Integrand size = 75, antiderivative size = 29 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=9-e^{\frac {e^{25-\frac {-3+x}{e^5 x}}}{3 x}}+x \] Output:
9+x-exp(1/3*exp(25-(-3+x)/exp(5)/x)/x)
Time = 0.74 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=-e^{\frac {e^{25+\frac {-1+\frac {3}{x}}{e^5}}}{3 x}}+x \] Input:
Integrate[(3*E^5*x^3 + E^(E^((3 - x + 25*E^5*x)/(E^5*x))/(3*x) + (3 - x + 25*E^5*x)/(E^5*x))*(3 + E^5*x))/(3*E^5*x^3),x]
Output:
-E^(E^(25 + (-1 + 3/x)/E^5)/(3*x)) + x
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.
\(\displaystyle \int \frac {\left (e^5 x+3\right ) \exp \left (\frac {25 e^5 x-x+3}{e^5 x}+\frac {e^{\frac {25 e^5 x-x+3}{e^5 x}}}{3 x}\right )+3 e^5 x^3}{3 e^5 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {3 e^5 x^3+\exp \left (\frac {25 e^5 x-x+3}{e^5 x}+\frac {e^{\frac {25 e^5 x-x+3}{e^5 x}}}{3 x}\right ) \left (e^5 x+3\right )}{x^3}dx}{3 e^5}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {\int \left (\frac {\exp \left (25 \left (1-\frac {1}{25 e^5}\right )+\frac {e^{-\frac {1}{e^5}+25+\frac {3}{e^5 x}}}{3 x}+\frac {3}{e^5 x}\right ) \left (e^5 x+3\right )}{x^3}+3 e^5\right )dx}{3 e^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\text {Subst}\left (\int \exp \left (\frac {1}{3} e^{\frac {3 x}{e^5}-\frac {1}{e^5}+25} x+\frac {3 x}{e^5}+5 \left (6-\frac {1}{5 e^5}\right )\right )dx,x,\frac {1}{x}\right )+3 \int \frac {\exp \left (25 \left (1-\frac {1}{25 e^5}\right )+\frac {e^{-\frac {1}{e^5}+25+\frac {3}{e^5 x}}}{3 x}+\frac {3}{e^5 x}\right )}{x^3}dx+3 e^5 x}{3 e^5}\) |
Input:
Int[(3*E^5*x^3 + E^(E^((3 - x + 25*E^5*x)/(E^5*x))/(3*x) + (3 - x + 25*E^5 *x)/(E^5*x))*(3 + E^5*x))/(3*E^5*x^3),x]
Output:
$Aborted
Time = 1.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
risch | \(x -{\mathrm e}^{\frac {{\mathrm e}^{\frac {\left (25 x \,{\mathrm e}^{5}+3-x \right ) {\mathrm e}^{-5}}{x}}}{3 x}}\) | \(28\) |
parts | \(x -{\mathrm e}^{\frac {{\mathrm e}^{\frac {\left (25 x \,{\mathrm e}^{5}+3-x \right ) {\mathrm e}^{-5}}{x}}}{3 x}}\) | \(30\) |
norman | \(\frac {x^{3}-x^{2} {\mathrm e}^{\frac {{\mathrm e}^{\frac {\left (25 x \,{\mathrm e}^{5}+3-x \right ) {\mathrm e}^{-5}}{x}}}{3 x}}}{x^{2}}\) | \(39\) |
parallelrisch | \(\frac {{\mathrm e}^{-5} \left (3 x \,{\mathrm e}^{5}-3 \,{\mathrm e}^{5} {\mathrm e}^{\frac {{\mathrm e}^{\frac {\left (25 x \,{\mathrm e}^{5}+3-x \right ) {\mathrm e}^{-5}}{x}}}{3 x}}\right )}{3}\) | \(42\) |
Input:
int(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((25*x*ex p(5)+3-x)/x/exp(5))/x)+3*x^3*exp(5))/x^3/exp(5),x,method=_RETURNVERBOSE)
Output:
x-exp(1/3*exp((25*x*exp(5)+3-x)/x*exp(-5))/x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx={\left (x e^{\left (\frac {{\left (25 \, x e^{5} - x + 3\right )} e^{\left (-5\right )}}{x}\right )} - e^{\left (\frac {{\left (75 \, x e^{5} - 3 \, x + e^{\left (\frac {{\left (25 \, x e^{5} - x + 3\right )} e^{\left (-5\right )}}{x} + 5\right )} + 9\right )} e^{\left (-5\right )}}{3 \, x}\right )}\right )} e^{\left (-\frac {{\left (25 \, x e^{5} - x + 3\right )} e^{\left (-5\right )}}{x}\right )} \] Input:
integrate(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((2 5*x*exp(5)+3-x)/x/exp(5))/x)+3*x^3*exp(5))/x^3/exp(5),x, algorithm="fricas ")
Output:
(x*e^((25*x*e^5 - x + 3)*e^(-5)/x) - e^(1/3*(75*x*e^5 - 3*x + e^((25*x*e^5 - x + 3)*e^(-5)/x + 5) + 9)*e^(-5)/x))*e^(-(25*x*e^5 - x + 3)*e^(-5)/x)
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=x - e^{\frac {e^{\frac {- x + 25 x e^{5} + 3}{x e^{5}}}}{3 x}} \] Input:
integrate(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((2 5*x*exp(5)+3-x)/x/exp(5))/x)+3*x**3*exp(5))/x**3/exp(5),x)
Output:
x - exp(exp((-x + 25*x*exp(5) + 3)*exp(-5)/x)/(3*x))
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx={\left (x e^{5} - e^{\left (\frac {e^{\left (\frac {3 \, e^{\left (-5\right )}}{x} - e^{\left (-5\right )} + 25\right )}}{3 \, x} + 5\right )}\right )} e^{\left (-5\right )} \] Input:
integrate(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((2 5*x*exp(5)+3-x)/x/exp(5))/x)+3*x^3*exp(5))/x^3/exp(5),x, algorithm="maxima ")
Output:
(x*e^5 - e^(1/3*e^(3*e^(-5)/x - e^(-5) + 25)/x + 5))*e^(-5)
\[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=\int { \frac {{\left (3 \, x^{3} e^{5} + {\left (x e^{5} + 3\right )} e^{\left (\frac {{\left (25 \, x e^{5} - x + 3\right )} e^{\left (-5\right )}}{x} + \frac {e^{\left (\frac {{\left (25 \, x e^{5} - x + 3\right )} e^{\left (-5\right )}}{x}\right )}}{3 \, x}\right )}\right )} e^{\left (-5\right )}}{3 \, x^{3}} \,d x } \] Input:
integrate(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((2 5*x*exp(5)+3-x)/x/exp(5))/x)+3*x^3*exp(5))/x^3/exp(5),x, algorithm="giac")
Output:
integrate(1/3*(3*x^3*e^5 + (x*e^5 + 3)*e^((25*x*e^5 - x + 3)*e^(-5)/x + 1/ 3*e^((25*x*e^5 - x + 3)*e^(-5)/x)/x))*e^(-5)/x^3, x)
Time = 1.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-5}}{x}}\,{\mathrm {e}}^{-{\mathrm {e}}^{-5}}\,{\mathrm {e}}^{25}}{3\,x}} \] Input:
int((exp(-5)*(x^3*exp(5) + (exp(exp((exp(-5)*(25*x*exp(5) - x + 3))/x)/(3* x))*exp((exp(-5)*(25*x*exp(5) - x + 3))/x)*(x*exp(5) + 3))/3))/x^3,x)
Output:
x - exp((exp((3*exp(-5))/x)*exp(-exp(-5))*exp(25))/(3*x))
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {3 e^5 x^3+e^{\frac {e^{\frac {3-x+25 e^5 x}{e^5 x}}}{3 x}+\frac {3-x+25 e^5 x}{e^5 x}} \left (3+e^5 x\right )}{3 e^5 x^3} \, dx=-e^{\frac {e^{\frac {3}{e^{5} x}} e^{25}}{3 e^{\frac {1}{e^{5}}} x}}+x \] Input:
int(1/3*((x*exp(5)+3)*exp((25*x*exp(5)+3-x)/x/exp(5))*exp(1/3*exp((25*x*ex p(5)+3-x)/x/exp(5))/x)+3*x^3*exp(5))/x^3/exp(5),x)
Output:
- e**((e**(3/(e**5*x))*e**25)/(3*e**(1/e**5)*x)) + x