Integrand size = 182, antiderivative size = 21 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10}{x}+\frac {1}{\left (\frac {1}{x}+\log (2 x-\log (2))\right )^2} \] Output:
1/(1/x+ln(2*x-ln(2)))^2+10/x
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=-2 \left (-\frac {5}{x}-\frac {x^2}{2 (1+x \log (2 x-\log (2)))^2}\right ) \] Input:
Integrate[(20*x - 4*x^4 + 4*x^5 + (-10 + 2*x^3)*Log[2] + (60*x^2 - 30*x*Lo g[2])*Log[2*x - Log[2]] + (60*x^3 - 30*x^2*Log[2])*Log[2*x - Log[2]]^2 + ( 20*x^4 - 10*x^3*Log[2])*Log[2*x - Log[2]]^3)/(-2*x^3 + x^2*Log[2] + (-6*x^ 4 + 3*x^3*Log[2])*Log[2*x - Log[2]] + (-6*x^5 + 3*x^4*Log[2])*Log[2*x - Lo g[2]]^2 + (-2*x^6 + x^5*Log[2])*Log[2*x - Log[2]]^3),x]
Output:
-2*(-5/x - x^2/(2*(1 + x*Log[2*x - Log[2]])^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.
| \(\displaystyle \int \frac {4 x^5-4 x^4+\left (2 x^3-10\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+20 x}{-2 x^3+x^2 \log (2)+\left (x^5 \log (2)-2 x^6\right ) \log ^3(2 x-\log (2))+\left (3 x^4 \log (2)-6 x^5\right ) \log ^2(2 x-\log (2))+\left (3 x^3 \log (2)-6 x^4\right ) \log (2 x-\log (2))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-4 x^5+4 x^4-\left (2 x^3-10\right ) \log (2)-\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))-\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))-\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))-20 x}{x^2 (2 x-\log (2)) (x \log (2 x-\log (2))+1)^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {10}{x^2}-\frac {2 x \left (2 x^2-2 x+\log (2)\right )}{(2 x-\log (2)) (x \log (2 x-\log (2))+1)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {x^2}{(x \log (2 x-\log (2))+1)^3}dx-\frac {1}{2} \log ^3(2) \int \frac {1}{(2 x-\log (2)) (x \log (2 x-\log (2))+1)^3}dx-\frac {1}{2} \log ^2(2) \int \frac {1}{(x \log (2 x-\log (2))+1)^3}dx+(2-\log (2)) \int \frac {x}{(x \log (2 x-\log (2))+1)^3}dx+\frac {10}{x}\) |
Input:
Int[(20*x - 4*x^4 + 4*x^5 + (-10 + 2*x^3)*Log[2] + (60*x^2 - 30*x*Log[2])* Log[2*x - Log[2]] + (60*x^3 - 30*x^2*Log[2])*Log[2*x - Log[2]]^2 + (20*x^4 - 10*x^3*Log[2])*Log[2*x - Log[2]]^3)/(-2*x^3 + x^2*Log[2] + (-6*x^4 + 3* x^3*Log[2])*Log[2*x - Log[2]] + (-6*x^5 + 3*x^4*Log[2])*Log[2*x - Log[2]]^ 2 + (-2*x^6 + x^5*Log[2])*Log[2*x - Log[2]]^3),x]
Output:
$Aborted
Time = 27.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {10}{x}+\frac {x^{2}}{\left (x \ln \left (2 x -\ln \left (2\right )\right )+1\right )^{2}}\) | \(26\) |
| parallelrisch | \(\frac {40+40 \ln \left (2 x -\ln \left (2\right )\right )^{2} x^{2}+4 x^{3}+80 x \ln \left (2 x -\ln \left (2\right )\right )}{4 x \left (\ln \left (2 x -\ln \left (2\right )\right )^{2} x^{2}+2 x \ln \left (2 x -\ln \left (2\right )\right )+1\right )}\) | \(72\) |
| derivativedivides | \(-\frac {-\frac {3 \ln \left (2\right )^{2} \left (2 x -\ln \left (2\right )\right )}{2}-40 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )-10 \ln \left (2\right )^{2} \ln \left (2 x -\ln \left (2\right )\right )^{2}-10 \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )^{2}-40 \ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )-20 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )-\frac {\left (2 x -\ln \left (2\right )\right )^{3}}{2}-\frac {3 \ln \left (2\right ) \left (2 x -\ln \left (2\right )\right )^{2}}{2}-40-\frac {\ln \left (2\right )^{3}}{2}}{x \left (\ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )+\ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )+2\right )^{2}}\) | \(183\) |
| default | \(-\frac {-\frac {3 \ln \left (2\right )^{2} \left (2 x -\ln \left (2\right )\right )}{2}-40 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )-10 \ln \left (2\right )^{2} \ln \left (2 x -\ln \left (2\right )\right )^{2}-10 \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )^{2}-40 \ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )-20 \ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )^{2} \left (2 x -\ln \left (2\right )\right )-\frac {\left (2 x -\ln \left (2\right )\right )^{3}}{2}-\frac {3 \ln \left (2\right ) \left (2 x -\ln \left (2\right )\right )^{2}}{2}-40-\frac {\ln \left (2\right )^{3}}{2}}{x \left (\ln \left (2\right ) \ln \left (2 x -\ln \left (2\right )\right )+\ln \left (2 x -\ln \left (2\right )\right ) \left (2 x -\ln \left (2\right )\right )+2\right )^{2}}\) | \(183\) |
Input:
int(((-10*x^3*ln(2)+20*x^4)*ln(2*x-ln(2))^3+(-30*x^2*ln(2)+60*x^3)*ln(2*x- ln(2))^2+(-30*x*ln(2)+60*x^2)*ln(2*x-ln(2))+(2*x^3-10)*ln(2)+4*x^5-4*x^4+2 0*x)/((x^5*ln(2)-2*x^6)*ln(2*x-ln(2))^3+(3*x^4*ln(2)-6*x^5)*ln(2*x-ln(2))^ 2+(3*x^3*ln(2)-6*x^4)*ln(2*x-ln(2))+x^2*ln(2)-2*x^3),x,method=_RETURNVERBO SE)
Output:
10/x+x^2/(x*ln(2*x-ln(2))+1)^2
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{3} + 20 \, x \log \left (2 \, x - \log \left (2\right )\right ) + 10}{x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + 2 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + x} \] Input:
integrate(((-10*x^3*log(2)+20*x^4)*log(2*x-log(2))^3+(-30*x^2*log(2)+60*x^ 3)*log(2*x-log(2))^2+(-30*x*log(2)+60*x^2)*log(2*x-log(2))+(2*x^3-10)*log( 2)+4*x^5-4*x^4+20*x)/((x^5*log(2)-2*x^6)*log(2*x-log(2))^3+(3*x^4*log(2)-6 *x^5)*log(2*x-log(2))^2+(3*x^3*log(2)-6*x^4)*log(2*x-log(2))+x^2*log(2)-2* x^3),x, algorithm="fricas")
Output:
(10*x^2*log(2*x - log(2))^2 + x^3 + 20*x*log(2*x - log(2)) + 10)/(x^3*log( 2*x - log(2))^2 + 2*x^2*log(2*x - log(2)) + x)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {x^{2}}{x^{2} \log {\left (2 x - \log {\left (2 \right )} \right )}^{2} + 2 x \log {\left (2 x - \log {\left (2 \right )} \right )} + 1} + \frac {10}{x} \] Input:
integrate(((-10*x**3*ln(2)+20*x**4)*ln(2*x-ln(2))**3+(-30*x**2*ln(2)+60*x* *3)*ln(2*x-ln(2))**2+(-30*x*ln(2)+60*x**2)*ln(2*x-ln(2))+(2*x**3-10)*ln(2) +4*x**5-4*x**4+20*x)/((x**5*ln(2)-2*x**6)*ln(2*x-ln(2))**3+(3*x**4*ln(2)-6 *x**5)*ln(2*x-ln(2))**2+(3*x**3*ln(2)-6*x**4)*ln(2*x-ln(2))+x**2*ln(2)-2*x **3),x)
Output:
x**2/(x**2*log(2*x - log(2))**2 + 2*x*log(2*x - log(2)) + 1) + 10/x
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.19 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {10 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{3} + 20 \, x \log \left (2 \, x - \log \left (2\right )\right ) + 10}{x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + 2 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + x} \] Input:
integrate(((-10*x^3*log(2)+20*x^4)*log(2*x-log(2))^3+(-30*x^2*log(2)+60*x^ 3)*log(2*x-log(2))^2+(-30*x*log(2)+60*x^2)*log(2*x-log(2))+(2*x^3-10)*log( 2)+4*x^5-4*x^4+20*x)/((x^5*log(2)-2*x^6)*log(2*x-log(2))^3+(3*x^4*log(2)-6 *x^5)*log(2*x-log(2))^2+(3*x^3*log(2)-6*x^4)*log(2*x-log(2))+x^2*log(2)-2* x^3),x, algorithm="maxima")
Output:
(10*x^2*log(2*x - log(2))^2 + x^3 + 20*x*log(2*x - log(2)) + 10)/(x^3*log( 2*x - log(2))^2 + 2*x^2*log(2*x - log(2)) + x)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (21) = 42\).
Time = 0.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 6.10 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {2 \, x^{4} - 2 \, x^{3} + x^{2} \log \left (2\right )}{2 \, x^{4} \log \left (2 \, x - \log \left (2\right )\right )^{2} - 2 \, x^{3} \log \left (2 \, x - \log \left (2\right )\right )^{2} + x^{2} \log \left (2\right ) \log \left (2 \, x - \log \left (2\right )\right )^{2} + 4 \, x^{3} \log \left (2 \, x - \log \left (2\right )\right ) - 4 \, x^{2} \log \left (2 \, x - \log \left (2\right )\right ) + 2 \, x \log \left (2\right ) \log \left (2 \, x - \log \left (2\right )\right ) + 2 \, x^{2} - 2 \, x + \log \left (2\right )} + \frac {10}{x} \] Input:
integrate(((-10*x^3*log(2)+20*x^4)*log(2*x-log(2))^3+(-30*x^2*log(2)+60*x^ 3)*log(2*x-log(2))^2+(-30*x*log(2)+60*x^2)*log(2*x-log(2))+(2*x^3-10)*log( 2)+4*x^5-4*x^4+20*x)/((x^5*log(2)-2*x^6)*log(2*x-log(2))^3+(3*x^4*log(2)-6 *x^5)*log(2*x-log(2))^2+(3*x^3*log(2)-6*x^4)*log(2*x-log(2))+x^2*log(2)-2* x^3),x, algorithm="giac")
Output:
(2*x^4 - 2*x^3 + x^2*log(2))/(2*x^4*log(2*x - log(2))^2 - 2*x^3*log(2*x - log(2))^2 + x^2*log(2)*log(2*x - log(2))^2 + 4*x^3*log(2*x - log(2)) - 4*x ^2*log(2*x - log(2)) + 2*x*log(2)*log(2*x - log(2)) + 2*x^2 - 2*x + log(2) ) + 10/x
Timed out. \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\int -\frac {{\ln \left (2\,x-\ln \left (2\right )\right )}^3\,\left (10\,x^3\,\ln \left (2\right )-20\,x^4\right )-\ln \left (2\right )\,\left (2\,x^3-10\right )-20\,x+{\ln \left (2\,x-\ln \left (2\right )\right )}^2\,\left (30\,x^2\,\ln \left (2\right )-60\,x^3\right )+\ln \left (2\,x-\ln \left (2\right )\right )\,\left (30\,x\,\ln \left (2\right )-60\,x^2\right )+4\,x^4-4\,x^5}{\ln \left (2\,x-\ln \left (2\right )\right )\,\left (3\,x^3\,\ln \left (2\right )-6\,x^4\right )+{\ln \left (2\,x-\ln \left (2\right )\right )}^3\,\left (x^5\,\ln \left (2\right )-2\,x^6\right )+{\ln \left (2\,x-\ln \left (2\right )\right )}^2\,\left (3\,x^4\,\ln \left (2\right )-6\,x^5\right )+x^2\,\ln \left (2\right )-2\,x^3} \,d x \] Input:
int(-(log(2*x - log(2))^3*(10*x^3*log(2) - 20*x^4) - log(2)*(2*x^3 - 10) - 20*x + log(2*x - log(2))^2*(30*x^2*log(2) - 60*x^3) + log(2*x - log(2))*( 30*x*log(2) - 60*x^2) + 4*x^4 - 4*x^5)/(log(2*x - log(2))*(3*x^3*log(2) - 6*x^4) + log(2*x - log(2))^3*(x^5*log(2) - 2*x^6) + log(2*x - log(2))^2*(3 *x^4*log(2) - 6*x^5) + x^2*log(2) - 2*x^3),x)
Output:
int(-(log(2*x - log(2))^3*(10*x^3*log(2) - 20*x^4) - log(2)*(2*x^3 - 10) - 20*x + log(2*x - log(2))^2*(30*x^2*log(2) - 60*x^3) + log(2*x - log(2))*( 30*x*log(2) - 60*x^2) + 4*x^4 - 4*x^5)/(log(2*x - log(2))*(3*x^3*log(2) - 6*x^4) + log(2*x - log(2))^3*(x^5*log(2) - 2*x^6) + log(2*x - log(2))^2*(3 *x^4*log(2) - 6*x^5) + x^2*log(2) - 2*x^3), x)
Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.00 \[ \int \frac {20 x-4 x^4+4 x^5+\left (-10+2 x^3\right ) \log (2)+\left (60 x^2-30 x \log (2)\right ) \log (2 x-\log (2))+\left (60 x^3-30 x^2 \log (2)\right ) \log ^2(2 x-\log (2))+\left (20 x^4-10 x^3 \log (2)\right ) \log ^3(2 x-\log (2))}{-2 x^3+x^2 \log (2)+\left (-6 x^4+3 x^3 \log (2)\right ) \log (2 x-\log (2))+\left (-6 x^5+3 x^4 \log (2)\right ) \log ^2(2 x-\log (2))+\left (-2 x^6+x^5 \log (2)\right ) \log ^3(2 x-\log (2))} \, dx=\frac {20 \,\mathrm {log}\left (\mathrm {log}\left (2\right )-2 x \right ) \mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right )^{2} x^{3}+40 \,\mathrm {log}\left (\mathrm {log}\left (2\right )-2 x \right ) \mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right ) x^{2}+20 \,\mathrm {log}\left (\mathrm {log}\left (2\right )-2 x \right ) x -20 \mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right )^{3} x^{3}-30 \mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right )^{2} x^{2}+x^{3}+10}{x \left (\mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right )^{2} x^{2}+2 \,\mathrm {log}\left (-\mathrm {log}\left (2\right )+2 x \right ) x +1\right )} \] Input:
int(((-10*x^3*log(2)+20*x^4)*log(2*x-log(2))^3+(-30*x^2*log(2)+60*x^3)*log (2*x-log(2))^2+(-30*x*log(2)+60*x^2)*log(2*x-log(2))+(2*x^3-10)*log(2)+4*x ^5-4*x^4+20*x)/((x^5*log(2)-2*x^6)*log(2*x-log(2))^3+(3*x^4*log(2)-6*x^5)* log(2*x-log(2))^2+(3*x^3*log(2)-6*x^4)*log(2*x-log(2))+x^2*log(2)-2*x^3),x )
Output:
(20*log(log(2) - 2*x)*log( - log(2) + 2*x)**2*x**3 + 40*log(log(2) - 2*x)* log( - log(2) + 2*x)*x**2 + 20*log(log(2) - 2*x)*x - 20*log( - log(2) + 2* x)**3*x**3 - 30*log( - log(2) + 2*x)**2*x**2 + x**3 + 10)/(x*(log( - log(2 ) + 2*x)**2*x**2 + 2*log( - log(2) + 2*x)*x + 1))