Integrand size = 210, antiderivative size = 26 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x-\frac {2 e^3}{3-5 e^{-3+\frac {1}{x}}-e^x+x} \]
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x+\frac {2 e^6}{5 e^{\frac {1}{x}}+e^{3+x}-e^3 (3+x)} \]
Integrate[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x )*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*E^3*x^2 - 2*x^3) + E^((1 - 3*x + x*L og[5])/x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^(( 2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^(( 1 - 3*x + x*Log[5])/x)*(-6*x^2 + 2*E^x*x^2 - 2*x^3)),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.
\(\displaystyle \int \frac {x^4+6 x^3+e^{2 x} x^2+2 e^3 x^2+9 x^2+x^2 e^{\frac {2 (-3 x+x \log (5)+1)}{x}}+e^x \left (-2 x^3-2 e^3 x^2-6 x^2\right )+\left (-2 x^3+2 e^x x^2-6 x^2+2 e^3\right ) e^{\frac {-3 x+x \log (5)+1}{x}}}{x^4+6 x^3+e^{2 x} x^2+9 x^2+x^2 e^{\frac {2 (-3 x+x \log (5)+1)}{x}}+e^x \left (-2 x^3-6 x^2\right )+\left (-2 x^3+2 e^x x^2-6 x^2\right ) e^{\frac {-3 x+x \log (5)+1}{x}}} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^4+6 x^3+e^{2 x} x^2+\left (9+2 e^3\right ) x^2+x^2 e^{\frac {2 (-3 x+x \log (5)+1)}{x}}+e^x \left (-2 x^3-2 e^3 x^2-6 x^2\right )+\left (-2 x^3+2 e^x x^2-6 x^2+2 e^3\right ) e^{\frac {-3 x+x \log (5)+1}{x}}}{x^4+6 x^3+e^{2 x} x^2+9 x^2+x^2 e^{\frac {2 (-3 x+x \log (5)+1)}{x}}+e^x \left (-2 x^3-6 x^2\right )+\left (-2 x^3+2 e^x x^2-6 x^2\right ) e^{\frac {-3 x+x \log (5)+1}{x}}}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^6 \left (x^4+6 x^3+e^{2 x} x^2+\left (9+2 e^3\right ) x^2+x^2 e^{\frac {2 (-3 x+x \log (5)+1)}{x}}+e^x \left (-2 x^3-2 e^3 x^2-6 x^2\right )+\left (-2 x^3+2 e^x x^2-6 x^2+2 e^3\right ) e^{\frac {-3 x+x \log (5)+1}{x}}\right )}{x^2 \left (e^3 x-e^{x+3}-5 e^{\frac {1}{x}}+3 e^3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^6 \int \frac {x^4+6 x^3+25 e^{\frac {2 (1-3 x)}{x}} x^2+e^{2 x} x^2+\left (9+2 e^3\right ) x^2+10 e^{\frac {1-3 x}{x}} \left (-x^3+e^x x^2-3 x^2+e^3\right )-2 e^x \left (x^3+e^3 x^2+3 x^2\right )}{x^2 \left (e^3 x-e^{x+3}-5 e^{\frac {1}{x}}+3 e^3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle e^6 \int \left (-\frac {2 \left (e^3 x^3-5 e^{\frac {1}{x}} x^2+2 e^3 x^2-5 e^{\frac {1}{x}}\right )}{x^2 \left (e^3 x-e^{x+3}-5 e^{\frac {1}{x}}+3 e^3\right )^2}-\frac {2}{-e^3 x+e^{x+3}+5 e^{\frac {1}{x}}-3 e^3}+\frac {1}{e^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^6 \left (10 \int \frac {e^{\frac {1}{x}}}{x^2 \left (-e^3 x+e^{x+3}+5 e^{\frac {1}{x}}-3 e^3\right )^2}dx-4 e^3 \int \frac {1}{\left (-e^3 x+e^{x+3}+5 e^{\frac {1}{x}}-3 e^3\right )^2}dx+10 \int \frac {e^{\frac {1}{x}}}{\left (-e^3 x+e^{x+3}+5 e^{\frac {1}{x}}-3 e^3\right )^2}dx-2 \int \frac {1}{-e^3 x+e^{x+3}+5 e^{\frac {1}{x}}-3 e^3}dx-2 e^3 \int \frac {x}{\left (e^3 x-e^{x+3}-5 e^{\frac {1}{x}}+3 e^3\right )^2}dx+\frac {x}{e^6}\right )\) |
Int[(9*x^2 + 2*E^3*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*E^3*x^2 - 2*x^3) + E^((1 - 3*x + x*Log[5]) /x)*(2*E^3 - 6*x^2 + 2*E^x*x^2 - 2*x^3))/(9*x^2 + E^(2*x)*x^2 + E^((2*(1 - 3*x + x*Log[5]))/x)*x^2 + 6*x^3 + x^4 + E^x*(-6*x^2 - 2*x^3) + E^((1 - 3* x + x*Log[5])/x)*(-6*x^2 + 2*E^x*x^2 - 2*x^3)),x]
3.15.31.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12
method | result | size |
risch | \(x -\frac {2 \,{\mathrm e}^{3}}{x -{\mathrm e}^{x}-5 \,{\mathrm e}^{-\frac {-1+3 x}{x}}+3}\) | \(29\) |
parallelrisch | \(-\frac {-x^{2}+{\mathrm e}^{x} x +x \,{\mathrm e}^{\frac {x \ln \left (5\right )-3 x +1}{x}}+2 \,{\mathrm e}^{3}-3 x}{x -{\mathrm e}^{x}-{\mathrm e}^{\frac {x \ln \left (5\right )-3 x +1}{x}}+3}\) | \(61\) |
int((x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^2)*exp( (x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x)+2*x^2*e xp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*ln(5)-3*x+1)/x)^2+(2*exp(x)*x^2-2*x^3-6 *x^2)*exp((x*ln(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+x^4+6*x^3+ 9*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\frac {x^{2} - x e^{x} - x e^{\left (\frac {x \log \left (5\right ) - 3 \, x + 1}{x}\right )} + 3 \, x - 2 \, e^{3}}{x - e^{x} - e^{\left (\frac {x \log \left (5\right ) - 3 \, x + 1}{x}\right )} + 3} \]
integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^ 2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x) +2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^ 2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+ x^4+6*x^3+9*x^2),x, algorithm=\
(x^2 - x*e^x - x*e^((x*log(5) - 3*x + 1)/x) + 3*x - 2*e^3)/(x - e^x - e^(( x*log(5) - 3*x + 1)/x) + 3)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x + \frac {2 e^{3}}{- x + e^{x} + e^{\frac {- 3 x + x \log {\left (5 \right )} + 1}{x}} - 3} \]
integrate((x**2*exp((x*ln(5)-3*x+1)/x)**2+(2*exp(x)*x**2+2*exp(3)-2*x**3-6 *x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)**2*x**2+(-2*x**2*exp(3)-2*x**3-6*x**2 )*exp(x)+2*x**2*exp(3)+x**4+6*x**3+9*x**2)/(x**2*exp((x*ln(5)-3*x+1)/x)**2 +(2*exp(x)*x**2-2*x**3-6*x**2)*exp((x*ln(5)-3*x+1)/x)+exp(x)**2*x**2+(-2*x **3-6*x**2)*exp(x)+x**4+6*x**3+9*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\frac {x^{2} e^{3} + 3 \, x e^{3} - x e^{\left (x + 3\right )} - 5 \, x e^{\frac {1}{x}} - 2 \, e^{6}}{x e^{3} + 3 \, e^{3} - e^{\left (x + 3\right )} - 5 \, e^{\frac {1}{x}}} \]
integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^ 2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x) +2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^ 2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+ x^4+6*x^3+9*x^2),x, algorithm=\
(x^2*e^3 + 3*x*e^3 - x*e^(x + 3) - 5*x*e^(1/x) - 2*e^6)/(x*e^3 + 3*e^3 - e ^(x + 3) - 5*e^(1/x))
Timed out. \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=\text {Timed out} \]
integrate((x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^2+2*exp(3)-2*x^3-6*x^ 2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^2*exp(3)-2*x^3-6*x^2)*exp(x) +2*x^2*exp(3)+x^4+6*x^3+9*x^2)/(x^2*exp((x*log(5)-3*x+1)/x)^2+(2*exp(x)*x^ 2-2*x^3-6*x^2)*exp((x*log(5)-3*x+1)/x)+exp(x)^2*x^2+(-2*x^3-6*x^2)*exp(x)+ x^4+6*x^3+9*x^2),x, algorithm=\
Time = 8.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {9 x^2+2 e^3 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 e^3 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (2 e^3-6 x^2+2 e^x x^2-2 x^3\right )}{9 x^2+e^{2 x} x^2+e^{\frac {2 (1-3 x+x \log (5))}{x}} x^2+6 x^3+x^4+e^x \left (-6 x^2-2 x^3\right )+e^{\frac {1-3 x+x \log (5)}{x}} \left (-6 x^2+2 e^x x^2-2 x^3\right )} \, dx=x-\frac {2\,{\mathrm {e}}^3}{x-{\mathrm {e}}^x-5\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-3}+3} \]
int((exp((x*log(5) - 3*x + 1)/x)*(2*exp(3) + 2*x^2*exp(x) - 6*x^2 - 2*x^3) + x^2*exp((2*(x*log(5) - 3*x + 1))/x) + x^2*exp(2*x) + 2*x^2*exp(3) - exp (x)*(2*x^2*exp(3) + 6*x^2 + 2*x^3) + 9*x^2 + 6*x^3 + x^4)/(x^2*exp((2*(x*l og(5) - 3*x + 1))/x) - exp(x)*(6*x^2 + 2*x^3) - exp((x*log(5) - 3*x + 1)/x )*(6*x^2 - 2*x^2*exp(x) + 2*x^3) + x^2*exp(2*x) + 9*x^2 + 6*x^3 + x^4),x)