Integrand size = 89, antiderivative size = 27 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\log \left (25+e^x-\frac {1}{5} x (-x+x (2 x-\log (x)))^2\right ) \] Output:
ln(25+exp(x)-1/5*x*(-x+(2*x-ln(x))*x)^2)
\[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx \] Input:
Integrate[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3* x^2*Log[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3*Log[x]^2),x]
Output:
Integrate[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3* x^2*Log[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3*Log[x]^2), 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.
\(\displaystyle \int \frac {20 x^4-20 x^3+5 x^2+3 x^2 \log ^2(x)+\left (8 x^2-16 x^3\right ) \log (x)-5 e^x}{4 x^5-4 x^4+x^3+x^3 \log ^2(x)+\left (2 x^3-4 x^4\right ) \log (x)-5 e^x-125} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x^5-24 x^4-4 x^4 \log (x)+21 x^3+x^3 \log ^2(x)+18 x^3 \log (x)-5 x^2-3 x^2 \log ^2(x)-8 x^2 \log (x)-125}{-4 x^5+4 x^4+4 x^4 \log (x)-x^3-x^3 \log ^2(x)-2 x^3 \log (x)+5 e^x+125}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -125 \int \frac {1}{-4 x^5+4 \log (x) x^4+4 x^4-\log ^2(x) x^3-2 \log (x) x^3-x^3+5 e^x+125}dx-21 \int \frac {x^3}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+24 \int \frac {x^4}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx-4 \int \frac {x^5}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx-18 \int \frac {x^3 \log (x)}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+4 \int \frac {x^4 \log (x)}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx-\int \frac {x^3 \log ^2(x)}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+5 \int \frac {x^2}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+8 \int \frac {x^2 \log (x)}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+3 \int \frac {x^2 \log ^2(x)}{4 x^5-4 \log (x) x^4-4 x^4+\log ^2(x) x^3+2 \log (x) x^3+x^3-5 e^x-125}dx+x\) |
Input:
Int[(-5*E^x + 5*x^2 - 20*x^3 + 20*x^4 + (8*x^2 - 16*x^3)*Log[x] + 3*x^2*Lo g[x]^2)/(-125 - 5*E^x + x^3 - 4*x^4 + 4*x^5 + (2*x^3 - 4*x^4)*Log[x] + x^3 *Log[x]^2),x]
Output:
$Aborted
Time = 0.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59
method | result | size |
risch | \(3 \ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}+\left (-4 x +2\right ) \ln \left (x \right )+\frac {4 x^{5}-4 x^{4}+x^{3}-5 \,{\mathrm e}^{x}-125}{x^{3}}\right )\) | \(43\) |
parallelrisch | \(\ln \left (x^{5}-x^{4} \ln \left (x \right )+\frac {x^{3} \ln \left (x \right )^{2}}{4}-x^{4}+\frac {x^{3} \ln \left (x \right )}{2}+\frac {x^{3}}{4}-\frac {5 \,{\mathrm e}^{x}}{4}-\frac {125}{4}\right )\) | \(44\) |
Input:
int((3*x^2*ln(x)^2+(-16*x^3+8*x^2)*ln(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/(x^ 3*ln(x)^2+(-4*x^4+2*x^3)*ln(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x,method=_RET URNVERBOSE)
Output:
3*ln(x)+ln(ln(x)^2+(-4*x+2)*ln(x)+(4*x^5-4*x^4+x^3-5*exp(x)-125)/x^3)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=3 \, \log \left (x\right ) + \log \left (\frac {4 \, x^{5} + x^{3} \log \left (x\right )^{2} - 4 \, x^{4} + x^{3} - 2 \, {\left (2 \, x^{4} - x^{3}\right )} \log \left (x\right ) - 5 \, e^{x} - 125}{x^{3}}\right ) \] Input:
integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5* x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*log(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="fricas")
Output:
3*log(x) + log((4*x^5 + x^3*log(x)^2 - 4*x^4 + x^3 - 2*(2*x^4 - x^3)*log(x ) - 5*e^x - 125)/x^3)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\log {\left (- \frac {4 x^{5}}{5} + \frac {4 x^{4} \log {\left (x \right )}}{5} + \frac {4 x^{4}}{5} - \frac {x^{3} \log {\left (x \right )}^{2}}{5} - \frac {2 x^{3} \log {\left (x \right )}}{5} - \frac {x^{3}}{5} + e^{x} + 25 \right )} \] Input:
integrate((3*x**2*ln(x)**2+(-16*x**3+8*x**2)*ln(x)-5*exp(x)+20*x**4-20*x** 3+5*x**2)/(x**3*ln(x)**2+(-4*x**4+2*x**3)*ln(x)-5*exp(x)+4*x**5-4*x**4+x** 3-125),x)
Output:
log(-4*x**5/5 + 4*x**4*log(x)/5 + 4*x**4/5 - x**3*log(x)**2/5 - 2*x**3*log (x)/5 - x**3/5 + exp(x) + 25)
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\log \left (-\frac {4}{5} \, x^{5} - \frac {1}{5} \, x^{3} \log \left (x\right )^{2} + \frac {4}{5} \, x^{4} - \frac {1}{5} \, x^{3} + \frac {2}{5} \, {\left (2 \, x^{4} - x^{3}\right )} \log \left (x\right ) + e^{x} + 25\right ) \] Input:
integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5* x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*log(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="maxima")
Output:
log(-4/5*x^5 - 1/5*x^3*log(x)^2 + 4/5*x^4 - 1/5*x^3 + 2/5*(2*x^4 - x^3)*lo g(x) + e^x + 25)
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\log \left (-4 \, x^{5} + 4 \, x^{4} \log \left (x\right ) - x^{3} \log \left (x\right )^{2} + 4 \, x^{4} - 2 \, x^{3} \log \left (x\right ) - x^{3} + 5 \, e^{x} + 125\right ) \] Input:
integrate((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5* x^2)/(x^3*log(x)^2+(-4*x^4+2*x^3)*log(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x, algorithm="giac")
Output:
log(-4*x^5 + 4*x^4*log(x) - x^3*log(x)^2 + 4*x^4 - 2*x^3*log(x) - x^3 + 5* e^x + 125)
Time = 1.95 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\ln \left (\ln \left (x\right )\,\left (2\,x^3-4\,x^4\right )-5\,{\mathrm {e}}^x+x^3\,{\ln \left (x\right )}^2+x^3-4\,x^4+4\,x^5-125\right ) \] Input:
int((log(x)*(8*x^2 - 16*x^3) - 5*exp(x) + 3*x^2*log(x)^2 + 5*x^2 - 20*x^3 + 20*x^4)/(log(x)*(2*x^3 - 4*x^4) - 5*exp(x) + x^3*log(x)^2 + x^3 - 4*x^4 + 4*x^5 - 125),x)
Output:
log(log(x)*(2*x^3 - 4*x^4) - 5*exp(x) + x^3*log(x)^2 + x^3 - 4*x^4 + 4*x^5 - 125)
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-5 e^x+5 x^2-20 x^3+20 x^4+\left (8 x^2-16 x^3\right ) \log (x)+3 x^2 \log ^2(x)}{-125-5 e^x+x^3-4 x^4+4 x^5+\left (2 x^3-4 x^4\right ) \log (x)+x^3 \log ^2(x)} \, dx=\mathrm {log}\left (5 e^{x}-\mathrm {log}\left (x \right )^{2} x^{3}+4 \,\mathrm {log}\left (x \right ) x^{4}-2 \,\mathrm {log}\left (x \right ) x^{3}-4 x^{5}+4 x^{4}-x^{3}+125\right ) \] Input:
int((3*x^2*log(x)^2+(-16*x^3+8*x^2)*log(x)-5*exp(x)+20*x^4-20*x^3+5*x^2)/( x^3*log(x)^2+(-4*x^4+2*x^3)*log(x)-5*exp(x)+4*x^5-4*x^4+x^3-125),x)
Output:
log(5*e**x - log(x)**2*x**3 + 4*log(x)*x**4 - 2*log(x)*x**3 - 4*x**5 + 4*x **4 - x**3 + 125)