Integrand size = 80, antiderivative size = 25 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=3+x-\frac {1+\log (5)}{x}+\frac {x}{\log \left (\frac {3 (13+x)}{x}\right )} \] Output:
3-(ln(5)+1)/x+x+x/ln(3/x*(13+x))
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=x+\frac {-1-\log (5)}{x}+\frac {x}{\log \left (3+\frac {39}{x}\right )} \] Input:
Integrate[(13*x^2 + (13*x^2 + x^3)*Log[(39 + 3*x)/x] + (13 + x + 13*x^2 + x^3 + (13 + x)*Log[5])*Log[(39 + 3*x)/x]^2)/((13*x^2 + x^3)*Log[(39 + 3*x) /x]^2),x]
Output:
x + (-1 - Log[5])/x + x/Log[3 + 39/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.
\(\displaystyle \int \frac {13 x^2+\left (x^3+13 x^2+x+(x+13) \log (5)+13\right ) \log ^2\left (\frac {3 x+39}{x}\right )+\left (x^3+13 x^2\right ) \log \left (\frac {3 x+39}{x}\right )}{\left (x^3+13 x^2\right ) \log ^2\left (\frac {3 x+39}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {13 x^2+\left (x^3+13 x^2+x+(x+13) \log (5)+13\right ) \log ^2\left (\frac {3 x+39}{x}\right )+\left (x^3+13 x^2\right ) \log \left (\frac {3 x+39}{x}\right )}{x^2 (x+13) \log ^2\left (\frac {3 x+39}{x}\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {x^2+1+\log (5)}{x^2}+\frac {13}{(x+13) \log ^2\left (\frac {39}{x}+3\right )}+\frac {1}{\log \left (\frac {39}{x}+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 13 \int \frac {1}{(x+13) \log ^2\left (3+\frac {39}{x}\right )}dx+\int \frac {1}{\log \left (3+\frac {39}{x}\right )}dx+x-\frac {1+\log (5)}{x}\) |
Input:
Int[(13*x^2 + (13*x^2 + x^3)*Log[(39 + 3*x)/x] + (13 + x + 13*x^2 + x^3 + (13 + x)*Log[5])*Log[(39 + 3*x)/x]^2)/((13*x^2 + x^3)*Log[(39 + 3*x)/x]^2) ,x]
Output:
$Aborted
Time = 1.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
risch | \(-\frac {-x^{2}+\ln \left (5\right )+1}{x}+\frac {x}{\ln \left (\frac {3 x +39}{x}\right )}\) | \(30\) |
norman | \(\frac {x^{2}+x^{2} \ln \left (\frac {3 x +39}{x}\right )+\left (-\ln \left (5\right )-1\right ) \ln \left (\frac {3 x +39}{x}\right )}{x \ln \left (\frac {3 x +39}{x}\right )}\) | \(52\) |
derivativedivides | \(-\frac {\left (-1+\left (-\frac {170}{169}-\frac {\ln \left (5\right )}{169}\right ) \ln \left (3+\frac {39}{x}\right )+\left (\frac {\ln \left (5\right )}{1521}+\frac {1}{1521}\right ) \ln \left (3+\frac {39}{x}\right ) \left (3+\frac {39}{x}\right )^{2}\right ) x}{\ln \left (3+\frac {39}{x}\right )}\) | \(55\) |
default | \(-\frac {\left (-1+\left (-\frac {170}{169}-\frac {\ln \left (5\right )}{169}\right ) \ln \left (3+\frac {39}{x}\right )+\left (\frac {\ln \left (5\right )}{1521}+\frac {1}{1521}\right ) \ln \left (3+\frac {39}{x}\right ) \left (3+\frac {39}{x}\right )^{2}\right ) x}{\ln \left (3+\frac {39}{x}\right )}\) | \(55\) |
parallelrisch | \(-\frac {-x^{2} \ln \left (\frac {3 x +39}{x}\right )+\ln \left (\frac {3 x +39}{x}\right ) \ln \left (5\right )-x^{2}+26 x \ln \left (\frac {3 x +39}{x}\right )+\ln \left (\frac {3 x +39}{x}\right )}{x \ln \left (\frac {3 x +39}{x}\right )}\) | \(70\) |
Input:
int((((x+13)*ln(5)+x^3+13*x^2+x+13)*ln((3*x+39)/x)^2+(x^3+13*x^2)*ln((3*x+ 39)/x)+13*x^2)/(x^3+13*x^2)/ln((3*x+39)/x)^2,x,method=_RETURNVERBOSE)
Output:
-(-x^2+ln(5)+1)/x+x/ln((3*x+39)/x)
Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {x^{2} + {\left (x^{2} - \log \left (5\right ) - 1\right )} \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )}{x \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )} \] Input:
integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)* log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)/log((3*x+39)/x)^2,x, algorithm="frica s")
Output:
(x^2 + (x^2 - log(5) - 1)*log(3*(x + 13)/x))/(x*log(3*(x + 13)/x))
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=x + \frac {x}{\log {\left (\frac {3 x + 39}{x} \right )}} + \frac {- \log {\left (5 \right )} - 1}{x} \] Input:
integrate((((x+13)*ln(5)+x**3+13*x**2+x+13)*ln((3*x+39)/x)**2+(x**3+13*x** 2)*ln((3*x+39)/x)+13*x**2)/(x**3+13*x**2)/ln((3*x+39)/x)**2,x)
Output:
x + x/log((3*x + 39)/x) + (-log(5) - 1)/x
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 224, normalized size of antiderivative = 8.96 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {2}{13} \, {\left ({\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \log \left (\frac {39}{x} + 3\right ) \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \log \left (x + 13\right ) + \log \left (x\right )\right )} \log \left (5\right ) - \frac {1}{13} \, {\left (\frac {13}{x} - \log \left (x + 13\right ) + \log \left (x\right )\right )} \log \left (5\right ) + \frac {\log \left (5\right ) \log \left (\frac {39}{x} + 3\right )^{2}}{13 \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )}} + \frac {2}{13} \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) - \frac {2}{13} \, \log \left (\frac {39}{x} + 3\right ) \log \left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right ) + \frac {\log \left (\frac {39}{x} + 3\right )^{2}}{13 \, {\left (\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )\right )}} + \frac {x {\left (\log \left (3\right ) + 1\right )} + x \log \left (x + 13\right ) - x \log \left (x\right )}{\log \left (3\right ) + \log \left (x + 13\right ) - \log \left (x\right )} - \frac {1}{x} - \frac {1}{13} \, \log \left (x + 13\right ) + \frac {1}{13} \, \log \left (x\right ) \] Input:
integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)* log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)/log((3*x+39)/x)^2,x, algorithm="maxim a")
Output:
2/13*((log(3) + log(x + 13) - log(x))*log(log(3) + log(x + 13) - log(x)) - log(39/x + 3)*log(log(3) + log(x + 13) - log(x)) - log(x + 13) + log(x))* log(5) - 1/13*(13/x - log(x + 13) + log(x))*log(5) + 1/13*log(5)*log(39/x + 3)^2/(log(3) + log(x + 13) - log(x)) + 2/13*(log(3) + log(x + 13) - log( x))*log(log(3) + log(x + 13) - log(x)) - 2/13*log(39/x + 3)*log(log(3) + l og(x + 13) - log(x)) + 1/13*log(39/x + 3)^2/(log(3) + log(x + 13) - log(x) ) + (x*(log(3) + 1) + x*log(x + 13) - x*log(x))/(log(3) + log(x + 13) - lo g(x)) - 1/x - 1/13*log(x + 13) + 1/13*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).
Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=-\frac {{\left (x + 13\right )} {\left (\log \left (5\right ) + 1\right )}}{13 \, x} + \frac {13}{\frac {{\left (x + 13\right )} \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )}{x} - \log \left (\frac {3 \, {\left (x + 13\right )}}{x}\right )} + \frac {13}{\frac {x + 13}{x} - 1} \] Input:
integrate((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)* log((3*x+39)/x)+13*x^2)/(x^3+13*x^2)/log((3*x+39)/x)^2,x, algorithm="giac" )
Output:
-1/13*(x + 13)*(log(5) + 1)/x + 13/((x + 13)*log(3*(x + 13)/x)/x - log(3*( x + 13)/x)) + 13/((x + 13)/x - 1)
Time = 3.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.44 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {x^5+\left (\frac {2\,\ln \left (5\right )}{13}+\frac {340}{13}\right )\,x^4+\left (\ln \left (125\right )+172\right )\,x^3+\left (-169\,\ln \left (5\right )-169\right )\,x}{x^4+26\,x^3+169\,x^2}+\frac {x^5+26\,x^4+169\,x^3}{169\,x^2\,\ln \left (\frac {3\,x+39}{x}\right )+26\,x^3\,\ln \left (\frac {3\,x+39}{x}\right )+x^4\,\ln \left (\frac {3\,x+39}{x}\right )} \] Input:
int((log((3*x + 39)/x)^2*(x + log(5)*(x + 13) + 13*x^2 + x^3 + 13) + log(( 3*x + 39)/x)*(13*x^2 + x^3) + 13*x^2)/(log((3*x + 39)/x)^2*(13*x^2 + x^3)) ,x)
Output:
(x^3*(log(125) + 172) - x*(169*log(5) + 169) + x^4*((2*log(5))/13 + 340/13 ) + x^5)/(169*x^2 + 26*x^3 + x^4) + (169*x^3 + 26*x^4 + x^5)/(169*x^2*log( (3*x + 39)/x) + 26*x^3*log((3*x + 39)/x) + x^4*log((3*x + 39)/x))
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {13 x^2+\left (13 x^2+x^3\right ) \log \left (\frac {39+3 x}{x}\right )+\left (13+x+13 x^2+x^3+(13+x) \log (5)\right ) \log ^2\left (\frac {39+3 x}{x}\right )}{\left (13 x^2+x^3\right ) \log ^2\left (\frac {39+3 x}{x}\right )} \, dx=\frac {-\mathrm {log}\left (\frac {3 x +39}{x}\right ) \mathrm {log}\left (5\right )+\mathrm {log}\left (\frac {3 x +39}{x}\right ) x^{2}-\mathrm {log}\left (\frac {3 x +39}{x}\right )+x^{2}}{\mathrm {log}\left (\frac {3 x +39}{x}\right ) x} \] Input:
int((((x+13)*log(5)+x^3+13*x^2+x+13)*log((3*x+39)/x)^2+(x^3+13*x^2)*log((3 *x+39)/x)+13*x^2)/(x^3+13*x^2)/log((3*x+39)/x)^2,x)
Output:
( - log((3*x + 39)/x)*log(5) + log((3*x + 39)/x)*x**2 - log((3*x + 39)/x) + x**2)/(log((3*x + 39)/x)*x)