Integrand size = 41, antiderivative size = 26 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=\frac {\left (e^{x/400}+x\right ) \left (4-x-x^2\right )}{3 x} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 14, 2230, 2225, 2208, 2209, 2207} \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=-\frac {1}{12} (2 x+1)^2-\frac {e^{x/400}}{3}-\frac {1}{3} e^{x/400} x+\frac {4 e^{x/400}}{3 x} \]
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Rule 12
Rule 14
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{x^2} \, dx}{1200} \\ & = \frac {\int \left (-400 (1+2 x)-\frac {e^{x/400} \left (1600-4 x+401 x^2+x^3\right )}{x^2}\right ) \, dx}{1200} \\ & = -\frac {1}{12} (1+2 x)^2-\frac {\int \frac {e^{x/400} \left (1600-4 x+401 x^2+x^3\right )}{x^2} \, dx}{1200} \\ & = -\frac {1}{12} (1+2 x)^2-\frac {\int \left (401 e^{x/400}+\frac {1600 e^{x/400}}{x^2}-\frac {4 e^{x/400}}{x}+e^{x/400} x\right ) \, dx}{1200} \\ & = -\frac {1}{12} (1+2 x)^2-\frac {\int e^{x/400} x \, dx}{1200}+\frac {1}{300} \int \frac {e^{x/400}}{x} \, dx-\frac {401 \int e^{x/400} \, dx}{1200}-\frac {4}{3} \int \frac {e^{x/400}}{x^2} \, dx \\ & = -\frac {401 e^{x/400}}{3}+\frac {4 e^{x/400}}{3 x}-\frac {1}{3} e^{x/400} x-\frac {1}{12} (1+2 x)^2+\frac {\operatorname {ExpIntegralEi}\left (\frac {x}{400}\right )}{300}-\frac {1}{300} \int \frac {e^{x/400}}{x} \, dx+\frac {1}{3} \int e^{x/400} \, dx \\ & = -\frac {e^{x/400}}{3}+\frac {4 e^{x/400}}{3 x}-\frac {1}{3} e^{x/400} x-\frac {1}{12} (1+2 x)^2 \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=-\frac {x}{3}-\frac {x^2}{3}-\frac {e^{x/400} \left (400-\frac {1600}{x}+400 x\right )}{1200} \]
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Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {x^{2}}{3}-\frac {x}{3}-\frac {\left (x^{2}+x -4\right ) {\mathrm e}^{\frac {x}{400}}}{3 x}\) | \(25\) |
derivativedivides | \(-\frac {x^{2}}{3}-\frac {x}{3}+\frac {4 \,{\mathrm e}^{\frac {x}{400}}}{3 x}-\frac {{\mathrm e}^{\frac {x}{400}} x}{3}-\frac {{\mathrm e}^{\frac {x}{400}}}{3}\) | \(32\) |
default | \(-\frac {x^{2}}{3}-\frac {x}{3}+\frac {4 \,{\mathrm e}^{\frac {x}{400}}}{3 x}-\frac {{\mathrm e}^{\frac {x}{400}} x}{3}-\frac {{\mathrm e}^{\frac {x}{400}}}{3}\) | \(32\) |
parts | \(-\frac {x^{2}}{3}-\frac {x}{3}+\frac {4 \,{\mathrm e}^{\frac {x}{400}}}{3 x}-\frac {{\mathrm e}^{\frac {x}{400}} x}{3}-\frac {{\mathrm e}^{\frac {x}{400}}}{3}\) | \(32\) |
norman | \(\frac {-\frac {x^{2}}{3}-\frac {x^{3}}{3}-\frac {{\mathrm e}^{\frac {x}{400}} x}{3}-\frac {{\mathrm e}^{\frac {x}{400}} x^{2}}{3}+\frac {4 \,{\mathrm e}^{\frac {x}{400}}}{3}}{x}\) | \(38\) |
parallelrisch | \(-\frac {400 x^{3}+400 \,{\mathrm e}^{\frac {x}{400}} x^{2}+400 x^{2}+400 \,{\mathrm e}^{\frac {x}{400}} x -1600 \,{\mathrm e}^{\frac {x}{400}}}{1200 x}\) | \(39\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=-\frac {x^{3} + x^{2} + {\left (x^{2} + x - 4\right )} e^{\left (\frac {1}{400} \, x\right )}}{3 \, x} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=- \frac {x^{2}}{3} - \frac {x}{3} + \frac {\left (- x^{2} - x + 4\right ) e^{\frac {x}{400}}}{3 x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=-\frac {1}{3} \, x^{2} - \frac {1}{3} \, {\left (x - 400\right )} e^{\left (\frac {1}{400} \, x\right )} - \frac {1}{3} \, x + \frac {1}{300} \, {\rm Ei}\left (\frac {1}{400} \, x\right ) - \frac {401}{3} \, e^{\left (\frac {1}{400} \, x\right )} - \frac {1}{300} \, \Gamma \left (-1, -\frac {1}{400} \, x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=-\frac {x^{3} + x^{2} e^{\left (\frac {1}{400} \, x\right )} + x^{2} + x e^{\left (\frac {1}{400} \, x\right )} - 4 \, e^{\left (\frac {1}{400} \, x\right )}}{3 \, x} \]
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Time = 8.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-400 x^2-800 x^3+e^{x/400} \left (-1600+4 x-401 x^2-x^3\right )}{1200 x^2} \, dx=\frac {4\,{\mathrm {e}}^{x/400}}{3\,x}-\frac {{\mathrm {e}}^{x/400}}{3}-x\,\left (\frac {{\mathrm {e}}^{x/400}}{3}+\frac {1}{3}\right )-\frac {x^2}{3} \]
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