Integrand size = 67, antiderivative size = 29 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {2}{25} \log (x) \left (x+\left (-\log (x)+\log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )^2\right ) \] Output:
2/25*ln(x)*(x+(ln(x^2/exp(2/5/x))-ln(x))^2)
Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(29)=58\).
Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.93 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {24 \log ^2(x)-10 x \log ^3(x)+\log ^2\left (x^2\right ) \left (6-10 x \log \left (x^2\right )+15 x \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )+6 \log (x) \left (5 x \log ^2\left (x^2\right )-2 \log \left (x^2\right ) \left (2+5 x \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )+5 x \left (x+\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )\right )}{375 x} \] Input:
Integrate[(10*x^2 + 10*x^2*Log[x] + (-8 - 10*x)*Log[x]^2 + 8*Log[x]*Log[x^ 2/E^(2/(5*x))] + 10*x*Log[x^2/E^(2/(5*x))]^2)/(125*x^2),x]
Output:
(24*Log[x]^2 - 10*x*Log[x]^3 + Log[x^2]^2*(6 - 10*x*Log[x^2] + 15*x*Log[x^ 2/E^(2/(5*x))]) + 6*Log[x]*(5*x*Log[x^2]^2 - 2*Log[x^2]*(2 + 5*x*Log[x^2/E ^(2/(5*x))]) + 5*x*(x + Log[x^2/E^(2/(5*x))]^2)))/(375*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.
| \(\displaystyle \int \frac {10 x^2+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x^2 \log (x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+(-10 x-8) \log ^2(x)}{125 x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{125} \int \frac {2 \left (5 \log (x) x^2+5 x^2+5 \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right ) x-(5 x+4) \log ^2(x)+4 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{125} \int \frac {5 \log (x) x^2+5 x^2+5 \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right ) x-(5 x+4) \log ^2(x)+4 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {2}{125} \int \left (\frac {5 \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x}+\frac {4 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x^2}+\frac {5 \log (x) x^2+5 x^2-5 \log ^2(x) x-4 \log ^2(x)}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{125} \left (5 \int \frac {\log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x}dx-\frac {4 (5 x+1)^2}{5 x^2}-\frac {2}{5 x^2}-\frac {4 (5 x+1)^2 \log (x)}{5 x^2}-\frac {4 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right ) \log (x)}{x}-\frac {4 \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{x}-\frac {5}{3} \log ^3(x)+\frac {4 \log ^2(x)}{x}+5 x \log (x)+\frac {8 \log (x)}{x}+20 \log (x)\right )\) |
Input:
Int[(10*x^2 + 10*x^2*Log[x] + (-8 - 10*x)*Log[x]^2 + 8*Log[x]*Log[x^2/E^(2 /(5*x))] + 10*x*Log[x^2/E^(2/(5*x))]^2)/(125*x^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(205\) vs. \(2(26)=52\).
Time = 0.38 (sec) , antiderivative size = 206, normalized size of antiderivative = 7.10
| method | result | size |
| default | \(\frac {2 x \ln \left (x \right )}{25}+\frac {8 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{125}-\frac {8 \ln \left (x \right )^{2}}{125 x}+\frac {2 {\left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right )}^{2} \ln \left (x \right )}{25}+\frac {2 \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )^{2} \ln \left (x \right )}{25}-\frac {4 \left (\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )-\frac {2}{5 x}\right ) \ln \left (x \right )^{2}}{25}+\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \ln \left (x \right )^{2}}{25}+\frac {2 \ln \left (x \right )^{3}}{25}-\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \left (\ln \left (x \right ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )-\frac {2 \ln \left (x \right )}{5 x}-\frac {2}{5 x}\right )}{25}\) | \(206\) |
| parts | \(\frac {2 x \ln \left (x \right )}{25}+\frac {8 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{125}-\frac {8 \ln \left (x \right )^{2}}{125 x}+\frac {2 {\left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right )}^{2} \ln \left (x \right )}{25}+\frac {2 \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )^{2} \ln \left (x \right )}{25}-\frac {4 \left (\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )-\frac {2}{5 x}\right ) \ln \left (x \right )^{2}}{25}+\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \ln \left (x \right )^{2}}{25}+\frac {2 \ln \left (x \right )^{3}}{25}-\frac {4 \left (\ln \left (x^{2} {\mathrm e}^{-\frac {2}{5 x}}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )\right ) \left (\ln \left (x \right ) \ln \left ({\mathrm e}^{\frac {2}{5 x}}\right )-\frac {2 \ln \left (x \right )}{5 x}-\frac {2}{5 x}\right )}{25}\) | \(206\) |
| risch | \(\text {Expression too large to display}\) | \(1286\) |
Input:
int(1/125*(10*x*ln(x^2/exp(2/5/x))^2+8*ln(x)*ln(x^2/exp(2/5/x))+(-10*x-8)* ln(x)^2+10*x^2*ln(x)+10*x^2)/x^2,x,method=_RETURNVERBOSE)
Output:
2/25*x*ln(x)+8/125*(ln(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))*(-ln(x)/x-1 /x)-8/125*ln(x)^2/x+2/25*(ln(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))^2*ln( x)+2/25*ln(exp(2/5/x))^2*ln(x)-4/25*(ln(exp(2/5/x))-2/5/x)*ln(x)^2+4/25*(l n(x^2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))*ln(x)^2+2/25*ln(x)^3-4/25*(ln(x^ 2/exp(2/5/x))-2*ln(x)+ln(exp(2/5/x)))*(ln(x)*ln(exp(2/5/x))-2/5*ln(x)/x-2/ 5/x)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {2 \, {\left (25 \, x^{2} \log \left (x\right )^{3} - 20 \, x \log \left (x\right )^{2} + {\left (25 \, x^{3} + 4\right )} \log \left (x\right )\right )}}{625 \, x^{2}} \] Input:
integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+( -10*x-8)*log(x)^2+10*x^2*log(x)+10*x^2)/x^2,x, algorithm="fricas")
Output:
2/625*(25*x^2*log(x)^3 - 20*x*log(x)^2 + (25*x^3 + 4)*log(x))/x^2
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {2 \log {\left (x \right )}^{3}}{25} - \frac {8 \log {\left (x \right )}^{2}}{125 x} + \frac {\left (50 x^{3} + 8\right ) \log {\left (x \right )}}{625 x^{2}} \] Input:
integrate(1/125*(10*x*ln(x**2/exp(2/5/x))**2+8*ln(x)*ln(x**2/exp(2/5/x))+( -10*x-8)*ln(x)**2+10*x**2*ln(x)+10*x**2)/x**2,x)
Output:
2*log(x)**3/25 - 8*log(x)**2/(125*x) + (50*x**3 + 8)*log(x)/(625*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (24) = 48\).
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.31 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=-\frac {2}{75} \, \log \left (x\right )^{3} + \frac {2}{25} \, x \log \left (x\right ) - \frac {8}{625} \, {\left (\frac {5 \, \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right )}{x} + \frac {10 \, x + 1}{x^{2}}\right )} \log \left (x\right ) + \frac {8 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 2\right )}}{125 \, x} + \frac {4 \, {\left (50 \, x^{2} \log \left (x\right )^{3} + 60 \, x \log \left (x\right ) + 60 \, x - 3\right )}}{1875 \, x^{2}} - \frac {4 \, {\left (20 \, x \log \left (x\right ) + 40 \, x - 1\right )}}{625 \, x^{2}} \] Input:
integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+( -10*x-8)*log(x)^2+10*x^2*log(x)+10*x^2)/x^2,x, algorithm="maxima")
Output:
-2/75*log(x)^3 + 2/25*x*log(x) - 8/625*(5*log(x^2*e^(-2/5/x))/x + (10*x + 1)/x^2)*log(x) + 8/125*(log(x)^2 + 2*log(x) + 2)/x + 4/1875*(50*x^2*log(x) ^3 + 60*x*log(x) + 60*x - 3)/x^2 - 4/625*(20*x*log(x) + 40*x - 1)/x^2
\[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\int { \frac {2 \, {\left (5 \, x \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right )^{2} + 5 \, x^{2} \log \left (x\right ) - {\left (5 \, x + 4\right )} \log \left (x\right )^{2} + 5 \, x^{2} + 4 \, \log \left (x^{2} e^{\left (-\frac {2}{5 \, x}\right )}\right ) \log \left (x\right )\right )}}{125 \, x^{2}} \,d x } \] Input:
integrate(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+( -10*x-8)*log(x)^2+10*x^2*log(x)+10*x^2)/x^2,x, algorithm="giac")
Output:
integrate(2/125*(5*x*log(x^2*e^(-2/5/x))^2 + 5*x^2*log(x) - (5*x + 4)*log( x)^2 + 5*x^2 + 4*log(x^2*e^(-2/5/x))*log(x))/x^2, x)
Time = 3.56 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {2\,\ln \left (x\right )\,\left (25\,x^3+25\,x^2\,{\ln \left (x^2\right )}^2-50\,x^2\,\ln \left (x^2\right )\,\ln \left (x\right )+25\,x^2\,{\ln \left (x\right )}^2-20\,x\,\ln \left (x^2\right )+20\,x\,\ln \left (x\right )+4\right )}{625\,x^2} \] Input:
int(((2*x^2*log(x))/25 + (2*x*log(x^2*exp(-2/(5*x)))^2)/25 + (8*log(x^2*ex p(-2/(5*x)))*log(x))/125 + (2*x^2)/25 - (log(x)^2*(10*x + 8))/125)/x^2,x)
Output:
(2*log(x)*(25*x^2*log(x)^2 - 20*x*log(x^2) + 20*x*log(x) + 25*x^3 + 25*x^2 *log(x^2)^2 - 50*x^2*log(x^2)*log(x) + 4))/(625*x^2)
\[ \int \frac {10 x^2+10 x^2 \log (x)+(-8-10 x) \log ^2(x)+8 \log (x) \log \left (e^{\left .-\frac {2}{5}\right /x} x^2\right )+10 x \log ^2\left (e^{\left .-\frac {2}{5}\right /x} x^2\right )}{125 x^2} \, dx=\frac {150 \left (\int \frac {\mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right )^{2}}{x}d x \right ) x^{2}+225 \mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right )^{2} x^{2}-900 \,\mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right ) \mathrm {log}\left (x \right ) x^{2}-120 \,\mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right ) \mathrm {log}\left (x \right ) x +1200 \,\mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right ) x^{2}+60 \,\mathrm {log}\left (\frac {x^{2}}{e^{\frac {2}{5 x}}}\right ) x -50 \mathrm {log}\left (x \right )^{3} x^{2}+900 \mathrm {log}\left (x \right )^{2} x^{2}+120 \mathrm {log}\left (x \right )^{2} x +150 \,\mathrm {log}\left (x \right ) x^{3}-2400 \,\mathrm {log}\left (x \right ) x^{2}-360 \,\mathrm {log}\left (x \right ) x -24 \,\mathrm {log}\left (x \right )+240 x}{1875 x^{2}} \] Input:
int(1/125*(10*x*log(x^2/exp(2/5/x))^2+8*log(x)*log(x^2/exp(2/5/x))+(-10*x- 8)*log(x)^2+10*x^2*log(x)+10*x^2)/x^2,x)
Output:
(150*int(log(x**2/e**(2/(5*x)))**2/x,x)*x**2 + 225*log(x**2/e**(2/(5*x)))* *2*x**2 - 900*log(x**2/e**(2/(5*x)))*log(x)*x**2 - 120*log(x**2/e**(2/(5*x )))*log(x)*x + 1200*log(x**2/e**(2/(5*x)))*x**2 + 60*log(x**2/e**(2/(5*x)) )*x - 50*log(x)**3*x**2 + 900*log(x)**2*x**2 + 120*log(x)**2*x + 150*log(x )*x**3 - 2400*log(x)*x**2 - 360*log(x)*x - 24*log(x) + 240*x)/(1875*x**2)