Integrand size = 118, antiderivative size = 22 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x+\log \left (\frac {1}{5} \left (3+\frac {3}{e^3}-x+\log (5+\log (x))\right )\right ) \] Output:
ln(-1/5*x+3/5/exp(3)+3/5+1/5*ln(5+ln(x)))+x
Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x+\log \left (3+3 e^3-e^3 x+e^3 \log (5+\log (x))\right ) \] Input:
Integrate[(15*x + E^3*(1 + 10*x - 5*x^2) + (3*x + E^3*(2*x - x^2))*Log[x] + (5*E^3*x + E^3*x*Log[x])*Log[5 + Log[x]])/(15*x + E^3*(15*x - 5*x^2) + ( 3*x + E^3*(3*x - x^2))*Log[x] + (5*E^3*x + E^3*x*Log[x])*Log[5 + Log[x]]), x]
Output:
x + Log[3 + 3*E^3 - E^3*x + E^3*Log[5 + Log[x]]]
Time = 1.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {7292, 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.
\(\displaystyle \int \frac {e^3 \left (-5 x^2+10 x+1\right )+\left (e^3 \left (2 x-x^2\right )+3 x\right ) \log (x)+15 x+\left (5 e^3 x+e^3 x \log (x)\right ) \log (\log (x)+5)}{e^3 \left (15 x-5 x^2\right )+\left (e^3 \left (3 x-x^2\right )+3 x\right ) \log (x)+15 x+\left (5 e^3 x+e^3 x \log (x)\right ) \log (\log (x)+5)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^3 \left (-5 x^2+10 x+1\right )+\left (e^3 \left (2 x-x^2\right )+3 x\right ) \log (x)+15 x+\left (5 e^3 x+e^3 x \log (x)\right ) \log (\log (x)+5)}{x (\log (x)+5) \left (-e^3 x+e^3 \log (\log (x)+5)+3 \left (1+e^3\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^3 (-5 x+x (-\log (x))+1)}{x (\log (x)+5) \left (-e^3 x+e^3 \log (\log (x)+5)+3 \left (1+e^3\right )\right )}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x+\log \left (-e^3 x+e^3 \log (\log (x)+5)+3 \left (1+e^3\right )\right )\) |
Input:
Int[(15*x + E^3*(1 + 10*x - 5*x^2) + (3*x + E^3*(2*x - x^2))*Log[x] + (5*E ^3*x + E^3*x*Log[x])*Log[5 + Log[x]])/(15*x + E^3*(15*x - 5*x^2) + (3*x + E^3*(3*x - x^2))*Log[x] + (5*E^3*x + E^3*x*Log[x])*Log[5 + Log[x]]),x]
Output:
x + Log[3*(1 + E^3) - E^3*x + E^3*Log[5 + Log[x]]]
Time = 1.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
method | result | size |
risch | \(x +\ln \left (-x +3+\ln \left (5+\ln \left (x \right )\right )+3 \,{\mathrm e}^{-3}\right )\) | \(18\) |
default | \(x +\ln \left (x \,{\mathrm e}^{3}-{\mathrm e}^{3} \ln \left (5+\ln \left (x \right )\right )-3 \,{\mathrm e}^{3}-3\right )\) | \(23\) |
norman | \(x +\ln \left (x \,{\mathrm e}^{3}-{\mathrm e}^{3} \ln \left (5+\ln \left (x \right )\right )-3 \,{\mathrm e}^{3}-3\right )\) | \(23\) |
parallelrisch | \(\ln \left (\left (x \,{\mathrm e}^{3}-{\mathrm e}^{3} \ln \left (5+\ln \left (x \right )\right )-3 \,{\mathrm e}^{3}-3\right ) {\mathrm e}^{-3}\right )+x\) | \(28\) |
Input:
int(((x*exp(3)*ln(x)+5*x*exp(3))*ln(5+ln(x))+((-x^2+2*x)*exp(3)+3*x)*ln(x) +(-5*x^2+10*x+1)*exp(3)+15*x)/((x*exp(3)*ln(x)+5*x*exp(3))*ln(5+ln(x))+((- x^2+3*x)*exp(3)+3*x)*ln(x)+(-5*x^2+15*x)*exp(3)+15*x),x,method=_RETURNVERB OSE)
Output:
x+ln(-x+3+ln(5+ln(x))+3*exp(-3))
Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x + \log \left (-{\left (x - 3\right )} e^{3} + e^{3} \log \left (\log \left (x\right ) + 5\right ) + 3\right ) \] Input:
integrate(((x*exp(3)*log(x)+5*x*exp(3))*log(5+log(x))+((-x^2+2*x)*exp(3)+3 *x)*log(x)+(-5*x^2+10*x+1)*exp(3)+15*x)/((x*exp(3)*log(x)+5*x*exp(3))*log( 5+log(x))+((-x^2+3*x)*exp(3)+3*x)*log(x)+(-5*x^2+15*x)*exp(3)+15*x),x, alg orithm="fricas")
Output:
x + log(-(x - 3)*e^3 + e^3*log(log(x) + 5) + 3)
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x + \log {\left (\frac {- x e^{3} + 3 + 3 e^{3}}{e^{3}} + \log {\left (\log {\left (x \right )} + 5 \right )} \right )} \] Input:
integrate(((x*exp(3)*ln(x)+5*x*exp(3))*ln(5+ln(x))+((-x**2+2*x)*exp(3)+3*x )*ln(x)+(-5*x**2+10*x+1)*exp(3)+15*x)/((x*exp(3)*ln(x)+5*x*exp(3))*ln(5+ln (x))+((-x**2+3*x)*exp(3)+3*x)*ln(x)+(-5*x**2+15*x)*exp(3)+15*x),x)
Output:
x + log((-x*exp(3) + 3 + 3*exp(3))*exp(-3) + log(log(x) + 5))
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x + \log \left (-{\left (x e^{3} - e^{3} \log \left (\log \left (x\right ) + 5\right ) - 3 \, e^{3} - 3\right )} e^{\left (-3\right )}\right ) \] Input:
integrate(((x*exp(3)*log(x)+5*x*exp(3))*log(5+log(x))+((-x^2+2*x)*exp(3)+3 *x)*log(x)+(-5*x^2+10*x+1)*exp(3)+15*x)/((x*exp(3)*log(x)+5*x*exp(3))*log( 5+log(x))+((-x^2+3*x)*exp(3)+3*x)*log(x)+(-5*x^2+15*x)*exp(3)+15*x),x, alg orithm="maxima")
Output:
x + log(-(x*e^3 - e^3*log(log(x) + 5) - 3*e^3 - 3)*e^(-3))
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x + \log \left (-x e^{3} + e^{3} \log \left (\log \left (x\right ) + 5\right ) + 3 \, e^{3} + 3\right ) \] Input:
integrate(((x*exp(3)*log(x)+5*x*exp(3))*log(5+log(x))+((-x^2+2*x)*exp(3)+3 *x)*log(x)+(-5*x^2+10*x+1)*exp(3)+15*x)/((x*exp(3)*log(x)+5*x*exp(3))*log( 5+log(x))+((-x^2+3*x)*exp(3)+3*x)*log(x)+(-5*x^2+15*x)*exp(3)+15*x),x, alg orithm="giac")
Output:
x + log(-x*e^3 + e^3*log(log(x) + 5) + 3*e^3 + 3)
Time = 3.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=x+\ln \left (3\,{\mathrm {e}}^{-3}-x+\ln \left (\ln \left (x\right )+5\right )+3\right ) \] Input:
int((15*x + exp(3)*(10*x - 5*x^2 + 1) + log(x)*(3*x + exp(3)*(2*x - x^2)) + log(log(x) + 5)*(5*x*exp(3) + x*exp(3)*log(x)))/(15*x + exp(3)*(15*x - 5 *x^2) + log(x)*(3*x + exp(3)*(3*x - x^2)) + log(log(x) + 5)*(5*x*exp(3) + x*exp(3)*log(x))),x)
Output:
x + log(3*exp(-3) - x + log(log(x) + 5) + 3)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {15 x+e^3 \left (1+10 x-5 x^2\right )+\left (3 x+e^3 \left (2 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))}{15 x+e^3 \left (15 x-5 x^2\right )+\left (3 x+e^3 \left (3 x-x^2\right )\right ) \log (x)+\left (5 e^3 x+e^3 x \log (x)\right ) \log (5+\log (x))} \, dx=\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )+5\right ) e^{3}-e^{3} x +3 e^{3}+3\right )+x \] Input:
int(((x*exp(3)*log(x)+5*x*exp(3))*log(5+log(x))+((-x^2+2*x)*exp(3)+3*x)*lo g(x)+(-5*x^2+10*x+1)*exp(3)+15*x)/((x*exp(3)*log(x)+5*x*exp(3))*log(5+log( x))+((-x^2+3*x)*exp(3)+3*x)*log(x)+(-5*x^2+15*x)*exp(3)+15*x),x)
Output:
log(log(log(x) + 5)*e**3 - e**3*x + 3*e**3 + 3) + x