Integrand size = 37, antiderivative size = 38 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=x+5 \left (1+e^x-\frac {-1+\frac {-e^4+e^x}{x}-x^2-\log (2)}{x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6, 14, 2230, 2225, 2208, 2209} \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=-\frac {5 e^x}{x^2}+\frac {5 e^4}{x^2}+6 x+5 e^x+\frac {5 (1+\log (2))}{x} \]
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Rule 6
Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \frac {-10 e^4+6 x^3+e^x \left (10-5 x+5 x^3\right )+x (-5-5 \log (2))}{x^3} \, dx \\ & = \int \left (\frac {5 e^x \left (2-x+x^3\right )}{x^3}+\frac {-10 e^4+6 x^3-5 x (1+\log (2))}{x^3}\right ) \, dx \\ & = 5 \int \frac {e^x \left (2-x+x^3\right )}{x^3} \, dx+\int \frac {-10 e^4+6 x^3-5 x (1+\log (2))}{x^3} \, dx \\ & = 5 \int \left (e^x+\frac {2 e^x}{x^3}-\frac {e^x}{x^2}\right ) \, dx+\int \left (6-\frac {10 e^4}{x^3}-\frac {5 (1+\log (2))}{x^2}\right ) \, dx \\ & = \frac {5 e^4}{x^2}+6 x+\frac {5 (1+\log (2))}{x}+5 \int e^x \, dx-5 \int \frac {e^x}{x^2} \, dx+10 \int \frac {e^x}{x^3} \, dx \\ & = 5 e^x+\frac {5 e^4}{x^2}-\frac {5 e^x}{x^2}+\frac {5 e^x}{x}+6 x+\frac {5 (1+\log (2))}{x}+5 \int \frac {e^x}{x^2} \, dx-5 \int \frac {e^x}{x} \, dx \\ & = 5 e^x+\frac {5 e^4}{x^2}-\frac {5 e^x}{x^2}+6 x-5 \operatorname {ExpIntegralEi}(x)+\frac {5 (1+\log (2))}{x}+5 \int \frac {e^x}{x} \, dx \\ & = 5 e^x+\frac {5 e^4}{x^2}-\frac {5 e^x}{x^2}+6 x+\frac {5 (1+\log (2))}{x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=\frac {5 e^4+5 e^x \left (-1+x^2\right )+x \left (5+6 x^2+\log (32)\right )}{x^2} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87
method | result | size |
parts | \(6 x +\frac {5 \ln \left (2\right )+5}{x}+\frac {5 \,{\mathrm e}^{4}}{x^{2}}-\frac {5 \,{\mathrm e}^{x}}{x^{2}}+5 \,{\mathrm e}^{x}\) | \(33\) |
norman | \(\frac {\left (5 \ln \left (2\right )+5\right ) x +6 x^{3}+5 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{4}-5 \,{\mathrm e}^{x}}{x^{2}}\) | \(34\) |
risch | \(6 x +\frac {\left (5 \ln \left (2\right )+5\right ) x +5 \,{\mathrm e}^{4}}{x^{2}}+\frac {5 \left (x^{2}-1\right ) {\mathrm e}^{x}}{x^{2}}\) | \(34\) |
parallelrisch | \(\frac {5 \,{\mathrm e}^{x} x^{2}+6 x^{3}+5 x \ln \left (2\right )+5 \,{\mathrm e}^{4}-5 \,{\mathrm e}^{x}+5 x}{x^{2}}\) | \(34\) |
default | \(6 x +\frac {5}{x}+\frac {5 \,{\mathrm e}^{4}}{x^{2}}-\frac {5 \,{\mathrm e}^{x}}{x^{2}}+\frac {5 \ln \left (2\right )}{x}+5 \,{\mathrm e}^{x}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=\frac {6 \, x^{3} + 5 \, {\left (x^{2} - 1\right )} e^{x} + 5 \, x \log \left (2\right ) + 5 \, x + 5 \, e^{4}}{x^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=6 x + \frac {\left (5 x^{2} - 5\right ) e^{x}}{x^{2}} + \frac {x \left (5 \log {\left (2 \right )} + 5\right ) + 5 e^{4}}{x^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=6 \, x + \frac {5 \, \log \left (2\right )}{x} + \frac {5}{x} + \frac {5 \, e^{4}}{x^{2}} + 5 \, e^{x} - 5 \, \Gamma \left (-1, -x\right ) - 10 \, \Gamma \left (-2, -x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=\frac {6 \, x^{3} + 5 \, x^{2} e^{x} + 5 \, x \log \left (2\right ) + 5 \, x + 5 \, e^{4} - 5 \, e^{x}}{x^{2}} \]
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Time = 9.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {-10 e^4-5 x+6 x^3+e^x \left (10-5 x+5 x^3\right )-5 x \log (2)}{x^3} \, dx=6\,x+5\,{\mathrm {e}}^x+\frac {5\,{\mathrm {e}}^4-5\,{\mathrm {e}}^x+x\,\left (\ln \left (32\right )+5\right )}{x^2} \]
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