Integrand size = 47, antiderivative size = 27 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=3+\frac {-1+x}{x}-\frac {1}{3} e^{3+e^5-x} x+x^3 \]
Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {1}{3} \left (-\frac {3}{x}-e^{3+e^5-x} x+3 x^3\right ) \]
Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {27, 25, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{e^5-x} \left (e^{x-e^5} \left (9 x^4+3\right )+e^3 \left (x^3-x^2\right )\right )}{3 x^2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {e^{e^5-x} \left (e^3 \left (x^2-x^3\right )-3 e^{x-e^5} \left (3 x^4+1\right )\right )}{x^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {e^{e^5-x} \left (e^3 \left (x^2-x^3\right )-3 e^{x-e^5} \left (3 x^4+1\right )\right )}{x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (-e^{-x+e^5+3} (x-1)-\frac {3 \left (3 x^4+1\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (3 x^3-e^{-x+e^5+3}+e^{-x+e^5+3} (1-x)-\frac {3}{x}\right )\) |
3.30.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
risch | \(x^{3}-\frac {1}{x}-\frac {x \,{\mathrm e}^{3+{\mathrm e}^{5}-x}}{3}\) | \(21\) |
norman | \(\frac {\left ({\mathrm e}^{-{\mathrm e}^{5}+x} x^{4}-\frac {x^{2} {\mathrm e}^{3}}{3}-{\mathrm e}^{-{\mathrm e}^{5}+x}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{x}\) | \(42\) |
parallelrisch | \(-\frac {\left (-3 \,{\mathrm e}^{-{\mathrm e}^{5}+x} x^{4}+x^{2} {\mathrm e}^{3}+3 \,{\mathrm e}^{-{\mathrm e}^{5}+x}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{3 x}\) | \(43\) |
parts | \(x^{3}-\frac {1}{x}+\frac {{\mathrm e}^{3} \left (-\left (-{\mathrm e}^{5}+x \right ) {\mathrm e}^{{\mathrm e}^{5}-x}-{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}\right )}{3}\) | \(45\) |
derivativedivides | \(\frac {{\mathrm e}^{3} \left ({\mathrm e}^{{\mathrm e}^{5}-x} \left (3 \,{\mathrm e}^{5}-x -1\right )+\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{15}}{x}-\left ({\mathrm e}^{15}+3 \,{\mathrm e}^{10}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}+\frac {{\mathrm e}^{15} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {1}{x}+\left (-{\mathrm e}^{5}+x \right )^{3}-3 \left (-{\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{5}+9 \left (-{\mathrm e}^{5}+x \right ) {\mathrm e}^{10}-\frac {3 \left (4 \,{\mathrm e}^{15} {\mathrm e}^{5}-3 \,{\mathrm e}^{20}\right )}{x}+\frac {9 \,{\mathrm e}^{20}}{x}+12 \,{\mathrm e}^{5} \left (\frac {\left (-{\mathrm e}^{5}+x \right )^{2}}{2}-2 \left (-{\mathrm e}^{5}+x \right ) {\mathrm e}^{5}-\frac {-3 \,{\mathrm e}^{10} {\mathrm e}^{5}+2 \,{\mathrm e}^{15}}{x}+3 \,{\mathrm e}^{10} \ln \left (x \right )\right )+18 \,{\mathrm e}^{10} \left (-{\mathrm e}^{5}+x -\frac {{\mathrm e}^{10}}{x}-2 \,{\mathrm e}^{5} \ln \left (x \right )\right )-\frac {{\mathrm e}^{3} \left (-{\mathrm e}^{{\mathrm e}^{5}-x}-\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{10}}{x}-\left (-{\mathrm e}^{10}-2 \,{\mathrm e}^{5}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {{\mathrm e}^{10} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {2 \,{\mathrm e}^{3} {\mathrm e}^{5} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}+{\mathrm e}^{3} {\mathrm e}^{5} \left (-{\mathrm e}^{{\mathrm e}^{5}-x}-\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{10}}{x}-\left (-{\mathrm e}^{10}-2 \,{\mathrm e}^{5}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )+{\mathrm e}^{10} {\mathrm e}^{3} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )\) | \(427\) |
default | \(\frac {{\mathrm e}^{3} \left ({\mathrm e}^{{\mathrm e}^{5}-x} \left (3 \,{\mathrm e}^{5}-x -1\right )+\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{15}}{x}-\left ({\mathrm e}^{15}+3 \,{\mathrm e}^{10}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}+\frac {{\mathrm e}^{15} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {1}{x}+\left (-{\mathrm e}^{5}+x \right )^{3}-3 \left (-{\mathrm e}^{5}+x \right )^{2} {\mathrm e}^{5}+9 \left (-{\mathrm e}^{5}+x \right ) {\mathrm e}^{10}-\frac {3 \left (4 \,{\mathrm e}^{15} {\mathrm e}^{5}-3 \,{\mathrm e}^{20}\right )}{x}+\frac {9 \,{\mathrm e}^{20}}{x}+12 \,{\mathrm e}^{5} \left (\frac {\left (-{\mathrm e}^{5}+x \right )^{2}}{2}-2 \left (-{\mathrm e}^{5}+x \right ) {\mathrm e}^{5}-\frac {-3 \,{\mathrm e}^{10} {\mathrm e}^{5}+2 \,{\mathrm e}^{15}}{x}+3 \,{\mathrm e}^{10} \ln \left (x \right )\right )+18 \,{\mathrm e}^{10} \left (-{\mathrm e}^{5}+x -\frac {{\mathrm e}^{10}}{x}-2 \,{\mathrm e}^{5} \ln \left (x \right )\right )-\frac {{\mathrm e}^{3} \left (-{\mathrm e}^{{\mathrm e}^{5}-x}-\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{10}}{x}-\left (-{\mathrm e}^{10}-2 \,{\mathrm e}^{5}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {{\mathrm e}^{10} {\mathrm e}^{3} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}-\frac {2 \,{\mathrm e}^{3} {\mathrm e}^{5} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )}{3}+{\mathrm e}^{3} {\mathrm e}^{5} \left (-{\mathrm e}^{{\mathrm e}^{5}-x}-\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{10}}{x}-\left (-{\mathrm e}^{10}-2 \,{\mathrm e}^{5}\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )+{\mathrm e}^{10} {\mathrm e}^{3} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (x \right )\right )\) | \(427\) |
int(1/3*((9*x^4+3)*exp(-exp(5)+x)+(x^3-x^2)*exp(3))/x^2/exp(-exp(5)+x),x,m ethod=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {3 \, x^{4} - x^{2} e^{\left (-x + e^{5} + 3\right )} - 3}{3 \, x} \]
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^{3} - \frac {x e^{3} e^{- x + e^{5}}}{3} - \frac {1}{x} \]
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^{3} - \frac {1}{3} \, {\left (x e^{\left (e^{5} + 3\right )} + e^{\left (e^{5} + 3\right )}\right )} e^{\left (-x\right )} - \frac {1}{x} + \frac {1}{3} \, e^{\left (-x + e^{5} + 3\right )} \]
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {3 \, {\left (x - e^{5}\right )}^{4} + 12 \, {\left (x - e^{5}\right )}^{3} e^{5} + 18 \, {\left (x - e^{5}\right )}^{2} e^{10} - {\left (x - e^{5}\right )}^{2} e^{\left (-x + e^{5} + 3\right )} + 9 \, {\left (x - e^{5}\right )} e^{15} - 2 \, {\left (x - e^{5}\right )} e^{\left (-x + e^{5} + 8\right )} - e^{\left (-x + e^{5} + 13\right )} - 3}{3 \, x} \]
1/3*(3*(x - e^5)^4 + 12*(x - e^5)^3*e^5 + 18*(x - e^5)^2*e^10 - (x - e^5)^ 2*e^(-x + e^5 + 3) + 9*(x - e^5)*e^15 - 2*(x - e^5)*e^(-x + e^5 + 8) - e^( -x + e^5 + 13) - 3)/x
Time = 12.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^3-\frac {1}{x}-\frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3} \]