Integrand size = 129, antiderivative size = 31 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=\frac {4 x^2}{e^{2 x}-x}-\log \left (3+e^{1-x} x\right ) \] Output:
4/(exp(2*x)-x)*x^2-ln(3+exp(1+ln(x)-x))
Time = 2.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=x+\frac {4 x^2}{e^{2 x}-x}-\log \left (3 e^x+e x\right ) \] Input:
Integrate[(-12*x^3 + E^(2*x)*(24*x^2 - 24*x^3) + E^(1 - x)*x*(E^(4*x)*(-1 + x) - x^2 - 3*x^3 + E^(2*x)*(2*x + 6*x^2 - 8*x^3)))/(3*E^(4*x)*x - 6*E^(2 *x)*x^2 + 3*x^3 + E^(1 - x)*x*(E^(4*x)*x - 2*E^(2*x)*x^2 + x^3)),x]
Output:
x + (4*x^2)/(E^(2*x) - x) - Log[3*E^x + E*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.
\(\displaystyle \int \frac {-12 x^3+e^{1-x} \left (-3 x^3-x^2+e^{2 x} \left (-8 x^3+6 x^2+2 x\right )+e^{4 x} (x-1)\right ) x+e^{2 x} \left (24 x^2-24 x^3\right )}{3 x^3-6 e^{2 x} x^2+e^{1-x} \left (x^3-2 e^{2 x} x^2+e^{4 x} x\right ) x+3 e^{4 x} x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^x \left (-12 x^3+e^{1-x} \left (-3 x^3-x^2+e^{2 x} \left (-8 x^3+6 x^2+2 x\right )+e^{4 x} (x-1)\right ) x+e^{2 x} \left (24 x^2-24 x^3\right )\right )}{\left (e^{2 x}-x\right )^2 x \left (e x+3 e^x\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {3 e^x (x-1)}{x \left (e x+3 e^x\right )}-\frac {4 e^{2 x} x (2 x-1)}{\left (e^{2 x}-x\right )^2}-\frac {3 x+1}{x}+\frac {4 e^{2 x}}{e^{2 x}-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {x^2}{\left (e^{2 x}-x\right )^2}dx+2 \int \frac {1}{e^{2 x}-x}dx+4 \int \frac {e^{2 x}}{e^{2 x}-x}dx+2 \int \frac {x}{\left (e^{2 x}-x\right )^2}dx-8 \int \frac {x}{e^{2 x}-x}dx+\frac {4 x^2}{e^{2 x}-x}-\frac {2 x}{e^{2 x}-x}-3 x-\log \left (\frac {3 e^x}{x}+e\right )-\log (x)\) |
Input:
Int[(-12*x^3 + E^(2*x)*(24*x^2 - 24*x^3) + E^(1 - x)*x*(E^(4*x)*(-1 + x) - x^2 - 3*x^3 + E^(2*x)*(2*x + 6*x^2 - 8*x^3)))/(3*E^(4*x)*x - 6*E^(2*x)*x^ 2 + 3*x^3 + E^(1 - x)*x*(E^(4*x)*x - 2*E^(2*x)*x^2 + x^3)),x]
Output:
$Aborted
Time = 1.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
risch | \(x -\frac {4 x^{2}}{x -{\mathrm e}^{2 x}}-\ln \left (\frac {x \,{\mathrm e}}{3}+{\mathrm e}^{x}\right )\) | \(29\) |
parallelrisch | \(\frac {-\ln \left (3+{\mathrm e}^{1+\ln \left (x \right )-x}\right ) x^{3}+\ln \left (3+{\mathrm e}^{1+\ln \left (x \right )-x}\right ) x^{2} {\mathrm e}^{2 x}-4 x^{4}}{x^{2} \left (x -{\mathrm e}^{2 x}\right )}\) | \(56\) |
Input:
int((((-1+x)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+ln(x) -x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)*e xp(1+ln(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x,method=_RETURNVERBOSE )
Output:
x-4*x^2/(x-exp(2*x))-ln(1/3*x*exp(1)+exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=\frac {4 \, x e^{\left (-2 \, x + 2 \, \log \left (x\right ) + 2\right )} - {\left (x e^{2} - e^{\left (-2 \, x + 2 \, \log \left (x\right ) + 2\right )}\right )} \log \left (e^{\left (-x + \log \left (x\right ) + 1\right )} + 3\right )}{x e^{2} - e^{\left (-2 \, x + 2 \, \log \left (x\right ) + 2\right )}} \] Input:
integrate((((-1+x)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1 +log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2 +x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="f ricas")
Output:
(4*x*e^(-2*x + 2*log(x) + 2) - (x*e^2 - e^(-2*x + 2*log(x) + 2))*log(e^(-x + log(x) + 1) + 3))/(x*e^2 - e^(-2*x + 2*log(x) + 2))
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=- 4 x - \frac {4 x}{x e^{- 2 x} - 1} - \log {\left (x \right )} - \log {\left (\frac {1}{\sqrt {e^{2 x}}} + \frac {3}{e x} \right )} \] Input:
integrate((((-1+x)*exp(2*x)**2+(-8*x**3+6*x**2+2*x)*exp(2*x)-3*x**3-x**2)* exp(1+ln(x)-x)+(-24*x**3+24*x**2)*exp(2*x)-12*x**3)/((x*exp(2*x)**2-2*exp( 2*x)*x**2+x**3)*exp(1+ln(x)-x)+3*x*exp(2*x)**2-6*exp(2*x)*x**2+3*x**3),x)
Output:
-4*x - 4*x/(x*exp(-2*x) - 1) - log(x) - log(1/sqrt(exp(2*x)) + 3*exp(-1)/x )
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=-\frac {3 \, x^{2} + x e^{\left (2 \, x\right )}}{x - e^{\left (2 \, x\right )}} - \log \left (\frac {1}{3} \, x e + e^{x}\right ) \] Input:
integrate((((-1+x)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1 +log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2 +x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="m axima")
Output:
-(3*x^2 + x*e^(2*x))/(x - e^(2*x)) - log(1/3*x*e + e^x)
Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=-\frac {7 \, x^{2} + x e^{\left (2 \, x\right )} + x \log \left (x e + 3 \, e^{x}\right ) - e^{\left (2 \, x\right )} \log \left (x e + 3 \, e^{x}\right )}{x - e^{\left (2 \, x\right )}} \] Input:
integrate((((-1+x)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1 +log(x)-x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2 +x^3)*exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x, algorithm="g iac")
Output:
-(7*x^2 + x*e^(2*x) + x*log(x*e + 3*e^x) - e^(2*x)*log(x*e + 3*e^x))/(x - e^(2*x))
Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=-\frac {x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{2\,x}\,\ln \left ({\mathrm {e}}^x+\frac {x\,\mathrm {e}}{3}\right )+x\,\ln \left ({\mathrm {e}}^x+\frac {x\,\mathrm {e}}{3}\right )+3\,x^2}{x-{\mathrm {e}}^{2\,x}} \] Input:
int((exp(log(x) - x + 1)*(exp(2*x)*(2*x + 6*x^2 - 8*x^3) + exp(4*x)*(x - 1 ) - x^2 - 3*x^3) + exp(2*x)*(24*x^2 - 24*x^3) - 12*x^3)/(3*x*exp(4*x) - 6* x^2*exp(2*x) + exp(log(x) - x + 1)*(x*exp(4*x) - 2*x^2*exp(2*x) + x^3) + 3 *x^3),x)
Output:
-(x*exp(2*x) - exp(2*x)*log(exp(x) + (x*exp(1))/3) + x*log(exp(x) + (x*exp (1))/3) + 3*x^2)/(x - exp(2*x))
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-12 x^3+e^{2 x} \left (24 x^2-24 x^3\right )+e^{1-x} x \left (e^{4 x} (-1+x)-x^2-3 x^3+e^{2 x} \left (2 x+6 x^2-8 x^3\right )\right )}{3 e^{4 x} x-6 e^{2 x} x^2+3 x^3+e^{1-x} x \left (e^{4 x} x-2 e^{2 x} x^2+x^3\right )} \, dx=\frac {-e^{2 x} \mathrm {log}\left (3 e^{x}+e x \right )+e^{2 x} x +\mathrm {log}\left (3 e^{x}+e x \right ) x +3 x^{2}}{e^{2 x}-x} \] Input:
int((((-1+x)*exp(2*x)^2+(-8*x^3+6*x^2+2*x)*exp(2*x)-3*x^3-x^2)*exp(1+log(x )-x)+(-24*x^3+24*x^2)*exp(2*x)-12*x^3)/((x*exp(2*x)^2-2*exp(2*x)*x^2+x^3)* exp(1+log(x)-x)+3*x*exp(2*x)^2-6*exp(2*x)*x^2+3*x^3),x)
Output:
( - e**(2*x)*log(3*e**x + e*x) + e**(2*x)*x + log(3*e**x + e*x)*x + 3*x**2 )/(e**(2*x) - x)