Integrand size = 79, antiderivative size = 24 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {1}{-x+e^{3-e} \left (1-\frac {3 x}{-2+x}\right )} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(24)=48\).
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1694, 12, 1828, 8} \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {2 e^{2 e}-e^{2 e} x}{-e^{2 e} x^2-2 e^e \left (e^3-e^e\right ) x-2 e^{3+e}} \]
[In]
[Out]
Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {e^{2 e} \left (e^6+e^{2 e}-4 e^{3+e}-2 e^e \left (e^3+e^e\right ) x+e^{2 e} x^2\right )}{\left (e^6+e^{2 e}-4 e^{3+e}-e^{2 e} x^2\right )^2} \, dx,x,\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )+x\right ) \\ & = e^{2 e} \text {Subst}\left (\int \frac {e^6+e^{2 e}-4 e^{3+e}-2 e^e \left (e^3+e^e\right ) x+e^{2 e} x^2}{\left (e^6+e^{2 e}-4 e^{3+e}-e^{2 e} x^2\right )^2} \, dx,x,\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )+x\right ) \\ & = \frac {e^{2 e} (2-x)}{2 e^{3+e}+2 e^e \left (e^3-e^e\right ) x+e^{2 e} x^2}-\frac {e^{2 e} \text {Subst}\left (\int 0 \, dx,x,\frac {1}{4} e^{6-2 e} \left (4 e^{-3+e}-4 e^{-6+2 e}\right )+x\right )}{2 \left (e^6+e^{2 e}-4 e^{3+e}\right )} \\ & = \frac {e^{2 e} (2-x)}{2 e^{3+e}+2 e^e \left (e^3-e^e\right ) x+e^{2 e} x^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {e^e (2-x)}{e^e (-2+x) x+2 e^3 (1+x)} \]
[In]
[Out]
Time = 1.45 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
gosper | \(-\frac {\left (-2+x \right ) {\mathrm e}^{{\mathrm e}-3}}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 x \,{\mathrm e}^{{\mathrm e}-3}+2 x +2}\) | \(35\) |
risch | \(\frac {-x \,{\mathrm e}^{{\mathrm e}-3}+2 \,{\mathrm e}^{{\mathrm e}-3}}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 x \,{\mathrm e}^{{\mathrm e}-3}+2 x +2}\) | \(42\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{2 \,{\mathrm e}-6} x^{2}-4 \,{\mathrm e}^{2 \,{\mathrm e}-6} x +6 x \,{\mathrm e}^{{\mathrm e}-3}}{2 \left ({\mathrm e}^{{\mathrm e}-3} x^{2}-2 x \,{\mathrm e}^{{\mathrm e}-3}+2 x +2\right )}\) | \(58\) |
norman | \(\frac {-{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-6} x^{2}+\left (2 \,{\mathrm e}^{2 \,{\mathrm e}} {\mathrm e}^{-6}-3 \,{\mathrm e}^{{\mathrm e}} {\mathrm e}^{-3}\right ) x}{{\mathrm e}^{{\mathrm e}-3} x^{2}-2 x \,{\mathrm e}^{{\mathrm e}-3}+2 x +2}\) | \(62\) |
default | \(\frac {{\mathrm e}^{{\mathrm e}-3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4+{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_Z}^{4}-\left (4 \,{\mathrm e}^{2 \,{\mathrm e}-6}-4 \,{\mathrm e}^{{\mathrm e}-3}\right ) \textit {\_Z}^{3}+\left (4 \,{\mathrm e}^{2 \,{\mathrm e}-6}-4 \,{\mathrm e}^{{\mathrm e}-3}+4\right ) \textit {\_Z}^{2}-\left (8 \,{\mathrm e}^{{\mathrm e}-3}-8\right ) \textit {\_Z} \right )}{\sum }\frac {\left ({\mathrm e}^{{\mathrm e}-3} \textit {\_R}^{2}-4 \textit {\_R} \,{\mathrm e}^{{\mathrm e}-3}+4 \,{\mathrm e}^{{\mathrm e}-3}-6\right ) \ln \left (x -\textit {\_R} \right )}{2+{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R}^{3}-3 \,{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R}^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}-6} \textit {\_R} +3 \,{\mathrm e}^{{\mathrm e}-3} \textit {\_R}^{2}-2 \textit {\_R} \,{\mathrm e}^{{\mathrm e}-3}-2 \,{\mathrm e}^{{\mathrm e}-3}+2 \textit {\_R}}\right )}{4}\) | \(179\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {{\left (x - 2\right )} e^{\left (e - 3\right )}}{{\left (x^{2} - 2 \, x\right )} e^{\left (e - 3\right )} + 2 \, x + 2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.60 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\frac {- x e^{e} + 2 e^{e}}{x^{2} e^{e} + x \left (- 2 e^{e} + 2 e^{3}\right ) + 2 e^{3}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {x e^{e} - 2 \, e^{e}}{x^{2} e^{e} + 2 \, x {\left (e^{3} - e^{e}\right )} + 2 \, e^{3}} \]
[In]
[Out]
\[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e - 6\right )} - 6 \, e^{\left (e - 3\right )}}{4 \, x^{2} + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (2 \, e - 6\right )} + 4 \, {\left (x^{3} - x^{2} - 2 \, x\right )} e^{\left (e - 3\right )} + 8 \, x + 4} \,d x } \]
[In]
[Out]
Time = 15.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-6 e^{-3+e}+e^{-6+2 e} \left (4-4 x+x^2\right )}{4+8 x+4 x^2+e^{-3+e} \left (-8 x-4 x^2+4 x^3\right )+e^{-6+2 e} \left (4 x^2-4 x^3+x^4\right )} \, dx=-\frac {x-2}{x^2+\left (2\,{\mathrm {e}}^{3-\mathrm {e}}-2\right )\,x+2\,{\mathrm {e}}^{3-\mathrm {e}}} \]
[In]
[Out]