Integrand size = 52, antiderivative size = 14 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (-e^{x^3}+x\right )}{x} \]
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Time = 0.62 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6874, 14, 2631} \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (x-e^{x^3}\right )}{x} \]
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Rule 14
Rule 2631
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1+3 x^3}{\left (e^{x^3}-x\right ) x}+\frac {3 x^3-\log \left (-e^{x^3}+x\right )}{x^2}\right ) \, dx \\ & = \int \frac {-1+3 x^3}{\left (e^{x^3}-x\right ) x} \, dx+\int \frac {3 x^3-\log \left (-e^{x^3}+x\right )}{x^2} \, dx \\ & = \int \left (-\frac {1}{\left (e^{x^3}-x\right ) x}+\frac {3 x^2}{e^{x^3}-x}\right ) \, dx+\int \left (3 x-\frac {\log \left (-e^{x^3}+x\right )}{x^2}\right ) \, dx \\ & = \frac {3 x^2}{2}+3 \int \frac {x^2}{e^{x^3}-x} \, dx-\int \frac {1}{\left (e^{x^3}-x\right ) x} \, dx-\int \frac {\log \left (-e^{x^3}+x\right )}{x^2} \, dx \\ & = \frac {3 x^2}{2}+\frac {\log \left (-e^{x^3}+x\right )}{x}+3 \int \frac {x^2}{e^{x^3}-x} \, dx-\int \frac {1}{\left (e^{x^3}-x\right ) x} \, dx-\int \frac {1-3 e^{x^3} x^2}{x \left (-e^{x^3}+x\right )} \, dx \\ & = \frac {3 x^2}{2}+\frac {\log \left (-e^{x^3}+x\right )}{x}+3 \int \frac {x^2}{e^{x^3}-x} \, dx-\int \frac {1}{\left (e^{x^3}-x\right ) x} \, dx-\int \left (3 x+\frac {-1+3 x^3}{\left (e^{x^3}-x\right ) x}\right ) \, dx \\ & = \frac {\log \left (-e^{x^3}+x\right )}{x}+3 \int \frac {x^2}{e^{x^3}-x} \, dx-\int \frac {1}{\left (e^{x^3}-x\right ) x} \, dx-\int \frac {-1+3 x^3}{\left (e^{x^3}-x\right ) x} \, dx \\ & = \frac {\log \left (-e^{x^3}+x\right )}{x}+3 \int \frac {x^2}{e^{x^3}-x} \, dx-\int \frac {1}{\left (e^{x^3}-x\right ) x} \, dx-\int \left (-\frac {1}{\left (e^{x^3}-x\right ) x}+\frac {3 x^2}{e^{x^3}-x}\right ) \, dx \\ & = \frac {\log \left (-e^{x^3}+x\right )}{x} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (-e^{x^3}+x\right )}{x} \]
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Time = 0.45 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {\ln \left (-{\mathrm e}^{x^{3}}+x \right )}{x}\) | \(14\) |
risch | \(\frac {\ln \left (-{\mathrm e}^{x^{3}}+x \right )}{x}\) | \(14\) |
parallelrisch | \(\frac {\ln \left (-{\mathrm e}^{x^{3}}+x \right )}{x}\) | \(14\) |
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (x - e^{\left (x^{3}\right )}\right )}{x} \]
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Time = 0.13 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log {\left (x - e^{x^{3}} \right )}}{x} \]
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Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (x - e^{\left (x^{3}\right )}\right )}{x} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\log \left (x - e^{\left (x^{3}\right )}\right )}{x} \]
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Time = 9.60 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {-x+3 e^{x^3} x^3+\left (-e^{x^3}+x\right ) \log \left (-e^{x^3}+x\right )}{e^{x^3} x^2-x^3} \, dx=\frac {\ln \left (x-{\mathrm {e}}^{x^3}\right )}{x} \]
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