Integrand size = 90, antiderivative size = 30 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=\log \left (\frac {1}{\left (3+e^{2 x} (7-x)-x\right ) \left (1+e^4+x^2\right )}\right ) \] Output:
ln(2/(2*(-x+7)*exp(x)^2-2*x+6)/(x^2+exp(4)+1))
Time = 4.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\log \left (3+7 e^{2 x}-x-e^{2 x} x\right )-\log \left (1+e^4+x^2\right ) \] Input:
Integrate[(-1 - E^4 + 6*x - 3*x^2 + E^(2*x)*(13 + E^4*(13 - 2*x) + 12*x + 11*x^2 - 2*x^3))/(-3 + E^4*(-3 + x) + x - 3*x^2 + x^3 + E^(2*x)*(-7 + E^4* (-7 + x) + x - 7*x^2 + x^3)),x]
Output:
-Log[3 + 7*E^(2*x) - x - E^(2*x)*x] - Log[1 + E^4 + x^2]
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.
\(\displaystyle \int \frac {-3 x^2+e^{2 x} \left (-2 x^3+11 x^2+12 x+e^4 (13-2 x)+13\right )+6 x-e^4-1}{x^3-3 x^2+e^{2 x} \left (x^3-7 x^2+x+e^4 (x-7)-7\right )+x+e^4 (x-3)-3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^2-e^{2 x} \left (-2 x^3+11 x^2+12 x+e^4 (13-2 x)+13\right )-6 x+e^4+1}{\left (-e^{2 x} x-x+7 e^{2 x}+3\right ) \left (x^2+e^4+1\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \left (x^2-10 x+23\right )}{(x-7) \left (e^{2 x} x+x-7 e^{2 x}-3\right )}+\frac {2 x^3-11 x^2-2 \left (6-e^4\right ) x-13 \left (1+e^4\right )}{(7-x) \left (x^2+e^4+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \int \frac {1}{e^{2 x} x+x-7 e^{2 x}-3}dx+4 \int \frac {1}{(x-7) \left (e^{2 x} x+x-7 e^{2 x}-3\right )}dx+2 \int \frac {x}{e^{2 x} x+x-7 e^{2 x}-3}dx-\log \left (x^2+e^4+1\right )-2 x-\log (7-x)\) |
Input:
Int[(-1 - E^4 + 6*x - 3*x^2 + E^(2*x)*(13 + E^4*(13 - 2*x) + 12*x + 11*x^2 - 2*x^3))/(-3 + E^4*(-3 + x) + x - 3*x^2 + x^3 + E^(2*x)*(-7 + E^4*(-7 + x) + x - 7*x^2 + x^3)),x]
Output:
$Aborted
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00
method | result | size |
norman | \(-\ln \left (x^{2}+{\mathrm e}^{4}+1\right )-\ln \left (x \,{\mathrm e}^{2 x}-7 \,{\mathrm e}^{2 x}+x -3\right )\) | \(30\) |
parallelrisch | \(-\ln \left (x^{2}+{\mathrm e}^{4}+1\right )-\ln \left (x \,{\mathrm e}^{2 x}-7 \,{\mathrm e}^{2 x}+x -3\right )\) | \(30\) |
risch | \(-\ln \left (x^{3}-7 x^{2}+\left ({\mathrm e}^{4}+1\right ) x -7 \,{\mathrm e}^{4}-7\right )-\ln \left ({\mathrm e}^{2 x}+\frac {-3+x}{-7+x}\right )\) | \(42\) |
Input:
int((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/ (((-7+x)*exp(4)+x^3-7*x^2+x-7)*exp(x)^2+(-3+x)*exp(4)+x^3-3*x^2+x-3),x,met hod=_RETURNVERBOSE)
Output:
-ln(x^2+exp(4)+1)-ln(x*exp(x)^2-7*exp(x)^2+x-3)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\log \left (x^{3} - 7 \, x^{2} + {\left (x - 7\right )} e^{4} + x - 7\right ) - \log \left (\frac {{\left (x - 7\right )} e^{\left (2 \, x\right )} + x - 3}{x - 7}\right ) \] Input:
integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6 *x-1)/(((-7+x)*exp(4)+x^3-7*x^2+x-7)*exp(x)^2+(-3+x)*exp(4)+x^3-3*x^2+x-3) ,x, algorithm="fricas")
Output:
-log(x^3 - 7*x^2 + (x - 7)*e^4 + x - 7) - log(((x - 7)*e^(2*x) + x - 3)/(x - 7))
Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=- \log {\left (e^{2 x} + \frac {x - 3}{x - 7} \right )} - \log {\left (x^{3} - 7 x^{2} + x \left (1 + e^{4}\right ) - 7 e^{4} - 7 \right )} \] Input:
integrate((((-2*x+13)*exp(4)-2*x**3+11*x**2+12*x+13)*exp(x)**2-exp(4)-3*x* *2+6*x-1)/(((-7+x)*exp(4)+x**3-7*x**2+x-7)*exp(x)**2+(-3+x)*exp(4)+x**3-3* x**2+x-3),x)
Output:
-log(exp(2*x) + (x - 3)/(x - 7)) - log(x**3 - 7*x**2 + x*(1 + exp(4)) - 7* exp(4) - 7)
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\log \left (x^{2} + e^{4} + 1\right ) - \log \left (x - 7\right ) - \log \left (\frac {{\left (x - 7\right )} e^{\left (2 \, x\right )} + x - 3}{x - 7}\right ) \] Input:
integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6 *x-1)/(((-7+x)*exp(4)+x^3-7*x^2+x-7)*exp(x)^2+(-3+x)*exp(4)+x^3-3*x^2+x-3) ,x, algorithm="maxima")
Output:
-log(x^2 + e^4 + 1) - log(x - 7) - log(((x - 7)*e^(2*x) + x - 3)/(x - 7))
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\log \left (x^{2} + e^{4} + 1\right ) - \log \left (x e^{\left (2 \, x\right )} + x - 7 \, e^{\left (2 \, x\right )} - 3\right ) \] Input:
integrate((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6 *x-1)/(((-7+x)*exp(4)+x^3-7*x^2+x-7)*exp(x)^2+(-3+x)*exp(4)+x^3-3*x^2+x-3) ,x, algorithm="giac")
Output:
-log(x^2 + e^4 + 1) - log(x*e^(2*x) + x - 7*e^(2*x) - 3)
Time = 3.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\ln \left (x^2+{\mathrm {e}}^4+1\right )-\ln \left (x-7\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}-3\right ) \] Input:
int(-(exp(4) - 6*x - exp(2*x)*(12*x + 11*x^2 - 2*x^3 - exp(4)*(2*x - 13) + 13) + 3*x^2 + 1)/(x + exp(2*x)*(x + exp(4)*(x - 7) - 7*x^2 + x^3 - 7) + e xp(4)*(x - 3) - 3*x^2 + x^3 - 3),x)
Output:
- log(exp(4) + x^2 + 1) - log(x - 7*exp(2*x) + x*exp(2*x) - 3)
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-1-e^4+6 x-3 x^2+e^{2 x} \left (13+e^4 (13-2 x)+12 x+11 x^2-2 x^3\right )}{-3+e^4 (-3+x)+x-3 x^2+x^3+e^{2 x} \left (-7+e^4 (-7+x)+x-7 x^2+x^3\right )} \, dx=-\mathrm {log}\left (e^{4}+x^{2}+1\right )-\mathrm {log}\left (e^{2 x} x -7 e^{2 x}+x -3\right ) \] Input:
int((((-2*x+13)*exp(4)-2*x^3+11*x^2+12*x+13)*exp(x)^2-exp(4)-3*x^2+6*x-1)/ (((-7+x)*exp(4)+x^3-7*x^2+x-7)*exp(x)^2+(-3+x)*exp(4)+x^3-3*x^2+x-3),x)
Output:
- (log(e**4 + x**2 + 1) + log(e**(2*x)*x - 7*e**(2*x) + x - 3))