Integrand size = 102, antiderivative size = 24 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log \left (\log \left (\frac {1}{\frac {1}{5}+x+e^{\frac {1}{6 (5-x)}} x}\right )\right ) \] Output:
ln(ln(1/(1/5+x+exp(1/6/(5-x))*x)))
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log \left (\log \left (\frac {5}{1+5 \left (1+e^{\frac {1}{30-6 x}}\right ) x}\right )\right ) \] Input:
Integrate[(-750 + 300*x - 30*x^2 + (-750 + 295*x - 30*x^2)/E^(-30 + 6*x)^( -1))/((150 + 690*x - 294*x^2 + 30*x^3 + (750*x - 300*x^2 + 30*x^3)/E^(-30 + 6*x)^(-1))*Log[5/(1 + 5*x + (5*x)/E^(-30 + 6*x)^(-1))]),x]
Output:
Log[Log[5/(1 + 5*(1 + E^(30 - 6*x)^(-1))*x)]]
Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {7235}
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.
\(\displaystyle \int \frac {-30 x^2+e^{-\frac {1}{6 x-30}} \left (-30 x^2+295 x-750\right )+300 x-750}{\left (30 x^3-294 x^2+e^{-\frac {1}{6 x-30}} \left (30 x^3-300 x^2+750 x\right )+690 x+150\right ) \log \left (\frac {5}{5 e^{-\frac {1}{6 x-30}} x+5 x+1}\right )} \, dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (\log \left (\frac {5}{5 e^{\frac {1}{6 (5-x)}} x+5 x+1}\right )\right )\) |
Input:
Int[(-750 + 300*x - 30*x^2 + (-750 + 295*x - 30*x^2)/E^(-30 + 6*x)^(-1))/( (150 + 690*x - 294*x^2 + 30*x^3 + (750*x - 300*x^2 + 30*x^3)/E^(-30 + 6*x) ^(-1))*Log[5/(1 + 5*x + (5*x)/E^(-30 + 6*x)^(-1))]),x]
Output:
Log[Log[5/(1 + 5*x + 5*E^(1/(6*(5 - x)))*x)]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 0.96 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\ln \left (\ln \left (\frac {1}{5}+x \left ({\mathrm e}^{-\frac {1}{6 \left (-5+x \right )}}+1\right )\right )\right )\) | \(17\) |
parallelrisch | \(\ln \left (\ln \left (\frac {5}{5 \,{\mathrm e}^{-\frac {1}{6 \left (-5+x \right )}} x +1+5 x}\right )\right )\) | \(23\) |
norman | \(\ln \left (\ln \left (\frac {5}{5 x \,{\mathrm e}^{-\frac {1}{6 x -30}}+1+5 x}\right )\right )\) | \(25\) |
default | \(\ln \left (-\ln \left (\frac {5}{5 x \,{\mathrm e}^{-\frac {1}{6 x -30}}+1+5 x}\right )\right )\) | \(27\) |
Input:
int(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x ^2+750*x)*exp(-1/(6*x-30))+30*x^3-294*x^2+690*x+150)/ln(5/(5*x*exp(-1/(6*x -30))+1+5*x)),x,method=_RETURNVERBOSE)
Output:
ln(ln(1/5+x*(exp(-1/6/(-5+x))+1)))
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log \left (\log \left (\frac {5}{5 \, x e^{\left (-\frac {1}{6 \, {\left (x - 5\right )}}\right )} + 5 \, x + 1}\right )\right ) \] Input:
integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3 -300*x^2+750*x)*exp(-1/(6*x-30))+30*x^3-294*x^2+690*x+150)/log(5/(5*x*exp( -1/(6*x-30))+1+5*x)),x, algorithm="fricas")
Output:
log(log(5/(5*x*e^(-1/6/(x - 5)) + 5*x + 1)))
Time = 0.52 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log {\left (\log {\left (\frac {5}{5 x + 5 x e^{- \frac {1}{6 x - 30}} + 1} \right )} \right )} \] Input:
integrate(((-30*x**2+295*x-750)*exp(-1/(6*x-30))-30*x**2+300*x-750)/((30*x **3-300*x**2+750*x)*exp(-1/(6*x-30))+30*x**3-294*x**2+690*x+150)/ln(5/(5*x *exp(-1/(6*x-30))+1+5*x)),x)
Output:
log(log(5/(5*x + 5*x*exp(-1/(6*x - 30)) + 1)))
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log \left (-\frac {6 \, x \log \left (5\right ) - 6 \, {\left (x - 5\right )} \log \left ({\left (5 \, x + 1\right )} e^{\left (\frac {1}{6 \, {\left (x - 5\right )}}\right )} + 5 \, x\right ) - 30 \, \log \left (5\right ) + 1}{6 \, {\left (x - 5\right )}}\right ) \] Input:
integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3 -300*x^2+750*x)*exp(-1/(6*x-30))+30*x^3-294*x^2+690*x+150)/log(5/(5*x*exp( -1/(6*x-30))+1+5*x)),x, algorithm="maxima")
Output:
log(-1/6*(6*x*log(5) - 6*(x - 5)*log((5*x + 1)*e^(1/6/(x - 5)) + 5*x) - 30 *log(5) + 1)/(x - 5))
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\log \left (-\log \left (\frac {5}{5 \, x e^{\left (-\frac {1}{6 \, {\left (x - 5\right )}}\right )} + 5 \, x + 1}\right )\right ) \] Input:
integrate(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3 -300*x^2+750*x)*exp(-1/(6*x-30))+30*x^3-294*x^2+690*x+150)/log(5/(5*x*exp( -1/(6*x-30))+1+5*x)),x, algorithm="giac")
Output:
log(-log(5/(5*x*e^(-1/6/(x - 5)) + 5*x + 1)))
Time = 3.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\ln \left (\ln \left (\frac {5}{5\,x+5\,x\,{\mathrm {e}}^{-\frac {1}{6\,x-30}}+1}\right )\right ) \] Input:
int(-(exp(-1/(6*x - 30))*(30*x^2 - 295*x + 750) - 300*x + 30*x^2 + 750)/(l og(5/(5*x + 5*x*exp(-1/(6*x - 30)) + 1))*(690*x + exp(-1/(6*x - 30))*(750* x - 300*x^2 + 30*x^3) - 294*x^2 + 30*x^3 + 150)),x)
Output:
log(log(5/(5*x + 5*x*exp(-1/(6*x - 30)) + 1)))
Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-750+300 x-30 x^2+e^{-\frac {1}{-30+6 x}} \left (-750+295 x-30 x^2\right )}{\left (150+690 x-294 x^2+30 x^3+e^{-\frac {1}{-30+6 x}} \left (750 x-300 x^2+30 x^3\right )\right ) \log \left (\frac {5}{1+5 x+5 e^{-\frac {1}{-30+6 x}} x}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {5 e^{\frac {1}{6 x -30}}}{5 e^{\frac {1}{6 x -30}} x +e^{\frac {1}{6 x -30}}+5 x}\right )\right ) \] Input:
int(((-30*x^2+295*x-750)*exp(-1/(6*x-30))-30*x^2+300*x-750)/((30*x^3-300*x ^2+750*x)*exp(-1/(6*x-30))+30*x^3-294*x^2+690*x+150)/log(5/(5*x*exp(-1/(6* x-30))+1+5*x)),x)
Output:
log(log((5*e**(1/(6*x - 30)))/(5*e**(1/(6*x - 30))*x + e**(1/(6*x - 30)) + 5*x)))