Integrand size = 72, antiderivative size = 24 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-\log ^2\left (x \left (1-e^2-x+16 x^5\right )\right ) \] Output:
x-ln((16*x^5-x-exp(1)^2+1)*x)^2
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-\log ^2\left (x \left (1-e^2-x+16 x^5\right )\right ) \] Input:
Integrate[(-x + E^2*x + x^2 - 16*x^6 + (2 - 2*E^2 - 4*x + 192*x^5)*Log[x - E^2*x - x^2 + 16*x^6])/(-x + E^2*x + x^2 - 16*x^6),x]
Output:
x - Log[x*(1 - E^2 - x + 16*x^5)]^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.
\(\displaystyle \int \frac {-16 x^6+x^2+\left (192 x^5-4 x-2 e^2+2\right ) \log \left (16 x^6-x^2-e^2 x+x\right )+e^2 x-x}{-16 x^6+x^2+e^2 x-x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-16 x^6+x^2+\left (192 x^5-4 x-2 e^2+2\right ) \log \left (16 x^6-x^2-e^2 x+x\right )+e^2 x-x}{-16 x^6+x^2+\left (e^2-1\right ) x}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-16 x^6+x^2+\left (192 x^5-4 x-2 e^2+2\right ) \log \left (16 x^6-x^2-e^2 x+x\right )+\left (e^2-1\right ) x}{-16 x^6+x^2+\left (e^2-1\right ) x}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-16 x^6+x^2+\left (192 x^5-4 x-2 e^2+2\right ) \log \left (16 x^6-x^2-e^2 x+x\right )+\left (e^2-1\right ) x}{x \left (-16 x^5+x+e^2-1\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (1-\frac {2 \left (96 x^5-2 x-e^2+1\right ) \log \left (x \left (16 x^5-x-e^2+1\right )\right )}{x \left (16 x^5-x-e^2+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {\int \frac {1}{-16 x^5+x+e^2-1}dx}{x}dx+2 \int \frac {\int \frac {1}{-16 x^5+x+e^2-1}dx}{-16 x^5+x+e^2-1}dx-2 \log \left (x \left (16 x^5-x-e^2+1\right )\right ) \int \frac {1}{-16 x^5+x+e^2-1}dx+2 \int \frac {\log (x)}{-16 x^5+x+e^2-1}dx-160 \int \frac {x^4 \int \frac {1}{-16 x^5+x+e^2-1}dx}{-16 x^5+x+e^2-1}dx-160 \int \frac {x^4 \log (x)}{-16 x^5+x+e^2-1}dx+160 \int \frac {x^4 \log \left (x \left (16 x^5-x-e^2+1\right )\right )}{-16 x^5+x+e^2-1}dx-2 \log \left (x \left (16 x^5-x-e^2+1\right )\right ) \log (x)+x+\log ^2(x)\) |
Input:
Int[(-x + E^2*x + x^2 - 16*x^6 + (2 - 2*E^2 - 4*x + 192*x^5)*Log[x - E^2*x - x^2 + 16*x^6])/(-x + E^2*x + x^2 - 16*x^6),x]
Output:
$Aborted
Time = 19.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
risch | \(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\) | \(25\) |
default | \(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\) | \(27\) |
norman | \(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\) | \(27\) |
parts | \(x -\ln \left (-{\mathrm e}^{2} x +16 x^{6}-x^{2}+x \right )^{2}\) | \(27\) |
Input:
int(((-2*exp(1)^2+192*x^5-4*x+2)*ln(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2-1 6*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-x),x,method=_RETURNVERBOSE)
Output:
x-ln(-exp(2)*x+16*x^6-x^2+x)^2
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=-\log \left (16 \, x^{6} - x^{2} - x e^{2} + x\right )^{2} + x \] Input:
integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp (1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-x),x, algorithm="fricas")
Output:
-log(16*x^6 - x^2 - x*e^2 + x)^2 + x
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x - \log {\left (16 x^{6} - x^{2} - x e^{2} + x \right )}^{2} \] Input:
integrate(((-2*exp(1)**2+192*x**5-4*x+2)*ln(-x*exp(1)**2+16*x**6-x**2+x)+x *exp(1)**2-16*x**6+x**2-x)/(x*exp(1)**2-16*x**6+x**2-x),x)
Output:
x - log(16*x**6 - x**2 - x*exp(2) + x)**2
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=-\log \left (16 \, x^{5} - x - e^{2} + 1\right )^{2} - 2 \, \log \left (16 \, x^{5} - x - e^{2} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + x \] Input:
integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp (1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-x),x, algorithm="maxima")
Output:
-log(16*x^5 - x - e^2 + 1)^2 - 2*log(16*x^5 - x - e^2 + 1)*log(x) - log(x) ^2 + x
\[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=\int { \frac {16 \, x^{6} - x^{2} - x e^{2} - 2 \, {\left (96 \, x^{5} - 2 \, x - e^{2} + 1\right )} \log \left (16 \, x^{6} - x^{2} - x e^{2} + x\right ) + x}{16 \, x^{6} - x^{2} - x e^{2} + x} \,d x } \] Input:
integrate(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp (1)^2-16*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-x),x, algorithm="giac")
Output:
integrate((16*x^6 - x^2 - x*e^2 - 2*(96*x^5 - 2*x - e^2 + 1)*log(16*x^6 - x^2 - x*e^2 + x) + x)/(16*x^6 - x^2 - x*e^2 + x), x)
Time = 3.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=x-{\ln \left (x-x\,{\mathrm {e}}^2-x^2+16\,x^6\right )}^2 \] Input:
int((x - x*exp(2) + log(x - x*exp(2) - x^2 + 16*x^6)*(4*x + 2*exp(2) - 192 *x^5 - 2) - x^2 + 16*x^6)/(x - x*exp(2) - x^2 + 16*x^6),x)
Output:
x - log(x - x*exp(2) - x^2 + 16*x^6)^2
Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-x+e^2 x+x^2-16 x^6+\left (2-2 e^2-4 x+192 x^5\right ) \log \left (x-e^2 x-x^2+16 x^6\right )}{-x+e^2 x+x^2-16 x^6} \, dx=-\mathrm {log}\left (16 x^{6}-e^{2} x -x^{2}+x \right )^{2}+x \] Input:
int(((-2*exp(1)^2+192*x^5-4*x+2)*log(-x*exp(1)^2+16*x^6-x^2+x)+x*exp(1)^2- 16*x^6+x^2-x)/(x*exp(1)^2-16*x^6+x^2-x),x)
Output:
- log( - e**2*x + 16*x**6 - x**2 + x)**2 + x