Integrand size = 127, antiderivative size = 28 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=\log \left (2-x+\left (x+\frac {5-x}{1-\log \left (\frac {5}{x}\right )}\right )^2\right ) \] Output:
ln(((5-x)/(1-ln(5/x))+x)^2+2-x)
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=-2 \log \left (1-\log \left (\frac {5}{x}\right )\right )+\log \left (27-x-4 \log \left (\frac {5}{x}\right )-8 x \log \left (\frac {5}{x}\right )+2 \log ^2\left (\frac {5}{x}\right )-x \log ^2\left (\frac {5}{x}\right )+x^2 \log ^2\left (\frac {5}{x}\right )\right ) \] Input:
Integrate[(50 - 9*x + (-3*x + 2*x^2)*Log[5/x] + (-7*x - 2*x^2)*Log[5/x]^2 + (-x + 2*x^2)*Log[5/x]^3)/(-27*x + x^2 + (31*x + 7*x^2)*Log[5/x] + (-6*x - 7*x^2 - x^3)*Log[5/x]^2 + (2*x - x^2 + x^3)*Log[5/x]^3),x]
Output:
-2*Log[1 - Log[5/x]] + Log[27 - x - 4*Log[5/x] - 8*x*Log[5/x] + 2*Log[5/x] ^2 - x*Log[5/x]^2 + x^2*Log[5/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.
| \(\displaystyle \int \frac {\left (2 x^2-x\right ) \log ^3\left (\frac {5}{x}\right )+\left (-2 x^2-7 x\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x^2-3 x\right ) \log \left (\frac {5}{x}\right )-9 x+50}{x^2+\left (7 x^2+31 x\right ) \log \left (\frac {5}{x}\right )+\left (x^3-x^2+2 x\right ) \log ^3\left (\frac {5}{x}\right )+\left (-x^3-7 x^2-6 x\right ) \log ^2\left (\frac {5}{x}\right )-27 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (2 x^2-x\right ) \log ^3\left (\frac {5}{x}\right )\right )-\left (-2 x^2-7 x\right ) \log ^2\left (\frac {5}{x}\right )-\left (2 x^2-3 x\right ) \log \left (\frac {5}{x}\right )+9 x-50}{x \left (1-\log \left (\frac {5}{x}\right )\right ) \left (x^2 \log ^2\left (\frac {5}{x}\right )-x-x \log ^2\left (\frac {5}{x}\right )+2 \log ^2\left (\frac {5}{x}\right )-8 x \log \left (\frac {5}{x}\right )-4 \log \left (\frac {5}{x}\right )+27\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x-1}{x^2-x+2}+\frac {-2 x^4 \log \left (\frac {5}{x}\right )+9 x^3+12 x^3 \log \left (\frac {5}{x}\right )-58 x^2-2 x^2 \log \left (\frac {5}{x}\right )+37 x-12 x \log \left (\frac {5}{x}\right )-8 \log \left (\frac {5}{x}\right )+8}{x \left (x^2-x+2\right ) \left (x^2 \log ^2\left (\frac {5}{x}\right )-x-x \log ^2\left (\frac {5}{x}\right )+2 \log ^2\left (\frac {5}{x}\right )-8 x \log \left (\frac {5}{x}\right )-4 \log \left (\frac {5}{x}\right )+27\right )}+\frac {2}{x \left (\log \left (\frac {5}{x}\right )-1\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 x-1}{x^2-x+2}+\frac {-2 x^4 \log \left (\frac {5}{x}\right )+9 x^3+12 x^3 \log \left (\frac {5}{x}\right )-58 x^2-2 x^2 \log \left (\frac {5}{x}\right )+37 x-12 x \log \left (\frac {5}{x}\right )-8 \log \left (\frac {5}{x}\right )+8}{x \left (x^2-x+2\right ) \left (x^2 \log ^2\left (\frac {5}{x}\right )-x-x \log ^2\left (\frac {5}{x}\right )+2 \log ^2\left (\frac {5}{x}\right )-8 x \log \left (\frac {5}{x}\right )-4 \log \left (\frac {5}{x}\right )+27\right )}+\frac {2}{x \left (\log \left (\frac {5}{x}\right )-1\right )}\right )dx\) |
Input:
Int[(50 - 9*x + (-3*x + 2*x^2)*Log[5/x] + (-7*x - 2*x^2)*Log[5/x]^2 + (-x + 2*x^2)*Log[5/x]^3)/(-27*x + x^2 + (31*x + 7*x^2)*Log[5/x] + (-6*x - 7*x^ 2 - x^3)*Log[5/x]^2 + (2*x - x^2 + x^3)*Log[5/x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(28)=56\).
Time = 1.55 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46
| method | result | size |
| norman | \(-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (x^{2} \ln \left (\frac {5}{x}\right )^{2}-x \ln \left (\frac {5}{x}\right )^{2}-8 x \ln \left (\frac {5}{x}\right )+2 \ln \left (\frac {5}{x}\right )^{2}-x -4 \ln \left (\frac {5}{x}\right )+27\right )\) | \(69\) |
| parallelrisch | \(-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (x^{2} \ln \left (\frac {5}{x}\right )^{2}-x \ln \left (\frac {5}{x}\right )^{2}-8 x \ln \left (\frac {5}{x}\right )+2 \ln \left (\frac {5}{x}\right )^{2}-x -4 \ln \left (\frac {5}{x}\right )+27\right )\) | \(69\) |
| risch | \(\ln \left (x^{2}-x +2\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\ln \left (\frac {5}{x}\right )^{2}-\frac {4 \left (1+2 x \right ) \ln \left (\frac {5}{x}\right )}{x^{2}-x +2}-\frac {x -27}{x^{2}-x +2}\right )\) | \(70\) |
| derivativedivides | \(-2 \ln \left (\frac {5}{x}\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\frac {50 \ln \left (\frac {5}{x}\right )^{2}}{x^{2}}-\frac {25 \ln \left (\frac {5}{x}\right )^{2}}{x}-\frac {100 \ln \left (\frac {5}{x}\right )}{x^{2}}+25 \ln \left (\frac {5}{x}\right )^{2}-\frac {200 \ln \left (\frac {5}{x}\right )}{x}+\frac {675}{x^{2}}-\frac {25}{x}\right )\) | \(91\) |
| default | \(-2 \ln \left (\frac {5}{x}\right )-2 \ln \left (\ln \left (\frac {5}{x}\right )-1\right )+\ln \left (\frac {50 \ln \left (\frac {5}{x}\right )^{2}}{x^{2}}-\frac {25 \ln \left (\frac {5}{x}\right )^{2}}{x}-\frac {100 \ln \left (\frac {5}{x}\right )}{x^{2}}+25 \ln \left (\frac {5}{x}\right )^{2}-\frac {200 \ln \left (\frac {5}{x}\right )}{x}+\frac {675}{x^{2}}-\frac {25}{x}\right )\) | \(91\) |
Input:
int(((2*x^2-x)*ln(5/x)^3+(-2*x^2-7*x)*ln(5/x)^2+(2*x^2-3*x)*ln(5/x)-9*x+50 )/((x^3-x^2+2*x)*ln(5/x)^3+(-x^3-7*x^2-6*x)*ln(5/x)^2+(7*x^2+31*x)*ln(5/x) +x^2-27*x),x,method=_RETURNVERBOSE)
Output:
-2*ln(ln(5/x)-1)+ln(x^2*ln(5/x)^2-x*ln(5/x)^2-8*x*ln(5/x)+2*ln(5/x)^2-x-4* ln(5/x)+27)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=\log \left (x^{2} - x + 2\right ) + \log \left (\frac {{\left (x^{2} - x + 2\right )} \log \left (\frac {5}{x}\right )^{2} - 4 \, {\left (2 \, x + 1\right )} \log \left (\frac {5}{x}\right ) - x + 27}{x^{2} - x + 2}\right ) - 2 \, \log \left (\log \left (\frac {5}{x}\right ) - 1\right ) \] Input:
integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/ x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31 *x)*log(5/x)+x^2-27*x),x, algorithm="fricas")
Output:
log(x^2 - x + 2) + log(((x^2 - x + 2)*log(5/x)^2 - 4*(2*x + 1)*log(5/x) - x + 27)/(x^2 - x + 2)) - 2*log(log(5/x) - 1)
Exception generated. \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((2*x**2-x)*ln(5/x)**3+(-2*x**2-7*x)*ln(5/x)**2+(2*x**2-3*x)*ln( 5/x)-9*x+50)/((x**3-x**2+2*x)*ln(5/x)**3+(-x**3-7*x**2-6*x)*ln(5/x)**2+(7* x**2+31*x)*ln(5/x)+x**2-27*x),x)
Output:
Exception raised: PolynomialError >> 1/(x**10 - 4*x**9 + 14*x**8 - 28*x**7 + 49*x**6 - 56*x**5 + 56*x**4 - 32*x**3 + 16*x**2) contains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.64 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=\log \left (x^{2} - x + 2\right ) + \log \left (\frac {x^{2} \log \left (5\right )^{2} + {\left (x^{2} - x + 2\right )} \log \left (x\right )^{2} - {\left (\log \left (5\right )^{2} + 8 \, \log \left (5\right ) + 1\right )} x + 2 \, \log \left (5\right )^{2} - 2 \, {\left (x^{2} \log \left (5\right ) - x {\left (\log \left (5\right ) + 4\right )} + 2 \, \log \left (5\right ) - 2\right )} \log \left (x\right ) - 4 \, \log \left (5\right ) + 27}{x^{2} - x + 2}\right ) - 2 \, \log \left (-\log \left (5\right ) + \log \left (x\right ) + 1\right ) \] Input:
integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/ x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31 *x)*log(5/x)+x^2-27*x),x, algorithm="maxima")
Output:
log(x^2 - x + 2) + log((x^2*log(5)^2 + (x^2 - x + 2)*log(x)^2 - (log(5)^2 + 8*log(5) + 1)*x + 2*log(5)^2 - 2*(x^2*log(5) - x*(log(5) + 4) + 2*log(5) - 2)*log(x) - 4*log(5) + 27)/(x^2 - x + 2)) - 2*log(-log(5) + log(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.21 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=\log \left (25 \, \log \left (\frac {5}{x}\right )^{2} - \frac {25 \, \log \left (\frac {5}{x}\right )^{2}}{x} - \frac {200 \, \log \left (\frac {5}{x}\right )}{x} + \frac {50 \, \log \left (\frac {5}{x}\right )^{2}}{x^{2}} - \frac {25}{x} - \frac {100 \, \log \left (\frac {5}{x}\right )}{x^{2}} + \frac {675}{x^{2}}\right ) - 2 \, \log \left (\frac {5}{x}\right ) - 2 \, \log \left (\log \left (\frac {5}{x}\right ) - 1\right ) \] Input:
integrate(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/ x)-9*x+50)/((x^3-x^2+2*x)*log(5/x)^3+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31 *x)*log(5/x)+x^2-27*x),x, algorithm="giac")
Output:
log(25*log(5/x)^2 - 25*log(5/x)^2/x - 200*log(5/x)/x + 50*log(5/x)^2/x^2 - 25/x - 100*log(5/x)/x^2 + 675/x^2) - 2*log(5/x) - 2*log(log(5/x) - 1)
Time = 3.42 (sec) , antiderivative size = 173, normalized size of antiderivative = 6.18 \[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx=4\,\ln \left (x-5\right )-2\,\ln \left (\frac {\left (\ln \left (\frac {5}{x}\right )-1\right )\,{\left (x-5\right )}^2\,\left (-4\,x^5+49\,x^4-166\,x^3+113\,x^2+112\,x+32\right )}{x^2\,{\left (x^2-x+2\right )}^4}\right )-7\,\ln \left (x^2-x+2\right )+2\,\ln \left (4\,x^5-49\,x^4+166\,x^3-113\,x^2-112\,x-32\right )+\ln \left (\frac {-x^2\,{\ln \left (\frac {5}{x}\right )}^2+x\,{\ln \left (\frac {5}{x}\right )}^2+8\,x\,\ln \left (\frac {5}{x}\right )+x-2\,{\ln \left (\frac {5}{x}\right )}^2+4\,\ln \left (\frac {5}{x}\right )-27}{x\,\left (x^2-x+2\right )}\right )-3\,\ln \left (x\right ) \] Input:
int(-(9*x + log(5/x)*(3*x - 2*x^2) + log(5/x)^3*(x - 2*x^2) + log(5/x)^2*( 7*x + 2*x^2) - 50)/(log(5/x)*(31*x + 7*x^2) - 27*x + log(5/x)^3*(2*x - x^2 + x^3) - log(5/x)^2*(6*x + 7*x^2 + x^3) + x^2),x)
Output:
4*log(x - 5) - 2*log(((log(5/x) - 1)*(x - 5)^2*(112*x + 113*x^2 - 166*x^3 + 49*x^4 - 4*x^5 + 32))/(x^2*(x^2 - x + 2)^4)) - 7*log(x^2 - x + 2) + 2*lo g(166*x^3 - 113*x^2 - 112*x - 49*x^4 + 4*x^5 - 32) + log((x + 4*log(5/x) - 2*log(5/x)^2 - x^2*log(5/x)^2 + 8*x*log(5/x) + x*log(5/x)^2 - 27)/(x*(x^2 - x + 2))) - 3*log(x)
\[ \int \frac {50-9 x+\left (-3 x+2 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-7 x-2 x^2\right ) \log ^2\left (\frac {5}{x}\right )+\left (-x+2 x^2\right ) \log ^3\left (\frac {5}{x}\right )}{-27 x+x^2+\left (31 x+7 x^2\right ) \log \left (\frac {5}{x}\right )+\left (-6 x-7 x^2-x^3\right ) \log ^2\left (\frac {5}{x}\right )+\left (2 x-x^2+x^3\right ) \log ^3\left (\frac {5}{x}\right )} \, dx =\text {Too large to display} \] Input:
int(((2*x^2-x)*log(5/x)^3+(-2*x^2-7*x)*log(5/x)^2+(2*x^2-3*x)*log(5/x)-9*x +50)/((x^3-x^2+2*x)*log(5/x)^3+(-x^3-7*x^2-6*x)*log(5/x)^2+(7*x^2+31*x)*lo g(5/x)+x^2-27*x),x)
Output:
- int(log(5/x)**3/(log(5/x)**3*x**2 - log(5/x)**3*x + 2*log(5/x)**3 - log (5/x)**2*x**2 - 7*log(5/x)**2*x - 6*log(5/x)**2 + 7*log(5/x)*x + 31*log(5/ x) + x - 27),x) - 7*int(log(5/x)**2/(log(5/x)**3*x**2 - log(5/x)**3*x + 2* log(5/x)**3 - log(5/x)**2*x**2 - 7*log(5/x)**2*x - 6*log(5/x)**2 + 7*log(5 /x)*x + 31*log(5/x) + x - 27),x) - 3*int(log(5/x)/(log(5/x)**3*x**2 - log( 5/x)**3*x + 2*log(5/x)**3 - log(5/x)**2*x**2 - 7*log(5/x)**2*x - 6*log(5/x )**2 + 7*log(5/x)*x + 31*log(5/x) + x - 27),x) + 2*int((log(5/x)**3*x)/(lo g(5/x)**3*x**2 - log(5/x)**3*x + 2*log(5/x)**3 - log(5/x)**2*x**2 - 7*log( 5/x)**2*x - 6*log(5/x)**2 + 7*log(5/x)*x + 31*log(5/x) + x - 27),x) - 2*in t((log(5/x)**2*x)/(log(5/x)**3*x**2 - log(5/x)**3*x + 2*log(5/x)**3 - log( 5/x)**2*x**2 - 7*log(5/x)**2*x - 6*log(5/x)**2 + 7*log(5/x)*x + 31*log(5/x ) + x - 27),x) + 2*int((log(5/x)*x)/(log(5/x)**3*x**2 - log(5/x)**3*x + 2* log(5/x)**3 - log(5/x)**2*x**2 - 7*log(5/x)**2*x - 6*log(5/x)**2 + 7*log(5 /x)*x + 31*log(5/x) + x - 27),x) + 50*int(1/(log(5/x)**3*x**3 - log(5/x)** 3*x**2 + 2*log(5/x)**3*x - log(5/x)**2*x**3 - 7*log(5/x)**2*x**2 - 6*log(5 /x)**2*x + 7*log(5/x)*x**2 + 31*log(5/x)*x + x**2 - 27*x),x) - 9*int(1/(lo g(5/x)**3*x**2 - log(5/x)**3*x + 2*log(5/x)**3 - log(5/x)**2*x**2 - 7*log( 5/x)**2*x - 6*log(5/x)**2 + 7*log(5/x)*x + 31*log(5/x) + x - 27),x)