Integrand size = 239, antiderivative size = 36 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\left (e^x x-\frac {2}{e^x+x}\right )^2+\log \left (-x+\frac {1}{5} \left (-\frac {2}{x}+x\right )\right ) \]
Time = 9.69 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {\left (-2+e^{2 x} x+e^x x^2\right )^2}{\left (e^x+x\right )^2}-\log (x)+\log \left (1+2 x^2\right ) \]
Integrate[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) + E^(4*x)*(-6*x^3 - 6*x^4 - 12*x^5 - 12*x^6) + E^x*(8*x - 3*x^2 + 16*x^3 + 10*x^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 - 12 *x^6 - 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 4*x^7 - 4*x^8))/(x^4 + 2*x^6 + E^(3*x)*(x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^ 4) + E^x*(3*x^3 + 6*x^5)),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.
\(\displaystyle \int \frac {2 x^5+15 x^3+e^{4 x} \left (-12 x^6-12 x^5-6 x^4-6 x^3\right )+e^x \left (8 x^6+10 x^4+16 x^3-3 x^2+8 x\right )+e^{5 x} \left (-4 x^5-4 x^4-2 x^3-2 x^2\right )+e^{3 x} \left (-12 x^7-12 x^6-6 x^5-6 x^4+8 x^3+2 x^2+4 x-1\right )+e^{2 x} \left (-4 x^8-4 x^7-2 x^6+6 x^5+8 x^4+10 x^3+4 x^2-3 x\right )+8 x}{2 x^6+x^4+e^{3 x} \left (2 x^3+x\right )+e^x \left (6 x^5+3 x^3\right )+e^{2 x} \left (6 x^4+3 x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^5+15 x^3+e^{4 x} \left (-12 x^6-12 x^5-6 x^4-6 x^3\right )+e^x \left (8 x^6+10 x^4+16 x^3-3 x^2+8 x\right )+e^{5 x} \left (-4 x^5-4 x^4-2 x^3-2 x^2\right )+e^{3 x} \left (-12 x^7-12 x^6-6 x^5-6 x^4+8 x^3+2 x^2+4 x-1\right )+e^{2 x} \left (-4 x^8-4 x^7-2 x^6+6 x^5+8 x^4+10 x^3+4 x^2-3 x\right )+8 x}{x \left (x+e^x\right )^3 \left (2 x^2+1\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {4 \left (x^3-x^2-2\right )}{\left (x+e^x\right )^2}+\frac {8 x^3+2 x^2+4 x-1}{x \left (2 x^2+1\right )}-\frac {8 (x-1)}{\left (x+e^x\right )^3}-2 e^{2 x} x (x+1)+\frac {4 (x-2) x}{x+e^x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {x^3}{\left (x+e^x\right )^2}dx+4 \int \frac {x^2}{\left (x+e^x\right )^2}dx+4 \int \frac {x^2}{x+e^x}dx-8 \int \frac {x}{x+e^x}dx-e^{2 x} x^2+\log \left (2 x^2+1\right )+4 x-\frac {4}{\left (x+e^x\right )^2}-\log (x)\) |
Int[(8*x + 15*x^3 + 2*x^5 + E^(5*x)*(-2*x^2 - 2*x^3 - 4*x^4 - 4*x^5) + E^( 4*x)*(-6*x^3 - 6*x^4 - 12*x^5 - 12*x^6) + E^x*(8*x - 3*x^2 + 16*x^3 + 10*x ^4 + 8*x^6) + E^(3*x)*(-1 + 4*x + 2*x^2 + 8*x^3 - 6*x^4 - 6*x^5 - 12*x^6 - 12*x^7) + E^(2*x)*(-3*x + 4*x^2 + 10*x^3 + 8*x^4 + 6*x^5 - 2*x^6 - 4*x^7 - 4*x^8))/(x^4 + 2*x^6 + E^(3*x)*(x + 2*x^3) + E^(2*x)*(3*x^2 + 6*x^4) + E ^x*(3*x^3 + 6*x^5)),x]
3.1.14.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25
method | result | size |
risch | \(4 x -\ln \left (x \right )+\ln \left (2 x^{2}+1\right )-{\mathrm e}^{2 x} x^{2}-\frac {4 \left (x^{3}+{\mathrm e}^{x} x^{2}+1\right )}{\left ({\mathrm e}^{x}+x \right )^{2}}\) | \(45\) |
parallelrisch | \(-\frac {{\mathrm e}^{2 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}+x^{2} {\mathrm e}^{4 x}+x^{2} \ln \left (x \right )+2 x \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x} \ln \left (x \right )-\ln \left (\frac {1}{2}+x^{2}\right ) x^{2}-2 \ln \left (\frac {1}{2}+x^{2}\right ) x \,{\mathrm e}^{x}-\ln \left (\frac {1}{2}+x^{2}\right ) {\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x^{2}+4-4 x \,{\mathrm e}^{2 x}}{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}\) | \(113\) |
int(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3)*exp( x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8-4*x^7 -2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3*x^2+8 *x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x)^2+(6 *x^5+3*x^3)*exp(x)+2*x^6+x^4),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (2 \, x^{2} + 1\right ) + {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \]
integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 )*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 -4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 *x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x )^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm=\
-(2*x^3*e^(3*x) + x^2*e^(4*x) - 4*x^2*e^x + (x^4 - 4*x)*e^(2*x) - (x^2 + 2 *x*e^x + e^(2*x))*log(2*x^2 + 1) + (x^2 + 2*x*e^x + e^(2*x))*log(x) + 4)/( x^2 + 2*x*e^x + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=- x^{2} e^{2 x} + 4 x - \log {\left (x \right )} + \log {\left (2 x^{2} + 1 \right )} + \frac {- 4 x^{3} - 4 x^{2} e^{x} - 4}{x^{2} + 2 x e^{x} + e^{2 x}} \]
integrate(((-4*x**5-4*x**4-2*x**3-2*x**2)*exp(x)**5+(-12*x**6-12*x**5-6*x* *4-6*x**3)*exp(x)**4+(-12*x**7-12*x**6-6*x**5-6*x**4+8*x**3+2*x**2+4*x-1)* exp(x)**3+(-4*x**8-4*x**7-2*x**6+6*x**5+8*x**4+10*x**3+4*x**2-3*x)*exp(x)* *2+(8*x**6+10*x**4+16*x**3-3*x**2+8*x)*exp(x)+2*x**5+15*x**3+8*x)/((2*x**3 +x)*exp(x)**3+(6*x**4+3*x**2)*exp(x)**2+(6*x**5+3*x**3)*exp(x)+2*x**6+x**4 ),x)
-x**2*exp(2*x) + 4*x - log(x) + log(2*x**2 + 1) + (-4*x**3 - 4*x**2*exp(x) - 4)/(x**2 + 2*x*exp(x) + exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.89 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} + {\left (x^{4} - 4 \, x\right )} e^{\left (2 \, x\right )} + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} + \log \left (2 \, x^{2} + 1\right ) - \log \left (x\right ) \]
integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 )*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 -4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 *x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x )^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm=\
-(2*x^3*e^(3*x) + x^2*e^(4*x) - 4*x^2*e^x + (x^4 - 4*x)*e^(2*x) + 4)/(x^2 + 2*x*e^x + e^(2*x)) + log(2*x^2 + 1) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (28) = 56\).
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.28 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=-\frac {x^{4} e^{\left (2 \, x\right )} + 2 \, x^{3} e^{\left (3 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 4 \, x^{2} e^{x} - x^{2} \log \left (2 \, x^{2} + 1\right ) - 2 \, x e^{x} \log \left (2 \, x^{2} + 1\right ) + x^{2} \log \left (x\right ) + 2 \, x e^{x} \log \left (x\right ) - 4 \, x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (2 \, x^{2} + 1\right ) + e^{\left (2 \, x\right )} \log \left (x\right ) + 4}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} \]
integrate(((-4*x^5-4*x^4-2*x^3-2*x^2)*exp(x)^5+(-12*x^6-12*x^5-6*x^4-6*x^3 )*exp(x)^4+(-12*x^7-12*x^6-6*x^5-6*x^4+8*x^3+2*x^2+4*x-1)*exp(x)^3+(-4*x^8 -4*x^7-2*x^6+6*x^5+8*x^4+10*x^3+4*x^2-3*x)*exp(x)^2+(8*x^6+10*x^4+16*x^3-3 *x^2+8*x)*exp(x)+2*x^5+15*x^3+8*x)/((2*x^3+x)*exp(x)^3+(6*x^4+3*x^2)*exp(x )^2+(6*x^5+3*x^3)*exp(x)+2*x^6+x^4),x, algorithm=\
-(x^4*e^(2*x) + 2*x^3*e^(3*x) + x^2*e^(4*x) - 4*x^2*e^x - x^2*log(2*x^2 + 1) - 2*x*e^x*log(2*x^2 + 1) + x^2*log(x) + 2*x*e^x*log(x) - 4*x*e^(2*x) - e^(2*x)*log(2*x^2 + 1) + e^(2*x)*log(x) + 4)/(x^2 + 2*x*e^x + e^(2*x))
Time = 0.54 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int \frac {8 x+15 x^3+2 x^5+e^{5 x} \left (-2 x^2-2 x^3-4 x^4-4 x^5\right )+e^{4 x} \left (-6 x^3-6 x^4-12 x^5-12 x^6\right )+e^x \left (8 x-3 x^2+16 x^3+10 x^4+8 x^6\right )+e^{3 x} \left (-1+4 x+2 x^2+8 x^3-6 x^4-6 x^5-12 x^6-12 x^7\right )+e^{2 x} \left (-3 x+4 x^2+10 x^3+8 x^4+6 x^5-2 x^6-4 x^7-4 x^8\right )}{x^4+2 x^6+e^{3 x} \left (x+2 x^3\right )+e^{2 x} \left (3 x^2+6 x^4\right )+e^x \left (3 x^3+6 x^5\right )} \, dx=4\,x+\ln \left (x^2+\frac {1}{2}\right )-\ln \left (x\right )-x^2\,{\mathrm {e}}^{2\,x}-\frac {4}{{\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^x+x^2}+\frac {4\,\left (x^2-x^3\right )}{\left (x+{\mathrm {e}}^x\right )\,\left (x-1\right )} \]
int((8*x - exp(3*x)*(6*x^4 - 2*x^2 - 8*x^3 - 4*x + 6*x^5 + 12*x^6 + 12*x^7 + 1) + exp(x)*(8*x - 3*x^2 + 16*x^3 + 10*x^4 + 8*x^6) - exp(2*x)*(3*x - 4 *x^2 - 10*x^3 - 8*x^4 - 6*x^5 + 2*x^6 + 4*x^7 + 4*x^8) - exp(5*x)*(2*x^2 + 2*x^3 + 4*x^4 + 4*x^5) - exp(4*x)*(6*x^3 + 6*x^4 + 12*x^5 + 12*x^6) + 15* x^3 + 2*x^5)/(exp(x)*(3*x^3 + 6*x^5) + exp(2*x)*(3*x^2 + 6*x^4) + exp(3*x) *(x + 2*x^3) + x^4 + 2*x^6),x)