Integrand size = 86, antiderivative size = 26 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 x^2}{e \log \left (\left (-4-x+\frac {5 (4+x)}{x}\right )^2\right )} \] Output:
3*x^2/exp(1)/ln((5*(4+x)/x-x-4)^2)
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 x^2}{e \log \left (\frac {\left (20+x-x^2\right )^2}{x^2}\right )} \] Input:
Integrate[(-120*x - 6*x^3 + (-120*x - 6*x^2 + 6*x^3)*Log[(400 + 40*x - 39* x^2 - 2*x^3 + x^4)/x^2])/(E*(-20 - x + x^2)*Log[(400 + 40*x - 39*x^2 - 2*x ^3 + x^4)/x^2]^2),x]
Output:
(3*x^2)/(E*Log[(20 + x - x^2)^2/x^2])
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.
\(\displaystyle \int \frac {-6 x^3+\left (6 x^3-6 x^2-120 x\right ) \log \left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )-120 x}{e \left (x^2-x-20\right ) \log ^2\left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {6 \left (x^3+20 x+\left (-x^3+x^2+20 x\right ) \log \left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )\right )}{\left (-x^2+x+20\right ) \log ^2\left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )}dx}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6 \int \frac {x^3+20 x+\left (-x^3+x^2+20 x\right ) \log \left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )}{\left (-x^2+x+20\right ) \log ^2\left (\frac {x^4-2 x^3-39 x^2+40 x+400}{x^2}\right )}dx}{e}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {6 \int \left (\frac {x}{\log \left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}-\frac {x \left (x^2+20\right )}{(x-5) (x+4) \log ^2\left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}\right )dx}{e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {6 \left (-\int \frac {1}{\log ^2\left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}dx-25 \int \frac {1}{(x-5) \log ^2\left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}dx-\int \frac {x}{\log ^2\left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}dx-16 \int \frac {1}{(x+4) \log ^2\left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}dx+\int \frac {x}{\log \left (\frac {\left (-x^2+x+20\right )^2}{x^2}\right )}dx\right )}{e}\) |
Input:
Int[(-120*x - 6*x^3 + (-120*x - 6*x^2 + 6*x^3)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x^4)/x^2])/(E*(-20 - x + x^2)*Log[(400 + 40*x - 39*x^2 - 2*x^3 + x ^4)/x^2]^2),x]
Output:
$Aborted
Time = 0.68 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27
method | result | size |
risch | \(\frac {3 x^{2} {\mathrm e}^{-1}}{\ln \left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right )}\) | \(33\) |
norman | \(\frac {3 x^{2} {\mathrm e}^{-1}}{\ln \left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right )}\) | \(35\) |
parallelrisch | \(\frac {3 x^{2} {\mathrm e}^{-1}}{\ln \left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right )}\) | \(35\) |
Input:
int(((6*x^3-6*x^2-120*x)*ln((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x)/ (x^2-x-20)/exp(1)/ln((x^4-2*x^3-39*x^2+40*x+400)/x^2)^2,x,method=_RETURNVE RBOSE)
Output:
3*x^2*exp(-1)/ln((x^4-2*x^3-39*x^2+40*x+400)/x^2)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 \, x^{2} e^{\left (-1\right )}}{\log \left (\frac {x^{4} - 2 \, x^{3} - 39 \, x^{2} + 40 \, x + 400}{x^{2}}\right )} \] Input:
integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3- 120*x)/(x^2-x-20)/exp(1)/log((x^4-2*x^3-39*x^2+40*x+400)/x^2)^2,x, algorit hm="fricas")
Output:
3*x^2*e^(-1)/log((x^4 - 2*x^3 - 39*x^2 + 40*x + 400)/x^2)
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 x^{2}}{e \log {\left (\frac {x^{4} - 2 x^{3} - 39 x^{2} + 40 x + 400}{x^{2}} \right )}} \] Input:
integrate(((6*x**3-6*x**2-120*x)*ln((x**4-2*x**3-39*x**2+40*x+400)/x**2)-6 *x**3-120*x)/(x**2-x-20)/exp(1)/ln((x**4-2*x**3-39*x**2+40*x+400)/x**2)**2 ,x)
Output:
3*x**2*exp(-1)/log((x**4 - 2*x**3 - 39*x**2 + 40*x + 400)/x**2)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 \, x^{2} e^{\left (-1\right )}}{2 \, {\left (\log \left (x + 4\right ) + \log \left (x - 5\right ) - \log \left (x\right )\right )}} \] Input:
integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3- 120*x)/(x^2-x-20)/exp(1)/log((x^4-2*x^3-39*x^2+40*x+400)/x^2)^2,x, algorit hm="maxima")
Output:
3/2*x^2*e^(-1)/(log(x + 4) + log(x - 5) - log(x))
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 \, x^{2} e^{\left (-1\right )}}{\log \left (\frac {x^{4} - 2 \, x^{3} - 39 \, x^{2} + 40 \, x + 400}{x^{2}}\right )} \] Input:
integrate(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3- 120*x)/(x^2-x-20)/exp(1)/log((x^4-2*x^3-39*x^2+40*x+400)/x^2)^2,x, algorit hm="giac")
Output:
3*x^2*e^(-1)/log((x^4 - 2*x^3 - 39*x^2 + 40*x + 400)/x^2)
Time = 2.71 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3\,x^2\,{\mathrm {e}}^{-1}+\frac {3\,x^2\,\ln \left (\frac {x^4-2\,x^3-39\,x^2+40\,x+400}{x^2}\right )\,{\mathrm {e}}^{-1}\,\left (-x^2+x+20\right )}{x^2+20}}{\ln \left (\frac {x^4-2\,x^3-39\,x^2+40\,x+400}{x^2}\right )}-3\,x\,{\mathrm {e}}^{-1}+3\,x^2\,{\mathrm {e}}^{-1}+\frac {60\,x+2400}{\mathrm {e}\,x^2+20\,\mathrm {e}} \] Input:
int((exp(-1)*(120*x + log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)*(120*x + 6*x^2 - 6*x^3) + 6*x^3))/(log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)^2 *(x - x^2 + 20)),x)
Output:
(3*x^2*exp(-1) + (3*x^2*log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2)*exp(- 1)*(x - x^2 + 20))/(x^2 + 20))/log((40*x - 39*x^2 - 2*x^3 + x^4 + 400)/x^2 ) - 3*x*exp(-1) + 3*x^2*exp(-1) + (60*x + 2400)/(20*exp(1) + x^2*exp(1))
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-120 x-6 x^3+\left (-120 x-6 x^2+6 x^3\right ) \log \left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )}{e \left (-20-x+x^2\right ) \log ^2\left (\frac {400+40 x-39 x^2-2 x^3+x^4}{x^2}\right )} \, dx=\frac {3 x^{2}}{\mathrm {log}\left (\frac {x^{4}-2 x^{3}-39 x^{2}+40 x +400}{x^{2}}\right ) e} \] Input:
int(((6*x^3-6*x^2-120*x)*log((x^4-2*x^3-39*x^2+40*x+400)/x^2)-6*x^3-120*x) /(x^2-x-20)/exp(1)/log((x^4-2*x^3-39*x^2+40*x+400)/x^2)^2,x)
Output:
(3*x**2)/(log((x**4 - 2*x**3 - 39*x**2 + 40*x + 400)/x**2)*e)