Integrand size = 146, antiderivative size = 26 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\log \left (e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}}-x\right ) \] Output:
ln(exp(-x^2/(ln(20*x^2)-exp(5)))-x)
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=-\frac {x^2}{-e^5+\log \left (20 x^2\right )}+\log \left (1-e^{\frac {x^2}{-e^5+\log \left (20 x^2\right )}} x\right ) \] Input:
Integrate[(-E^10 + 2*E^5*Log[20*x^2] - Log[20*x^2]^2 + (2*x + 2*E^5*x - 2* x*Log[20*x^2])/E^(x^2/(-E^5 + Log[20*x^2])))/(-(E^10*x) + 2*E^5*x*Log[20*x ^2] - x*Log[20*x^2]^2 + (E^10 - 2*E^5*Log[20*x^2] + Log[20*x^2]^2)/E^(x^2/ (-E^5 + Log[20*x^2]))),x]
Output:
-(x^2/(-E^5 + Log[20*x^2])) + Log[1 - E^(x^2/(-E^5 + Log[20*x^2]))*x]
Time = 1.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7292, 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.
\(\displaystyle \int \frac {-\log ^2\left (20 x^2\right )+2 e^5 \log \left (20 x^2\right )+e^{-\frac {x^2}{\log \left (20 x^2\right )-e^5}} \left (-2 x \log \left (20 x^2\right )+2 e^5 x+2 x\right )-e^{10}}{-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{\log \left (20 x^2\right )-e^5}} \left (\log ^2\left (20 x^2\right )-2 e^5 \log \left (20 x^2\right )+e^{10}\right )+2 e^5 x \log \left (20 x^2\right )-e^{10} x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-\log ^2\left (20 x^2\right )+2 e^5 \log \left (20 x^2\right )+e^{-\frac {x^2}{\log \left (20 x^2\right )-e^5}} \left (-2 x \log \left (20 x^2\right )+2 e^5 x+2 x\right )-e^{10}}{\left (e^{\frac {x^2}{e^5-\log \left (20 x^2\right )}}-x\right ) \left (e^5-\log \left (20 x^2\right )\right )^2}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (e^{\frac {x^2}{e^5-\log \left (20 x^2\right )}}-x\right )\) |
Input:
Int[(-E^10 + 2*E^5*Log[20*x^2] - Log[20*x^2]^2 + (2*x + 2*E^5*x - 2*x*Log[ 20*x^2])/E^(x^2/(-E^5 + Log[20*x^2])))/(-(E^10*x) + 2*E^5*x*Log[20*x^2] - x*Log[20*x^2]^2 + (E^10 - 2*E^5*Log[20*x^2] + Log[20*x^2]^2)/E^(x^2/(-E^5 + Log[20*x^2]))),x]
Output:
Log[E^(x^2/(E^5 - Log[20*x^2])) - x]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 2.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\ln \left ({\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{5}-\ln \left (20 x^{2}\right )}}-x \right )\) | \(24\) |
default | \(\ln \left (x -{\mathrm e}^{-\frac {x^{2}}{\ln \left (20 x^{2}\right )-{\mathrm e}^{5}}}\right )\) | \(25\) |
norman | \(\ln \left (x -{\mathrm e}^{-\frac {x^{2}}{\ln \left (20 x^{2}\right )-{\mathrm e}^{5}}}\right )\) | \(25\) |
parallelrisch | \(\ln \left (x -{\mathrm e}^{-\frac {x^{2}}{\ln \left (20 x^{2}\right )-{\mathrm e}^{5}}}\right )\) | \(25\) |
Input:
int(((-2*x*ln(20*x^2)+2*x*exp(5)+2*x)*exp(-x^2/(ln(20*x^2)-exp(5)))-ln(20* x^2)^2+2*exp(5)*ln(20*x^2)-exp(5)^2)/((ln(20*x^2)^2-2*exp(5)*ln(20*x^2)+ex p(5)^2)*exp(-x^2/(ln(20*x^2)-exp(5)))-x*ln(20*x^2)^2+2*x*exp(5)*ln(20*x^2) -x*exp(5)^2),x,method=_RETURNVERBOSE)
Output:
ln(exp(x^2/(exp(5)-ln(20*x^2)))-x)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\log \left (-x + e^{\left (\frac {x^{2}}{e^{5} - \log \left (20 \, x^{2}\right )}\right )}\right ) \] Input:
integrate(((-2*x*log(20*x^2)+2*x*exp(5)+2*x)*exp(-x^2/(log(20*x^2)-exp(5)) )-log(20*x^2)^2+2*exp(5)*log(20*x^2)-exp(5)^2)/((log(20*x^2)^2-2*exp(5)*lo g(20*x^2)+exp(5)^2)*exp(-x^2/(log(20*x^2)-exp(5)))-x*log(20*x^2)^2+2*x*exp (5)*log(20*x^2)-x*exp(5)^2),x, algorithm="fricas")
Output:
log(-x + e^(x^2/(e^5 - log(20*x^2))))
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\log {\left (- x + e^{- \frac {x^{2}}{\log {\left (20 x^{2} \right )} - e^{5}}} \right )} \] Input:
integrate(((-2*x*ln(20*x**2)+2*x*exp(5)+2*x)*exp(-x**2/(ln(20*x**2)-exp(5) ))-ln(20*x**2)**2+2*exp(5)*ln(20*x**2)-exp(5)**2)/((ln(20*x**2)**2-2*exp(5 )*ln(20*x**2)+exp(5)**2)*exp(-x**2/(ln(20*x**2)-exp(5)))-x*ln(20*x**2)**2+ 2*x*exp(5)*ln(20*x**2)-x*exp(5)**2),x)
Output:
log(-x + exp(-x**2/(log(20*x**2) - exp(5))))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {x^{2}}{e^{5} - \log \left (5\right ) - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right )} + \log \left (x\right ) + \log \left (\frac {x e^{\left (-\frac {x^{2}}{e^{5} - \log \left (5\right ) - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right )}\right )} - 1}{x}\right ) \] Input:
integrate(((-2*x*log(20*x^2)+2*x*exp(5)+2*x)*exp(-x^2/(log(20*x^2)-exp(5)) )-log(20*x^2)^2+2*exp(5)*log(20*x^2)-exp(5)^2)/((log(20*x^2)^2-2*exp(5)*lo g(20*x^2)+exp(5)^2)*exp(-x^2/(log(20*x^2)-exp(5)))-x*log(20*x^2)^2+2*x*exp (5)*log(20*x^2)-x*exp(5)^2),x, algorithm="maxima")
Output:
x^2/(e^5 - log(5) - 2*log(2) - 2*log(x)) + log(x) + log((x*e^(-x^2/(e^5 - log(5) - 2*log(2) - 2*log(x))) - 1)/x)
Time = 0.76 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\log \left (x - e^{\left (\frac {x^{2}}{e^{5} - \log \left (20 \, x^{2}\right )}\right )}\right ) \] Input:
integrate(((-2*x*log(20*x^2)+2*x*exp(5)+2*x)*exp(-x^2/(log(20*x^2)-exp(5)) )-log(20*x^2)^2+2*exp(5)*log(20*x^2)-exp(5)^2)/((log(20*x^2)^2-2*exp(5)*lo g(20*x^2)+exp(5)^2)*exp(-x^2/(log(20*x^2)-exp(5)))-x*log(20*x^2)^2+2*x*exp (5)*log(20*x^2)-x*exp(5)^2),x, algorithm="giac")
Output:
log(x - e^(x^2/(e^5 - log(20*x^2))))
Time = 4.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\ln \left ({\mathrm {e}}^{\frac {x^2}{{\mathrm {e}}^5-\ln \left (20\,x^2\right )}}-x\right ) \] Input:
int((exp(10) + log(20*x^2)^2 - exp(x^2/(exp(5) - log(20*x^2)))*(2*x + 2*x* exp(5) - 2*x*log(20*x^2)) - 2*exp(5)*log(20*x^2))/(x*exp(10) - exp(x^2/(ex p(5) - log(20*x^2)))*(exp(10) + log(20*x^2)^2 - 2*exp(5)*log(20*x^2)) + x* log(20*x^2)^2 - 2*x*exp(5)*log(20*x^2)),x)
Output:
log(exp(x^2/(exp(5) - log(20*x^2))) - x)
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {-e^{10}+2 e^5 \log \left (20 x^2\right )-\log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (2 x+2 e^5 x-2 x \log \left (20 x^2\right )\right )}{-e^{10} x+2 e^5 x \log \left (20 x^2\right )-x \log ^2\left (20 x^2\right )+e^{-\frac {x^2}{-e^5+\log \left (20 x^2\right )}} \left (e^{10}-2 e^5 \log \left (20 x^2\right )+\log ^2\left (20 x^2\right )\right )} \, dx=\frac {\mathrm {log}\left (e^{\frac {x^{2}}{\mathrm {log}\left (20 x^{2}\right )-e^{5}}} x -1\right ) \mathrm {log}\left (20 x^{2}\right )-\mathrm {log}\left (e^{\frac {x^{2}}{\mathrm {log}\left (20 x^{2}\right )-e^{5}}} x -1\right ) e^{5}-x^{2}}{\mathrm {log}\left (20 x^{2}\right )-e^{5}} \] Input:
int(((-2*x*log(20*x^2)+2*x*exp(5)+2*x)*exp(-x^2/(log(20*x^2)-exp(5)))-log( 20*x^2)^2+2*exp(5)*log(20*x^2)-exp(5)^2)/((log(20*x^2)^2-2*exp(5)*log(20*x ^2)+exp(5)^2)*exp(-x^2/(log(20*x^2)-exp(5)))-x*log(20*x^2)^2+2*x*exp(5)*lo g(20*x^2)-x*exp(5)^2),x)
Output:
(log(e**(x**2/(log(20*x**2) - e**5))*x - 1)*log(20*x**2) - log(e**(x**2/(l og(20*x**2) - e**5))*x - 1)*e**5 - x**2)/(log(20*x**2) - e**5)