Integrand size = 95, antiderivative size = 23 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=5+x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \] Output:
ln(x)*x^2*(x-1-ln(-exp(x)+x^4))+5
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \] Input:
Integrate[(x^5 - x^6 + E^x*(-x + x^2) + (6*x^5 - 3*x^6 + E^x*(-2*x + 2*x^2 ))*Log[x] + (-(E^x*x) + x^5 + (-2*E^x*x + 2*x^5)*Log[x])*Log[-E^x + x^4])/ (E^x - x^4),x]
Output:
x^2*Log[x]*(-1 + x - Log[-E^x + x^4])
Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(23)=46\).
Time = 3.58 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {-x^6+x^5+e^x \left (x^2-x\right )+\left (x^5+\left (2 x^5-2 e^x x\right ) \log (x)-e^x x\right ) \log \left (x^4-e^x\right )+\left (-3 x^6+6 x^5+e^x \left (2 x^2-2 x\right )\right ) \log (x)}{e^x-x^4} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (x (2 \log (x)+1) \left (-\log \left (x^4-e^x\right )+x-1\right )+\frac {(x-4) x^5 \log (x)}{x^4-e^x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x^3}{3}+\frac {1}{3} x^3 \log (x)+\frac {x^2}{2}-x^2 \log (x) \log \left (x^4-e^x\right )-\frac {1}{6} \left (3 x^2-2 x^3\right ) (2 \log (x)+1)\) |
Input:
Int[(x^5 - x^6 + E^x*(-x + x^2) + (6*x^5 - 3*x^6 + E^x*(-2*x + 2*x^2))*Log [x] + (-(E^x*x) + x^5 + (-2*E^x*x + 2*x^5)*Log[x])*Log[-E^x + x^4])/(E^x - x^4),x]
Output:
x^2/2 - x^3/3 + (x^3*Log[x])/3 - ((3*x^2 - 2*x^3)*(1 + 2*Log[x]))/6 - x^2* Log[x]*Log[-E^x + x^4]
Time = 5.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17
method | result | size |
risch | \(-x^{2} \ln \left (x \right ) \ln \left (-{\mathrm e}^{x}+x^{4}\right )+\left (-1+x \right ) x^{2} \ln \left (x \right )\) | \(27\) |
parallelrisch | \(-x^{2} \ln \left (x \right ) \ln \left (-{\mathrm e}^{x}+x^{4}\right )+x^{3} \ln \left (x \right )-x^{2} \ln \left (x \right )\) | \(31\) |
Input:
int((((-2*exp(x)*x+2*x^5)*ln(x)-exp(x)*x+x^5)*ln(-exp(x)+x^4)+((2*x^2-2*x) *exp(x)-3*x^6+6*x^5)*ln(x)+(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x,method=_ RETURNVERBOSE)
Output:
-x^2*ln(x)*ln(-exp(x)+x^4)+(-1+x)*x^2*ln(x)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=-x^{2} \log \left (x^{4} - e^{x}\right ) \log \left (x\right ) + {\left (x^{3} - x^{2}\right )} \log \left (x\right ) \] Input:
integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2* x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x , algorithm="fricas")
Output:
-x^2*log(x^4 - e^x)*log(x) + (x^3 - x^2)*log(x)
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=- x^{2} \log {\left (x \right )} \log {\left (x^{4} - e^{x} \right )} + \left (x^{3} - x^{2}\right ) \log {\left (x \right )} \] Input:
integrate((((-2*exp(x)*x+2*x**5)*ln(x)-exp(x)*x+x**5)*ln(-exp(x)+x**4)+((2 *x**2-2*x)*exp(x)-3*x**6+6*x**5)*ln(x)+(x**2-x)*exp(x)-x**6+x**5)/(exp(x)- x**4),x)
Output:
-x**2*log(x)*log(x**4 - exp(x)) + (x**3 - x**2)*log(x)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=-x^{2} \log \left (x^{4} - e^{x}\right ) \log \left (x\right ) + {\left (x^{3} - x^{2}\right )} \log \left (x\right ) \] Input:
integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2* x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x , algorithm="maxima")
Output:
-x^2*log(x^4 - e^x)*log(x) + (x^3 - x^2)*log(x)
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=x^{3} \log \left (x\right ) - x^{2} \log \left (x^{4} - e^{x}\right ) \log \left (x\right ) - x^{2} \log \left (x\right ) \] Input:
integrate((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2* x^2-2*x)*exp(x)-3*x^6+6*x^5)*log(x)+(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x , algorithm="giac")
Output:
x^3*log(x) - x^2*log(x^4 - e^x)*log(x) - x^2*log(x)
Time = 0.58 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=-x^2\,\ln \left (x\right )\,\left (\ln \left (x^4-{\mathrm {e}}^x\right )-x+1\right ) \] Input:
int(-(log(x^4 - exp(x))*(log(x)*(2*x*exp(x) - 2*x^5) + x*exp(x) - x^5) + e xp(x)*(x - x^2) + log(x)*(exp(x)*(2*x - 2*x^2) - 6*x^5 + 3*x^6) - x^5 + x^ 6)/(exp(x) - x^4),x)
Output:
-x^2*log(x)*(log(x^4 - exp(x)) - x + 1)
\[ \int \frac {x^5-x^6+e^x \left (-x+x^2\right )+\left (6 x^5-3 x^6+e^x \left (-2 x+2 x^2\right )\right ) \log (x)+\left (-e^x x+x^5+\left (-2 e^x x+2 x^5\right ) \log (x)\right ) \log \left (-e^x+x^4\right )}{e^x-x^4} \, dx=-\left (\int \frac {x^{6}}{e^{x}-x^{4}}d x \right )+\int \frac {x^{5}}{e^{x}-x^{4}}d x +\int \frac {e^{x} x^{2}}{e^{x}-x^{4}}d x -2 \left (\int \frac {e^{x} \mathrm {log}\left (-e^{x}+x^{4}\right ) \mathrm {log}\left (x \right ) x}{e^{x}-x^{4}}d x \right )-\left (\int \frac {e^{x} \mathrm {log}\left (-e^{x}+x^{4}\right ) x}{e^{x}-x^{4}}d x \right )+2 \left (\int \frac {e^{x} \mathrm {log}\left (x \right ) x^{2}}{e^{x}-x^{4}}d x \right )-2 \left (\int \frac {e^{x} \mathrm {log}\left (x \right ) x}{e^{x}-x^{4}}d x \right )-\left (\int \frac {e^{x} x}{e^{x}-x^{4}}d x \right )+\int \frac {\mathrm {log}\left (-e^{x}+x^{4}\right ) x^{5}}{e^{x}-x^{4}}d x +2 \left (\int \frac {\mathrm {log}\left (-e^{x}+x^{4}\right ) \mathrm {log}\left (x \right ) x^{5}}{e^{x}-x^{4}}d x \right )-3 \left (\int \frac {\mathrm {log}\left (x \right ) x^{6}}{e^{x}-x^{4}}d x \right )+6 \left (\int \frac {\mathrm {log}\left (x \right ) x^{5}}{e^{x}-x^{4}}d x \right ) \] Input:
int((((-2*exp(x)*x+2*x^5)*log(x)-exp(x)*x+x^5)*log(-exp(x)+x^4)+((2*x^2-2* x)*exp(x)-3*x^6+6*x^5)*log(x)+(x^2-x)*exp(x)-x^6+x^5)/(exp(x)-x^4),x)
Output:
- int(x**6/(e**x - x**4),x) + int(x**5/(e**x - x**4),x) + int((e**x*x**2) /(e**x - x**4),x) - 2*int((e**x*log( - e**x + x**4)*log(x)*x)/(e**x - x**4 ),x) - int((e**x*log( - e**x + x**4)*x)/(e**x - x**4),x) + 2*int((e**x*log (x)*x**2)/(e**x - x**4),x) - 2*int((e**x*log(x)*x)/(e**x - x**4),x) - int( (e**x*x)/(e**x - x**4),x) + int((log( - e**x + x**4)*x**5)/(e**x - x**4),x ) + 2*int((log( - e**x + x**4)*log(x)*x**5)/(e**x - x**4),x) - 3*int((log( x)*x**6)/(e**x - x**4),x) + 6*int((log(x)*x**5)/(e**x - x**4),x)