Integrand size = 178, antiderivative size = 32 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=\frac {\frac {2}{x}-\frac {2 \left (5-e^5\right )}{x-\left (e^x+x\right )^2}}{e^2} \]
Time = 5.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=-\frac {2 \left (-\frac {1}{x}+\frac {-5+e^5}{e^{2 x}-x+2 e^x x+x^2}\right )}{e^2} \]
Integrate[(-2*E^(4*x) - 8*E^(3*x)*x + 8*x^2 - 16*x^3 - 2*x^4 + E^(2*x)*(4* x - 32*x^2 + 4*E^5*x^2) + E^5*(-2*x^2 + 4*x^3) + E^x*(-12*x^2 - 28*x^3 + E ^5*(4*x^2 + 4*x^3)))/(E^(2 + 4*x)*x^2 + 4*E^(2 + 3*x)*x^3 + E^(2 + 2*x)*(- 2*x^3 + 6*x^4) + E^(2 + x)*(-4*x^4 + 4*x^5) + E^2*(x^4 - 2*x^5 + x^6)),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.
\(\displaystyle \int \frac {-2 x^4-16 x^3+8 x^2+e^{2 x} \left (4 e^5 x^2-32 x^2+4 x\right )+e^5 \left (4 x^3-2 x^2\right )+e^x \left (-28 x^3-12 x^2+e^5 \left (4 x^3+4 x^2\right )\right )-8 e^{3 x} x-2 e^{4 x}}{4 e^{3 x+2} x^3+e^{4 x+2} x^2+e^{x+2} \left (4 x^5-4 x^4\right )+e^{2 x+2} \left (6 x^4-2 x^3\right )+e^2 \left (x^6-2 x^5+x^4\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^4-16 x^3+8 x^2+e^{2 x} \left (4 e^5 x^2-32 x^2+4 x\right )+e^5 \left (4 x^3-2 x^2\right )+e^x \left (-28 x^3-12 x^2+e^5 \left (4 x^3+4 x^2\right )\right )-8 e^{3 x} x-2 e^{4 x}}{e^2 x^2 \left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {2 \left (x^4+8 x^3-4 x^2+4 e^{3 x} x+e^{4 x}-2 e^{2 x} \left (e^5 x^2-8 x^2+x\right )+e^5 \left (x^2-2 x^3\right )+2 e^x \left (7 x^3+3 x^2-e^5 \left (x^3+x^2\right )\right )\right )}{x^2 \left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx}{e^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \int \frac {x^4+8 x^3-4 x^2+4 e^{3 x} x+e^{4 x}-2 e^{2 x} \left (e^5 x^2-8 x^2+x\right )+e^5 \left (x^2-2 x^3\right )+2 e^x \left (7 x^3+3 x^2-e^5 \left (x^3+x^2\right )\right )}{x^2 \left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx}{e^2}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2 \int \left (\frac {\left (-5+e^5\right ) \left (2 x^2+2 e^x x-4 x-2 e^x+1\right )}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}-\frac {2 \left (-5+e^5\right )}{x^2+2 e^x x-x+e^{2 x}}+\frac {1}{x^2}\right )dx}{e^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (-\left (\left (5-e^5\right ) \int \frac {1}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx\right )+2 \left (5-e^5\right ) \int \frac {e^x}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx+4 \left (5-e^5\right ) \int \frac {x}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx-2 \left (5-e^5\right ) \int \frac {e^x x}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx-2 \left (5-e^5\right ) \int \frac {x^2}{\left (x^2+2 e^x x-x+e^{2 x}\right )^2}dx+2 \left (5-e^5\right ) \int \frac {1}{x^2+2 e^x x-x+e^{2 x}}dx-\frac {1}{x}\right )}{e^2}\) |
Int[(-2*E^(4*x) - 8*E^(3*x)*x + 8*x^2 - 16*x^3 - 2*x^4 + E^(2*x)*(4*x - 32 *x^2 + 4*E^5*x^2) + E^5*(-2*x^2 + 4*x^3) + E^x*(-12*x^2 - 28*x^3 + E^5*(4* x^2 + 4*x^3)))/(E^(2 + 4*x)*x^2 + 4*E^(2 + 3*x)*x^3 + E^(2 + 2*x)*(-2*x^3 + 6*x^4) + E^(2 + x)*(-4*x^4 + 4*x^5) + E^2*(x^4 - 2*x^5 + x^6)),x]
3.26.10.3.1 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 = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66
method | result | size |
parallelrisch | \(-\frac {\left (2 x \,{\mathrm e}^{5}-2 x^{2}-4 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{2 x}-8 x \right ) {\mathrm e}^{-2}}{x \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}-x \right )}\) | \(53\) |
risch | \(\frac {2 \,{\mathrm e}^{-2}}{x}-\frac {2 \,{\mathrm e}^{-2} {\mathrm e}^{5}}{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}-x}+\frac {10 \,{\mathrm e}^{-2}}{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}-x}\) | \(55\) |
int((-2*exp(x)^4-8*x*exp(x)^3+(4*x^2*exp(5)-32*x^2+4*x)*exp(x)^2+((4*x^3+4 *x^2)*exp(5)-28*x^3-12*x^2)*exp(x)+(4*x^3-2*x^2)*exp(5)-2*x^4-16*x^3+8*x^2 )/(x^2*exp(2)*exp(x)^4+4*x^3*exp(2)*exp(x)^3+(6*x^4-2*x^3)*exp(2)*exp(x)^2 +(4*x^5-4*x^4)*exp(2)*exp(x)+(x^6-2*x^5+x^4)*exp(2)),x,method=_RETURNVERBO SE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=-\frac {2 \, {\left (x e^{9} - {\left (x^{2} + 4 \, x\right )} e^{4} - 2 \, x e^{\left (x + 4\right )} - e^{\left (2 \, x + 4\right )}\right )}}{2 \, x^{2} e^{\left (x + 6\right )} + {\left (x^{3} - x^{2}\right )} e^{6} + x e^{\left (2 \, x + 6\right )}} \]
integrate((-2*exp(x)^4-8*x*exp(x)^3+(4*x^2*exp(5)-32*x^2+4*x)*exp(x)^2+((4 *x^3+4*x^2)*exp(5)-28*x^3-12*x^2)*exp(x)+(4*x^3-2*x^2)*exp(5)-2*x^4-16*x^3 +8*x^2)/(x^2*exp(2)*exp(x)^4+4*x^3*exp(2)*exp(x)^3+(6*x^4-2*x^3)*exp(2)*ex p(x)^2+(4*x^5-4*x^4)*exp(2)*exp(x)+(x^6-2*x^5+x^4)*exp(2)),x, algorithm=\
-2*(x*e^9 - (x^2 + 4*x)*e^4 - 2*x*e^(x + 4) - e^(2*x + 4))/(2*x^2*e^(x + 6 ) + (x^3 - x^2)*e^6 + x*e^(2*x + 6))
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=\frac {10 - 2 e^{5}}{x^{2} e^{2} + 2 x e^{2} e^{x} - x e^{2} + e^{2} e^{2 x}} + \frac {2}{x e^{2}} \]
integrate((-2*exp(x)**4-8*x*exp(x)**3+(4*x**2*exp(5)-32*x**2+4*x)*exp(x)** 2+((4*x**3+4*x**2)*exp(5)-28*x**3-12*x**2)*exp(x)+(4*x**3-2*x**2)*exp(5)-2 *x**4-16*x**3+8*x**2)/(x**2*exp(2)*exp(x)**4+4*x**3*exp(2)*exp(x)**3+(6*x* *4-2*x**3)*exp(2)*exp(x)**2+(4*x**5-4*x**4)*exp(2)*exp(x)+(x**6-2*x**5+x** 4)*exp(2)),x)
(10 - 2*exp(5))/(x**2*exp(2) + 2*x*exp(2)*exp(x) - x*exp(2) + exp(2)*exp(2 *x)) + 2*exp(-2)/x
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=\frac {2 \, {\left (x^{2} - x {\left (e^{5} - 4\right )} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}}{x^{3} e^{2} - x^{2} e^{2} + 2 \, x^{2} e^{\left (x + 2\right )} + x e^{\left (2 \, x + 2\right )}} \]
integrate((-2*exp(x)^4-8*x*exp(x)^3+(4*x^2*exp(5)-32*x^2+4*x)*exp(x)^2+((4 *x^3+4*x^2)*exp(5)-28*x^3-12*x^2)*exp(x)+(4*x^3-2*x^2)*exp(5)-2*x^4-16*x^3 +8*x^2)/(x^2*exp(2)*exp(x)^4+4*x^3*exp(2)*exp(x)^3+(6*x^4-2*x^3)*exp(2)*ex p(x)^2+(4*x^5-4*x^4)*exp(2)*exp(x)+(x^6-2*x^5+x^4)*exp(2)),x, algorithm=\
2*(x^2 - x*(e^5 - 4) + 2*x*e^x + e^(2*x))/(x^3*e^2 - x^2*e^2 + 2*x^2*e^(x + 2) + x*e^(2*x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=\frac {2 \, {\left (x^{2} e^{2} - 2 \, x e^{7} + 9 \, x e^{2} + 2 \, x e^{\left (x + 2\right )} + e^{\left (2 \, x + 2\right )}\right )}}{x^{3} e^{4} - x^{2} e^{4} + 2 \, x^{2} e^{\left (x + 4\right )} + x e^{\left (2 \, x + 4\right )}} \]
integrate((-2*exp(x)^4-8*x*exp(x)^3+(4*x^2*exp(5)-32*x^2+4*x)*exp(x)^2+((4 *x^3+4*x^2)*exp(5)-28*x^3-12*x^2)*exp(x)+(4*x^3-2*x^2)*exp(5)-2*x^4-16*x^3 +8*x^2)/(x^2*exp(2)*exp(x)^4+4*x^3*exp(2)*exp(x)^3+(6*x^4-2*x^3)*exp(2)*ex p(x)^2+(4*x^5-4*x^4)*exp(2)*exp(x)+(x^6-2*x^5+x^4)*exp(2)),x, algorithm=\
2*(x^2*e^2 - 2*x*e^7 + 9*x*e^2 + 2*x*e^(x + 2) + e^(2*x + 2))/(x^3*e^4 - x ^2*e^4 + 2*x^2*e^(x + 4) + x*e^(2*x + 4))
Time = 13.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81 \[ \int \frac {-2 e^{4 x}-8 e^{3 x} x+8 x^2-16 x^3-2 x^4+e^{2 x} \left (4 x-32 x^2+4 e^5 x^2\right )+e^5 \left (-2 x^2+4 x^3\right )+e^x \left (-12 x^2-28 x^3+e^5 \left (4 x^2+4 x^3\right )\right )}{e^{2+4 x} x^2+4 e^{2+3 x} x^3+e^{2+2 x} \left (-2 x^3+6 x^4\right )+e^{2+x} \left (-4 x^4+4 x^5\right )+e^2 \left (x^4-2 x^5+x^6\right )} \, dx=\frac {2\,{\mathrm {e}}^{2\,x}+x\,\left (4\,{\mathrm {e}}^x-2\,{\mathrm {e}}^5+8\right )+2\,x^2}{x\,{\mathrm {e}}^{2\,x+2}+2\,x^2\,{\mathrm {e}}^{x+2}-x^2\,{\mathrm {e}}^2+x^3\,{\mathrm {e}}^2} \]
int(-(2*exp(4*x) + 8*x*exp(3*x) + exp(5)*(2*x^2 - 4*x^3) + exp(x)*(12*x^2 - exp(5)*(4*x^2 + 4*x^3) + 28*x^3) - exp(2*x)*(4*x + 4*x^2*exp(5) - 32*x^2 ) - 8*x^2 + 16*x^3 + 2*x^4)/(exp(2)*(x^4 - 2*x^5 + x^6) - exp(2)*exp(x)*(4 *x^4 - 4*x^5) - exp(2*x)*exp(2)*(2*x^3 - 6*x^4) + x^2*exp(4*x)*exp(2) + 4* x^3*exp(3*x)*exp(2)),x)