Integrand size = 42, antiderivative size = 28 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=-\frac {3}{2}+e^x+x+\frac {1}{4} \left (\frac {3}{x}+\left (-e^x+x\right ) \log (x)\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 14, 2326, 2332} \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=x+\frac {3}{4 x}+\frac {1}{4} x \log (x)+\frac {e^x (4 x-x \log (x))}{4 x} \]
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Rule 12
Rule 14
Rule 2326
Rule 2332
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{x^2} \, dx \\ & = \frac {1}{4} \int \left (-\frac {e^x (1-4 x+x \log (x))}{x}+\frac {-3+5 x^2+x^2 \log (x)}{x^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^x (1-4 x+x \log (x))}{x} \, dx\right )+\frac {1}{4} \int \frac {-3+5 x^2+x^2 \log (x)}{x^2} \, dx \\ & = \frac {e^x (4 x-x \log (x))}{4 x}+\frac {1}{4} \int \left (\frac {-3+5 x^2}{x^2}+\log (x)\right ) \, dx \\ & = \frac {e^x (4 x-x \log (x))}{4 x}+\frac {1}{4} \int \frac {-3+5 x^2}{x^2} \, dx+\frac {1}{4} \int \log (x) \, dx \\ & = -\frac {x}{4}+\frac {1}{4} x \log (x)+\frac {e^x (4 x-x \log (x))}{4 x}+\frac {1}{4} \int \left (5-\frac {3}{x^2}\right ) \, dx \\ & = \frac {3}{4 x}+x+\frac {1}{4} x \log (x)+\frac {e^x (4 x-x \log (x))}{4 x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=\frac {1}{4} \left (4 e^x+\frac {3}{x}+4 x-e^x \log (x)+x \log (x)\right ) \]
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Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {{\mathrm e}^{x} \ln \left (x \right )}{4}+{\mathrm e}^{x}+x +\frac {3}{4 x}+\frac {x \ln \left (x \right )}{4}\) | \(21\) |
parts | \(-\frac {{\mathrm e}^{x} \ln \left (x \right )}{4}+{\mathrm e}^{x}+x +\frac {3}{4 x}+\frac {x \ln \left (x \right )}{4}\) | \(21\) |
norman | \(\frac {\frac {3}{4}+x^{2}+{\mathrm e}^{x} x +\frac {x^{2} \ln \left (x \right )}{4}-\frac {x \,{\mathrm e}^{x} \ln \left (x \right )}{4}}{x}\) | \(28\) |
risch | \(\frac {\left (x -{\mathrm e}^{x}\right ) \ln \left (x \right )}{4}+\frac {4 x^{2}+4 \,{\mathrm e}^{x} x +3}{4 x}\) | \(29\) |
parallelrisch | \(-\frac {x \,{\mathrm e}^{x} \ln \left (x \right )-x^{2} \ln \left (x \right )-4 \,{\mathrm e}^{x} x -4 x^{2}-3}{4 x}\) | \(31\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=\frac {4 \, x^{2} + 4 \, x e^{x} + {\left (x^{2} - x e^{x}\right )} \log \left (x\right ) + 3}{4 \, x} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=\frac {x \log {\left (x \right )}}{4} + x + \frac {\left (4 - \log {\left (x \right )}\right ) e^{x}}{4} + \frac {3}{4 x} \]
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Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=\frac {1}{4} \, x \log \left (x\right ) - \frac {1}{4} \, e^{x} \log \left (x\right ) + x + \frac {3}{4 \, x} + e^{x} \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx=\frac {x^{2} \log \left (x\right ) - x e^{x} \log \left (x\right ) + 4 \, x^{2} + 4 \, x e^{x} + 3}{4 \, x} \]
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Time = 14.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {-3+5 x^2+e^x \left (-x+4 x^2\right )+\left (x^2-e^x x^2\right ) \log (x)}{4 x^2} \, dx={\mathrm {e}}^x-\frac {{\mathrm {e}}^x\,\ln \left (x\right )}{4}+x\,\left (\frac {\ln \left (x\right )}{4}+1\right )+\frac {3}{4\,x} \]
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