Integrand size = 79, antiderivative size = 30 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {x^2}{2}-\log \left (e^x+4 e^{3 x+\frac {30 \log (x)}{x}}\right ) \]
[Out]
\[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {-x^2+x^3+4 e^{2 x} x^{30/x} \left (-30-3 x^2+x^3+30 \log (x)\right )}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx \\ & = \int \left (\frac {2 \left (15+x^2-15 \log (x)\right )}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )}+\frac {-30-3 x^2+x^3+30 \log (x)}{x^2}\right ) \, dx \\ & = 2 \int \frac {15+x^2-15 \log (x)}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+\int \frac {-30-3 x^2+x^3+30 \log (x)}{x^2} \, dx \\ & = 2 \int \left (\frac {1}{1+4 e^{2 x} x^{30/x}}+\frac {15}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )}-\frac {15 \log (x)}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )}\right ) \, dx+\int \left (\frac {-30-3 x^2+x^3}{x^2}+\frac {30 \log (x)}{x^2}\right ) \, dx \\ & = 2 \int \frac {1}{1+4 e^{2 x} x^{30/x}} \, dx+30 \int \frac {1}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+30 \int \frac {\log (x)}{x^2} \, dx-30 \int \frac {\log (x)}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+\int \frac {-30-3 x^2+x^3}{x^2} \, dx \\ & = -\frac {30}{x}-\frac {30 \log (x)}{x}+2 \int \frac {1}{1+4 e^{2 x} x^{30/x}} \, dx+30 \int \frac {1}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+30 \int \frac {\int \frac {1}{x^2+4 e^{2 x} x^{2+\frac {30}{x}}} \, dx}{x} \, dx-(30 \log (x)) \int \frac {1}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+\int \left (-3-\frac {30}{x^2}+x\right ) \, dx \\ & = -3 x+\frac {x^2}{2}-\frac {30 \log (x)}{x}+2 \int \frac {1}{1+4 e^{2 x} x^{30/x}} \, dx+30 \int \frac {1}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx+30 \int \frac {\int \frac {1}{x^2+4 e^{2 x} x^{2+\frac {30}{x}}} \, dx}{x} \, dx-(30 \log (x)) \int \frac {1}{x^2 \left (1+4 e^{2 x} x^{30/x}\right )} \, dx \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=-x+\frac {x^2}{2}-\log \left (1+4 e^{2 x} x^{30/x}\right ) \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(\frac {x^{2}}{2}-\ln \left ({\mathrm e}^{x}+4 \,{\mathrm e}^{\frac {30 \ln \left (x \right )+3 x^{2}}{x}}\right )\) | \(29\) |
risch | \(-\frac {30 \ln \left (x \right )}{x}-3 x +\frac {x^{2}}{2}+\frac {30 \ln \left (x \right )+3 x^{2}}{x}-\ln \left (\frac {{\mathrm e}^{x}}{4}+x^{\frac {30}{x}} {\mathrm e}^{3 x}\right )\) | \(51\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {1}{2} \, x^{2} - \log \left (e^{x} + 4 \, e^{\left (\frac {3 \, {\left (x^{2} + 10 \, \log \left (x\right )\right )}}{x}\right )}\right ) \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {x^{2}}{2} - \log {\left (\frac {e^{x}}{4} + e^{\frac {3 x^{2} + 30 \log {\left (x \right )}}{x}} \right )} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {1}{2} \, x^{2} - 3 \, x - \log \left (\frac {1}{4} \, {\left (4 \, e^{\left (2 \, x + \frac {30 \, \log \left (x\right )}{x}\right )} + 1\right )} e^{\left (-2 \, x\right )}\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {1}{2} \, x^{2} - \log \left (e^{x} + 4 \, e^{\left (\frac {3 \, {\left (x^{2} + 10 \, \log \left (x\right )\right )}}{x}\right )}\right ) \]
[In]
[Out]
Time = 11.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^x \left (-x^2+x^3\right )+e^{\frac {3 x^2+30 \log (x)}{x}} \left (-120-12 x^2+4 x^3+120 \log (x)\right )}{e^x x^2+4 e^{\frac {3 x^2+30 \log (x)}{x}} x^2} \, dx=\frac {x^2}{2}-\ln \left (\frac {{\mathrm {e}}^x}{4}+x^{30/x}\,{\mathrm {e}}^{3\,x}\right ) \]
[In]
[Out]