Integrand size = 105, antiderivative size = 28 \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=\log \left (\left (\frac {2 \left (e^5+x\right )}{5+x}-\frac {e^2}{x^2 \log ^2(x)}\right )^2\right ) \] Output:
ln((2*(exp(5)+x)/(5+x)-exp(2)/ln(x)^2/x^2)^2)
Timed out. \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=\text {\$Aborted} \] Input:
Integrate[(E^2*(100 + 40*x + 4*x^2) + E^2*(100 + 40*x + 4*x^2)*Log[x] + (2 0*x^3 - 4*E^5*x^3)*Log[x]^3)/(E^2*(-25*x - 10*x^2 - x^3)*Log[x] + (10*x^4 + 2*x^5 + E^5*(10*x^3 + 2*x^4))*Log[x]^3),x]
Output:
$Aborted
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.
| \(\displaystyle \int \frac {\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)+e^2 \left (4 x^2+40 x+100\right )+e^2 \left (4 x^2+40 x+100\right ) \log (x)}{e^2 \left (-x^3-10 x^2-25 x\right ) \log (x)+\left (2 x^5+10 x^4+e^5 \left (2 x^4+10 x^3\right )\right ) \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)\right )-e^2 \left (4 x^2+40 x+100\right )-e^2 \left (4 x^2+40 x+100\right ) \log (x)}{x (x+5) \log (x) \left (-2 x^3 \log ^2(x)-2 e^5 x^2 \log ^2(x)+e^2 x+5 e^2\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (-4 x^4 \log (x)-8 e^5 x^3 \log (x)-2 e^2 x^2-4 e^{10} x^2 \log (x)-15 e^2 \left (1+\frac {e^5}{15}\right ) x-10 e^7\right )}{x \left (x+e^5\right ) \left (-2 x^3 \log ^2(x)-2 e^5 x^2 \log ^2(x)+e^2 x+5 e^2\right )}-\frac {2 \left (e^5-5\right )}{(x+5) \left (x+e^5\right )}-\frac {4}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 e^2 \int \frac {1}{2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2}dx+20 e^2 \int \frac {1}{x \left (2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2\right )}dx+2 e^2 \left (15+e^5\right ) \int \frac {1}{\left (x+e^5\right ) \left (2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2\right )}dx-4 e^7 \int \frac {1}{\left (x+e^5\right ) \left (2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2\right )}dx-20 e^2 \int \frac {1}{\left (x+e^5\right ) \left (2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2\right )}dx+8 e^5 \int \frac {x \log (x)}{2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2}dx+8 \int \frac {x^2 \log (x)}{2 \log ^2(x) x^3+2 e^5 \log ^2(x) x^2-e^2 x-5 e^2}dx-2 \log (x+5)+2 \log \left (x+e^5\right )-4 \log (\log (x))\) |
Input:
Int[(E^2*(100 + 40*x + 4*x^2) + E^2*(100 + 40*x + 4*x^2)*Log[x] + (20*x^3 - 4*E^5*x^3)*Log[x]^3)/(E^2*(-25*x - 10*x^2 - x^3)*Log[x] + (10*x^4 + 2*x^ 5 + E^5*(10*x^3 + 2*x^4))*Log[x]^3),x]
Output:
$Aborted
Time = 1.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57
| method | result | size |
| risch | \(2 \ln \left ({\mathrm e}^{5}+x \right )-2 \ln \left (5+x \right )-4 \ln \left (\ln \left (x \right )\right )+2 \ln \left (\ln \left (x \right )^{2}-\frac {{\mathrm e}^{2} \left (5+x \right )}{2 \left ({\mathrm e}^{5}+x \right ) x^{2}}\right )\) | \(44\) |
| parallelrisch | \(-4 \ln \left (\ln \left (x \right )\right )-2 \ln \left (5+x \right )+2 \ln \left (x^{2} \ln \left (x \right )^{2} {\mathrm e}^{5}+x^{3} \ln \left (x \right )^{2}-\frac {{\mathrm e}^{2} x}{2}-\frac {5 \,{\mathrm e}^{2}}{2}\right )-4 \ln \left (x \right )\) | \(48\) |
| default | \(-4 \ln \left (x \right )-2 \ln \left (5+x \right )+2 \ln \left (2 x^{2} \ln \left (x \right )^{2} {\mathrm e}^{5}+2 x^{3} \ln \left (x \right )^{2}-{\mathrm e}^{2} x -5 \,{\mathrm e}^{2}\right )-4 \ln \left (\ln \left (x \right )\right )\) | \(50\) |
| norman | \(-4 \ln \left (x \right )-2 \ln \left (5+x \right )+2 \ln \left (2 x^{2} \ln \left (x \right )^{2} {\mathrm e}^{5}+2 x^{3} \ln \left (x \right )^{2}-{\mathrm e}^{2} x -5 \,{\mathrm e}^{2}\right )-4 \ln \left (\ln \left (x \right )\right )\) | \(50\) |
Input:
int(((-4*x^3*exp(5)+20*x^3)*ln(x)^3+(4*x^2+40*x+100)*exp(2)*ln(x)+(4*x^2+4 0*x+100)*exp(2))/(((2*x^4+10*x^3)*exp(5)+2*x^5+10*x^4)*ln(x)^3+(-x^3-10*x^ 2-25*x)*exp(2)*ln(x)),x,method=_RETURNVERBOSE)
Output:
2*ln(exp(5)+x)-2*ln(5+x)-4*ln(ln(x))+2*ln(ln(x)^2-1/2*exp(2)*(5+x)/(exp(5) +x)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \left (x\right )^{2} - {\left (x + 5\right )} e^{2}}{x^{3} + x^{2} e^{5}}\right ) - 4 \, \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+ (4*x^2+40*x+100)*exp(2))/(((2*x^4+10*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(- x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="fricas")
Output:
2*log(x + e^5) - 2*log(x + 5) + 2*log((2*(x^3 + x^2*e^5)*log(x)^2 - (x + 5 )*e^2)/(x^3 + x^2*e^5)) - 4*log(log(x))
Exception generated. \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((-4*x**3*exp(5)+20*x**3)*ln(x)**3+(4*x**2+40*x+100)*exp(2)*ln(x )+(4*x**2+40*x+100)*exp(2))/(((2*x**4+10*x**3)*exp(5)+2*x**5+10*x**4)*ln(x )**3+(-x**3-10*x**2-25*x)*exp(2)*ln(x)),x)
Output:
Exception raised: PolynomialError >> 1/(4*x**10 + 16*x**9*exp(5) + 24*x**8 *exp(10) + 16*x**7*exp(15) + 4*x**6*exp(20)) contains an element of the se t of generators.
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=2 \, \log \left (x + e^{5}\right ) - 2 \, \log \left (x + 5\right ) + 2 \, \log \left (\frac {2 \, {\left (x^{3} + x^{2} e^{5}\right )} \log \left (x\right )^{2} - x e^{2} - 5 \, e^{2}}{2 \, {\left (x^{3} + x^{2} e^{5}\right )}}\right ) - 4 \, \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+ (4*x^2+40*x+100)*exp(2))/(((2*x^4+10*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(- x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="maxima")
Output:
2*log(x + e^5) - 2*log(x + 5) + 2*log(1/2*(2*(x^3 + x^2*e^5)*log(x)^2 - x* e^2 - 5*e^2)/(x^3 + x^2*e^5)) - 4*log(log(x))
Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=2 \, \log \left (2 \, x^{3} \log \left (x\right )^{2} + 2 \, x^{2} e^{5} \log \left (x\right )^{2} - x e^{2} - 5 \, e^{2}\right ) - 2 \, \log \left (x + 5\right ) - 4 \, \log \left (x\right ) - 4 \, \log \left (\log \left (x\right )\right ) \] Input:
integrate(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+ (4*x^2+40*x+100)*exp(2))/(((2*x^4+10*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(- x^3-10*x^2-25*x)*exp(2)*log(x)),x, algorithm="giac")
Output:
2*log(2*x^3*log(x)^2 + 2*x^2*e^5*log(x)^2 - x*e^2 - 5*e^2) - 2*log(x + 5) - 4*log(x) - 4*log(log(x))
Timed out. \[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx=\int \frac {\left (20\,x^3-4\,x^3\,{\mathrm {e}}^5\right )\,{\ln \left (x\right )}^3+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )\,\ln \left (x\right )+{\mathrm {e}}^2\,\left (4\,x^2+40\,x+100\right )}{{\ln \left (x\right )}^3\,\left ({\mathrm {e}}^5\,\left (2\,x^4+10\,x^3\right )+10\,x^4+2\,x^5\right )-{\mathrm {e}}^2\,\ln \left (x\right )\,\left (x^3+10\,x^2+25\,x\right )} \,d x \] Input:
int((exp(2)*(40*x + 4*x^2 + 100) - log(x)^3*(4*x^3*exp(5) - 20*x^3) + exp( 2)*log(x)*(40*x + 4*x^2 + 100))/(log(x)^3*(exp(5)*(10*x^3 + 2*x^4) + 10*x^ 4 + 2*x^5) - exp(2)*log(x)*(25*x + 10*x^2 + x^3)),x)
Output:
int((exp(2)*(40*x + 4*x^2 + 100) - log(x)^3*(4*x^3*exp(5) - 20*x^3) + exp( 2)*log(x)*(40*x + 4*x^2 + 100))/(log(x)^3*(exp(5)*(10*x^3 + 2*x^4) + 10*x^ 4 + 2*x^5) - exp(2)*log(x)*(25*x + 10*x^2 + x^3)), x)
\[ \int \frac {e^2 \left (100+40 x+4 x^2\right )+e^2 \left (100+40 x+4 x^2\right ) \log (x)+\left (20 x^3-4 e^5 x^3\right ) \log ^3(x)}{e^2 \left (-25 x-10 x^2-x^3\right ) \log (x)+\left (10 x^4+2 x^5+e^5 \left (10 x^3+2 x^4\right )\right ) \log ^3(x)} \, dx =\text {Too large to display} \] Input:
int(((-4*x^3*exp(5)+20*x^3)*log(x)^3+(4*x^2+40*x+100)*exp(2)*log(x)+(4*x^2 +40*x+100)*exp(2))/(((2*x^4+10*x^3)*exp(5)+2*x^5+10*x^4)*log(x)^3+(-x^3-10 *x^2-25*x)*exp(2)*log(x)),x)
Output:
4*( - int((log(x)**2*x**2)/(2*log(x)**2*e**5*x**3 + 10*log(x)**2*e**5*x**2 + 2*log(x)**2*x**4 + 10*log(x)**2*x**3 - e**2*x**2 - 10*e**2*x - 25*e**2) ,x)*e**5 + 5*int((log(x)**2*x**2)/(2*log(x)**2*e**5*x**3 + 10*log(x)**2*e* *5*x**2 + 2*log(x)**2*x**4 + 10*log(x)**2*x**3 - e**2*x**2 - 10*e**2*x - 2 5*e**2),x) + int(x/(2*log(x)**3*e**5*x**3 + 10*log(x)**3*e**5*x**2 + 2*log (x)**3*x**4 + 10*log(x)**3*x**3 - log(x)*e**2*x**2 - 10*log(x)*e**2*x - 25 *log(x)*e**2),x)*e**2 + int(x/(2*log(x)**2*e**5*x**3 + 10*log(x)**2*e**5*x **2 + 2*log(x)**2*x**4 + 10*log(x)**2*x**3 - e**2*x**2 - 10*e**2*x - 25*e* *2),x)*e**2 + 25*int(1/(2*log(x)**3*e**5*x**4 + 10*log(x)**3*e**5*x**3 + 2 *log(x)**3*x**5 + 10*log(x)**3*x**4 - log(x)*e**2*x**3 - 10*log(x)*e**2*x* *2 - 25*log(x)*e**2*x),x)*e**2 + 10*int(1/(2*log(x)**3*e**5*x**3 + 10*log( x)**3*e**5*x**2 + 2*log(x)**3*x**4 + 10*log(x)**3*x**3 - log(x)*e**2*x**2 - 10*log(x)*e**2*x - 25*log(x)*e**2),x)*e**2 + 25*int(1/(2*log(x)**2*e**5* x**4 + 10*log(x)**2*e**5*x**3 + 2*log(x)**2*x**5 + 10*log(x)**2*x**4 - e** 2*x**3 - 10*e**2*x**2 - 25*e**2*x),x)*e**2 + 10*int(1/(2*log(x)**2*e**5*x* *3 + 10*log(x)**2*e**5*x**2 + 2*log(x)**2*x**4 + 10*log(x)**2*x**3 - e**2* x**2 - 10*e**2*x - 25*e**2),x)*e**2)