Integrand size = 47, antiderivative size = 27 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=3+\frac {-1+x}{x}-\frac {1}{3} e^{3+e^5-x} x+x^3 \] Output:
(-1+x)/x-1/3/exp(-exp(5)+x)*x*exp(3)+3+x^3
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {1}{3} \left (-\frac {3}{x}-e^{3+e^5-x} x+3 x^3\right ) \] Input:
Integrate[(E^(E^5 - x)*(E^3*(-x^2 + x^3) + E^(-E^5 + x)*(3 + 9*x^4)))/(3*x ^2),x]
Output:
(-3/x - E^(3 + E^5 - x)*x + 3*x^3)/3
Time = 0.49 (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 |
\(\displaystyle \frac {1}{3} \left (3 x^3-e^{-x+e^5+3}+e^{-x+e^5+3} (1-x)-\frac {3}{x}\right )\) |
Input:
Int[(E^(E^5 - x)*(E^3*(-x^2 + x^3) + E^(-E^5 + x)*(3 + 9*x^4)))/(3*x^2),x]
Output:
(-E^(3 + E^5 - x) + E^(3 + E^5 - x)*(1 - x) - 3/x + 3*x^3)/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.37 (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 | \(x^{3}-\frac {1}{x}+\frac {{\mathrm e}^{3} \left (-{\mathrm e}^{{\mathrm e}^{5}-x} \left (-{\mathrm e}^{5}+x \right )-{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}\right )}{3}\) | \(45\) |
orering | \(\frac {\left (x^{6}-2 x^{5}-6 x^{4}-x^{2}+2 x +2\right ) \left (\left (9 x^{4}+3\right ) {\mathrm e}^{-{\mathrm e}^{5}+x}+\left (x^{3}-x^{2}\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{3 x \left (3 x^{6}-6 x^{4}+x^{2}-4 x +2\right )}+\frac {x^{2} \left (x^{5}+2 x^{4}-x +2\right ) \left (\frac {\left (36 x^{3} {\mathrm e}^{-{\mathrm e}^{5}+x}+\left (9 x^{4}+3\right ) {\mathrm e}^{-{\mathrm e}^{5}+x}+\left (3 x^{2}-2 x \right ) {\mathrm e}^{3}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{3 x^{2}}-\frac {2 \left (\left (9 x^{4}+3\right ) {\mathrm e}^{-{\mathrm e}^{5}+x}+\left (x^{3}-x^{2}\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{3 x^{3}}-\frac {\left (\left (9 x^{4}+3\right ) {\mathrm e}^{-{\mathrm e}^{5}+x}+\left (x^{3}-x^{2}\right ) {\mathrm e}^{3}\right ) {\mathrm e}^{{\mathrm e}^{5}-x}}{3 x^{2}}\right )}{3 x^{6}-6 x^{4}+x^{2}-4 x +2}\) | \(263\) |
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 {expIntegral}_{1}\left (x \right )\right )}{3}+\frac {{\mathrm e}^{3} {\mathrm e}^{15} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (x \right )\right )}{3}-\frac {1}{x}+\left (-{\mathrm e}^{5}+x \right )^{3}-3 \,{\mathrm e}^{5} \left (-{\mathrm e}^{5}+x \right )^{2}+9 \,{\mathrm e}^{10} \left (-{\mathrm e}^{5}+x \right )-\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 \,{\mathrm e}^{5} \left (-{\mathrm e}^{5}+x \right )-\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 {expIntegral}_{1}\left (x \right )\right )}{3}-\frac {{\mathrm e}^{3} {\mathrm e}^{10} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{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 {expIntegral}_{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 {expIntegral}_{1}\left (x \right )\right )+{\mathrm e}^{3} {\mathrm e}^{10} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{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 {expIntegral}_{1}\left (x \right )\right )}{3}+\frac {{\mathrm e}^{3} {\mathrm e}^{15} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (x \right )\right )}{3}-\frac {1}{x}+\left (-{\mathrm e}^{5}+x \right )^{3}-3 \,{\mathrm e}^{5} \left (-{\mathrm e}^{5}+x \right )^{2}+9 \,{\mathrm e}^{10} \left (-{\mathrm e}^{5}+x \right )-\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 \,{\mathrm e}^{5} \left (-{\mathrm e}^{5}+x \right )-\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 {expIntegral}_{1}\left (x \right )\right )}{3}-\frac {{\mathrm e}^{3} {\mathrm e}^{10} \left (-\frac {{\mathrm e}^{{\mathrm e}^{5}-x}}{x}+{\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{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 {expIntegral}_{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 {expIntegral}_{1}\left (x \right )\right )+{\mathrm e}^{3} {\mathrm e}^{10} \left (\frac {{\mathrm e}^{{\mathrm e}^{5}-x} {\mathrm e}^{5}}{x}-\left ({\mathrm e}^{5}+1\right ) {\mathrm e}^{{\mathrm e}^{5}} \operatorname {expIntegral}_{1}\left (x \right )\right )\) | \(427\) |
Input:
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)
Output:
x^3-1/x-1/3*x*exp(3+exp(5)-x)
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {3 \, x^{4} - x^{2} e^{\left (-x + e^{5} + 3\right )} - 3}{3 \, x} \] Input:
integrate(1/3*((9*x^4+3)*exp(-exp(5)+x)+(x^3-x^2)*exp(3))/x^2/exp(-exp(5)+ x),x, algorithm="fricas")
Output:
1/3*(3*x^4 - x^2*e^(-x + e^5 + 3) - 3)/x
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^{3} - \frac {x e^{3} e^{- x + e^{5}}}{3} - \frac {1}{x} \] Input:
integrate(1/3*((9*x**4+3)*exp(-exp(5)+x)+(x**3-x**2)*exp(3))/x**2/exp(-exp (5)+x),x)
Output:
x**3 - x*exp(3)*exp(-x + exp(5))/3 - 1/x
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^{3} - \frac {1}{3} \, {\left (x e^{\left (e^{5} + 3\right )} + e^{\left (e^{5} + 3\right )}\right )} e^{\left (-x\right )} - \frac {1}{x} + \frac {1}{3} \, e^{\left (-x + e^{5} + 3\right )} \] Input:
integrate(1/3*((9*x^4+3)*exp(-exp(5)+x)+(x^3-x^2)*exp(3))/x^2/exp(-exp(5)+ x),x, algorithm="maxima")
Output:
x^3 - 1/3*(x*e^(e^5 + 3) + e^(e^5 + 3))*e^(-x) - 1/x + 1/3*e^(-x + e^5 + 3 )
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.52 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {3 \, {\left (x - e^{5}\right )}^{4} + 12 \, {\left (x - e^{5}\right )}^{3} e^{5} + 18 \, {\left (x - e^{5}\right )}^{2} e^{10} - {\left (x - e^{5}\right )}^{2} e^{\left (-x + e^{5} + 3\right )} + 9 \, {\left (x - e^{5}\right )} e^{15} - 2 \, {\left (x - e^{5}\right )} e^{\left (-x + e^{5} + 8\right )} - e^{\left (-x + e^{5} + 13\right )} - 3}{3 \, x} \] Input:
integrate(1/3*((9*x^4+3)*exp(-exp(5)+x)+(x^3-x^2)*exp(3))/x^2/exp(-exp(5)+ x),x, algorithm="giac")
Output:
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 = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=x^3-\frac {1}{x}-\frac {x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3\,{\mathrm {e}}^{{\mathrm {e}}^5}}{3} \] Input:
int(-(exp(exp(5) - x)*((exp(3)*(x^2 - x^3))/3 - (exp(x - exp(5))*(9*x^4 + 3))/3))/x^2,x)
Output:
x^3 - 1/x - (x*exp(-x)*exp(3)*exp(exp(5)))/3
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{e^5-x} \left (e^3 \left (-x^2+x^3\right )+e^{-e^5+x} \left (3+9 x^4\right )\right )}{3 x^2} \, dx=\frac {-e^{e^{5}} e^{3} x^{2}+3 e^{x} x^{4}-3 e^{x}}{3 e^{x} x} \] Input:
int(1/3*((9*x^4+3)*exp(-exp(5)+x)+(x^3-x^2)*exp(3))/x^2/exp(-exp(5)+x),x)
Output:
( - e**(e**5)*e**3*x**2 + 3*e**x*x**4 - 3*e**x)/(3*e**x*x)