Integrand size = 20, antiderivative size = 15 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=e^{\frac {1}{x^2}}+3 x-\log (5)+\log (x) \]
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Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 2240, 45} \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=e^{\frac {1}{x^2}}+3 x+\log (x) \]
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Rule 14
Rule 45
Rule 2240
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{\frac {1}{x^2}}}{x^3}+\frac {1+3 x}{x}\right ) \, dx \\ & = -\left (2 \int \frac {e^{\frac {1}{x^2}}}{x^3} \, dx\right )+\int \frac {1+3 x}{x} \, dx \\ & = e^{\frac {1}{x^2}}+\int \left (3+\frac {1}{x}\right ) \, dx \\ & = e^{\frac {1}{x^2}}+3 x+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=e^{\frac {1}{x^2}}+3 x+\log (x) \]
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Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73
method | result | size |
risch | \(3 x +\ln \left (x \right )+{\mathrm e}^{\frac {1}{x^{2}}}\) | \(11\) |
parallelrisch | \(3 x +\ln \left (x \right )+{\mathrm e}^{\frac {1}{x^{2}}}\) | \(11\) |
parts | \(3 x +\ln \left (x \right )+{\mathrm e}^{\frac {1}{x^{2}}}\) | \(11\) |
derivativedivides | \(-\ln \left (\frac {1}{x}\right )+3 x +{\mathrm e}^{\frac {1}{x^{2}}}\) | \(15\) |
default | \(-\ln \left (\frac {1}{x}\right )+3 x +{\mathrm e}^{\frac {1}{x^{2}}}\) | \(15\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {1}{x^{2}}}+3 x^{3}}{x^{2}}+\ln \left (x \right )\) | \(22\) |
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Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=3 \, x + e^{\left (\frac {1}{x^{2}}\right )} + \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=3 x + e^{\frac {1}{x^{2}}} + \log {\left (x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=3 \, x + e^{\left (\frac {1}{x^{2}}\right )} + \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=3 \, x + e^{\left (\frac {1}{x^{2}}\right )} + \log \left (x\right ) \]
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Time = 12.74 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-2 e^{\frac {1}{x^2}}+x^2+3 x^3}{x^3} \, dx=3\,x+{\mathrm {e}}^{\frac {1}{x^2}}+\ln \left (x\right ) \]
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