Integrand size = 61, antiderivative size = 32 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {\left (e^{-1-x}+\frac {2 e^x}{x}+\frac {x}{4}\right ) (5+x (5+x))}{x} \]
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Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2230, 2225, 2208, 2209, 2207} \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {x^2}{4}+\frac {10 e^x}{x^2}+e^{-x-1} x+\frac {5 x}{4}+5 e^{-x-1}+2 e^x+\frac {5 e^{-x-1}}{x}+\frac {10 e^x}{x} \]
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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 {1}{4} \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{x^3} \, dx \\ & = \frac {1}{4} \int \left (5+2 x-\frac {4 e^{-1-x} \left (5+5 x+4 x^2+x^3\right )}{x^2}+\frac {8 e^x \left (-10+5 x^2+x^3\right )}{x^3}\right ) \, dx \\ & = \frac {5 x}{4}+\frac {x^2}{4}+2 \int \frac {e^x \left (-10+5 x^2+x^3\right )}{x^3} \, dx-\int \frac {e^{-1-x} \left (5+5 x+4 x^2+x^3\right )}{x^2} \, dx \\ & = \frac {5 x}{4}+\frac {x^2}{4}+2 \int \left (e^x-\frac {10 e^x}{x^3}+\frac {5 e^x}{x}\right ) \, dx-\int \left (4 e^{-1-x}+\frac {5 e^{-1-x}}{x^2}+\frac {5 e^{-1-x}}{x}+e^{-1-x} x\right ) \, dx \\ & = \frac {5 x}{4}+\frac {x^2}{4}+2 \int e^x \, dx-4 \int e^{-1-x} \, dx-5 \int \frac {e^{-1-x}}{x^2} \, dx-5 \int \frac {e^{-1-x}}{x} \, dx+10 \int \frac {e^x}{x} \, dx-20 \int \frac {e^x}{x^3} \, dx-\int e^{-1-x} x \, dx \\ & = 4 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4}-\frac {5 \operatorname {ExpIntegralEi}(-x)}{e}+10 \operatorname {ExpIntegralEi}(x)+5 \int \frac {e^{-1-x}}{x} \, dx-10 \int \frac {e^x}{x^2} \, dx-\int e^{-1-x} \, dx \\ & = 5 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {10 e^x}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4}+10 \operatorname {ExpIntegralEi}(x)-10 \int \frac {e^x}{x} \, dx \\ & = 5 e^{-1-x}+2 e^x+\frac {10 e^x}{x^2}+\frac {5 e^{-1-x}}{x}+\frac {10 e^x}{x}+\frac {5 x}{4}+e^{-1-x} x+\frac {x^2}{4} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {1}{4} \left (8 e^x \left (1+\frac {5}{x^2}+\frac {5}{x}\right )+5 x+x^2-4 e^{-x} \left (-\frac {5}{e}-\frac {5}{e x}-\frac {x}{e}\right )\right ) \]
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Time = 0.65 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {x^{2}}{4}+\frac {5 x}{4}+\frac {2 \left (x^{2}+5 x +5\right ) {\mathrm e}^{x}}{x^{2}}+\frac {\left (x^{2}+5 x +5\right ) {\mathrm e}^{-1-x}}{x}\) | \(43\) |
parts | \(\frac {5 x}{4}+\frac {x^{2}}{4}+4 \,{\mathrm e}^{-1-x}+\frac {5 \,{\mathrm e}^{-1-x}}{x}-{\mathrm e}^{-1-x} \left (-1-x \right )+\frac {10 \,{\mathrm e}^{x}}{x^{2}}+\frac {10 \,{\mathrm e}^{x}}{x}+2 \,{\mathrm e}^{x}\) | \(60\) |
parallelrisch | \(\frac {4 x^{3} {\mathrm e}^{-1-x}+x^{4}+8 \,{\mathrm e}^{x} x^{2}+20 \,{\mathrm e}^{-1-x} x^{2}+5 x^{3}+40 \,{\mathrm e}^{x} x +20 x \,{\mathrm e}^{-1-x}+40 \,{\mathrm e}^{x}}{4 x^{2}}\) | \(62\) |
norman | \(\frac {\left ({\mathrm e}^{-1} x^{3}+10 \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{-1} x +10 x \,{\mathrm e}^{2 x}+5 x^{2} {\mathrm e}^{-1}+\frac {5 \,{\mathrm e}^{x} x^{3}}{4}+\frac {{\mathrm e}^{x} x^{4}}{4}+2 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{-x}}{x^{2}}\) | \(70\) |
default | \(\frac {x^{2}}{4}+\frac {5 x}{4}+4 \,{\mathrm e}^{-1} {\mathrm e}^{-x}+\frac {10 \,{\mathrm e}^{x}}{x^{2}}+\frac {10 \,{\mathrm e}^{x}}{x}-5 \,{\mathrm e}^{-1} \left (-\frac {{\mathrm e}^{-x}}{x}+\operatorname {Ei}_{1}\left (x \right )\right )+5 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (x \right )-{\mathrm e}^{-1} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )+2 \,{\mathrm e}^{x}\) | \(78\) |
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Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {{\left (4 \, x^{3} + 20 \, x^{2} + 8 \, {\left (x^{2} + 5 \, x + 5\right )} e^{\left (2 \, x + 1\right )} + {\left (x^{4} + 5 \, x^{3}\right )} e^{\left (x + 1\right )} + 20 \, x\right )} e^{\left (-x - 1\right )}}{4 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {x^{2}}{4} + \frac {5 x}{4} + \frac {\left (x^{4} + 5 x^{3} + 5 x^{2}\right ) e^{- x} + \left (2 e x^{3} + 10 e x^{2} + 10 e x\right ) e^{x}}{e x^{3}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {1}{4} \, x^{2} - 5 \, {\rm Ei}\left (-x\right ) e^{\left (-1\right )} + {\left (x + 1\right )} e^{\left (-x - 1\right )} + 5 \, e^{\left (-1\right )} \Gamma \left (-1, x\right ) + \frac {5}{4} \, x + 10 \, {\rm Ei}\left (x\right ) + 2 \, e^{x} + 4 \, e^{\left (-x - 1\right )} + 20 \, \Gamma \left (-2, -x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {{\left (x^{4} e + 5 \, x^{3} e + 4 \, x^{3} e^{\left (-x\right )} + 20 \, x^{2} e^{\left (-x\right )} + 8 \, x^{2} e^{\left (x + 1\right )} + 20 \, x e^{\left (-x\right )} + 40 \, x e^{\left (x + 1\right )} + 40 \, e^{\left (x + 1\right )}\right )} e^{\left (-1\right )}}{4 \, x^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {5 x^3+2 x^4+e^x \left (-80+40 x^2+8 x^3\right )+e^{-1-x} \left (-20 x-20 x^2-16 x^3-4 x^4\right )}{4 x^3} \, dx=\frac {5\,x}{4}+5\,{\mathrm {e}}^{-x-1}+2\,{\mathrm {e}}^x+\frac {10\,{\mathrm {e}}^x}{x}+\frac {10\,{\mathrm {e}}^x}{x^2}+x\,{\mathrm {e}}^{-x-1}+\frac {5\,{\mathrm {e}}^{-x-1}}{x}+\frac {x^2}{4} \]
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