+3 e x-2 e^x x-6 e x^2+2 e x^3\right )}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {32 e^2 (-1+x) x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}\right ) \, dx \\ & = \frac {1}{x}+\left (32 e^2\right ) \int \frac {(-1+x) x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx-\left (16 e^3\right ) \int \frac {x^3 \left (4 e^x+3 e x-2 e^x x-6 e x^2+2 e x^3\right )}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx \\ & = \frac {1}{x}+\left (32 e^2\right ) \int \left (-\frac {x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}+\frac {x^2}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4}\right ) \, dx-\left (16 e^3\right ) \int \left (\frac {4 e^x x^3}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {3 e x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}-\frac {2 e^x x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}-\frac {6 e x^5}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}+\frac {2 e x^6}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2}\right ) \, dx \\ & = \frac {1}{x}-\left (32 e^2\right ) \int \frac {x}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx+\left (32 e^2\right ) \int \frac {x^2}{e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4} \, dx+\left (32 e^3\right ) \int \frac {e^x x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (64 e^3\right ) \int \frac {e^x x^3}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (32 e^4\right ) \int \frac {x^6}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx-\left (48 e^4\right ) \int \frac {x^4}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx+\left (96 e^4\right ) \int \frac {x^5}{\left (e^{2 x}-2 e^{1+x} x^2-e^2 x^3+e^2 x^4\right )^2} \, dx \\ \end{align*}
Time = 8.79 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {1}{x}-\frac {16 e^2 x^2}{e^{2 x}-2 e^{1+x} x^2+e^2 (-1+x) x^3} \]
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Time = 0.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {1}{x}-\frac {16}{\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-2 \,{\mathrm e}^{-1+x}+x^{2}-x}\) | \(32\) |
parallelrisch | \(\frac {x^{2}-2 x \,{\mathrm e}^{x -\ln \left (x \right )-1}+\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-17 x}{x \left (\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-2 x \,{\mathrm e}^{x -\ln \left (x \right )-1}+x^{2}-x \right )}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{2} - 2 \, x e^{\left (x - \log \left (x\right ) - 1\right )} - 17 \, x + e^{\left (2 \, x - 2 \, \log \left (x\right ) - 2\right )}}{x^{3} - 2 \, x^{2} e^{\left (x - \log \left (x\right ) - 1\right )} - x^{2} + x e^{\left (2 \, x - 2 \, \log \left (x\right ) - 2\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=- \frac {16 x^{2}}{x^{4} - x^{3} - 2 x^{2} e^{x - 1} + e^{2 x - 2}} + \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{4} e^{2} - 17 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.33 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{4} e^{2} - 33 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \]
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Timed out. \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\int -\frac {{\mathrm {e}}^{4\,x-4\,\ln \left (x\right )-4}+{\mathrm {e}}^{2\,x-2\,\ln \left (x\right )-2}\,\left (30\,x-26\,x^2\right )+{\mathrm {e}}^{x-\ln \left (x\right )-1}\,\left (28\,x^3+4\,x^2\right )-4\,x\,{\mathrm {e}}^{3\,x-3\,\ln \left (x\right )-3}+17\,x^2-34\,x^3+x^4}{x^2\,{\mathrm {e}}^{4\,x-4\,\ln \left (x\right )-4}-4\,x^3\,{\mathrm {e}}^{3\,x-3\,\ln \left (x\right )-3}-{\mathrm {e}}^{2\,x-2\,\ln \left (x\right )-2}\,\left (2\,x^3-6\,x^4\right )+{\mathrm {e}}^{x-\ln \left (x\right )-1}\,\left (4\,x^4-4\,x^5\right )+x^4-2\,x^5+x^6} \,d x \]
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