Integrand size = 47, antiderivative size = 22 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=-2+\frac {1}{x}+e^{2+e^x x^4} \left (-2+x^2\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(22)=44\).
Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {14, 2326} \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=\frac {1}{x}-\frac {e^{e^x x^4+2} x \left (-e^x x^5-4 e^x x^4+2 e^x x^3+8 e^x x^2\right )}{e^x x^4+4 e^x x^3} \]
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Rule 14
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^2}+e^{2+e^x x^4} x \left (2-8 e^x x^2-2 e^x x^3+4 e^x x^4+e^x x^5\right )\right ) \, dx \\ & = \frac {1}{x}+\int e^{2+e^x x^4} x \left (2-8 e^x x^2-2 e^x x^3+4 e^x x^4+e^x x^5\right ) \, dx \\ & = \frac {1}{x}-\frac {e^{2+e^x x^4} x \left (8 e^x x^2+2 e^x x^3-4 e^x x^4-e^x x^5\right )}{4 e^x x^3+e^x x^4} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=\frac {1}{x}+e^{2+e^x x^4} \left (-2+x^2\right ) \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {1}{x}+\left (x^{2}-2\right ) {\mathrm e}^{{\mathrm e}^{x} x^{4}+2}\) | \(20\) |
parallelrisch | \(\frac {{\mathrm e}^{{\mathrm e}^{x} x^{4}+2} x^{3}-2 x \,{\mathrm e}^{{\mathrm e}^{x} x^{4}+2}+1}{x}\) | \(32\) |
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=\frac {{\left (x^{3} - 2 \, x\right )} e^{\left (x^{4} e^{x} + 2\right )} + 1}{x} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=\left (x^{2} - 2\right ) e^{x^{4} e^{x} + 2} + \frac {1}{x} \]
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx={\left (x^{2} e^{2} - 2 \, e^{2}\right )} e^{\left (x^{4} e^{x}\right )} + \frac {1}{x} \]
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\[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx=\int { \frac {{\left (2 \, x^{3} + {\left (x^{8} + 4 \, x^{7} - 2 \, x^{6} - 8 \, x^{5}\right )} e^{x}\right )} e^{\left (x^{4} e^{x} + 2\right )} - 1}{x^{2}} \,d x } \]
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Time = 13.88 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-1+e^{2+e^x x^4} \left (2 x^3+e^x \left (-8 x^5-2 x^6+4 x^7+x^8\right )\right )}{x^2} \, dx={\mathrm {e}}^{x^4\,{\mathrm {e}}^x+2}\,\left (x^2-2\right )+\frac {1}{x} \]
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