Integrand size = 60, antiderivative size = 23 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=\frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right ) \] Output:
1/2*x*ln(5/x/ln(5/(exp(1)+4))+4)
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=\frac {1}{2} x \log \left (4+\frac {5}{x \log \left (\frac {5}{4+e}\right )}\right ) \] Input:
Integrate[(-5 + (5 + 4*x*Log[5/(4 + E)])*Log[(5 + 4*x*Log[5/(4 + E)])/(x*L og[5/(4 + E)])])/(10 + 8*x*Log[5/(4 + E)]),x]
Output:
(x*Log[4 + 5/(x*Log[5/(4 + E)])])/2
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {\left (4 x \log \left (\frac {5}{4+e}\right )+5\right ) \log \left (\frac {4 x \log \left (\frac {5}{4+e}\right )+5}{x \log \left (\frac {5}{4+e}\right )}\right )-5}{8 x \log \left (\frac {5}{4+e}\right )+10} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{2} \log \left (\frac {5}{x \log \left (\frac {5}{4+e}\right )}+4\right )-\frac {5}{2 \left (4 x \log \left (\frac {5}{4+e}\right )+5\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} x \log \left (\frac {5}{x \log \left (\frac {5}{4+e}\right )}+4\right )\) |
Input:
Int[(-5 + (5 + 4*x*Log[5/(4 + E)])*Log[(5 + 4*x*Log[5/(4 + E)])/(x*Log[5/( 4 + E)])])/(10 + 8*x*Log[5/(4 + E)]),x]
Output:
(x*Log[4 + 5/(x*Log[5/(4 + E)])])/2
Time = 0.70 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48
method | result | size |
norman | \(\frac {x \ln \left (\frac {4 x \ln \left (\frac {5}{{\mathrm e}+4}\right )+5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}\right )}{2}\) | \(34\) |
parallelrisch | \(\frac {x \ln \left (\frac {4 x \ln \left (\frac {5}{{\mathrm e}+4}\right )+5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}\right )}{2}\) | \(34\) |
risch | \(\frac {x \ln \left (\frac {4 x \left (\ln \left (5\right )-\ln \left ({\mathrm e}+4\right )\right )+5}{x \left (\ln \left (5\right )-\ln \left ({\mathrm e}+4\right )\right )}\right )}{2}\) | \(36\) |
derivativedivides | \(-\frac {-\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \ln \left (\frac {5}{{\mathrm e}+4}\right ) x}{4}+\frac {5 \ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right )}{4}}{2 \ln \left (\frac {5}{{\mathrm e}+4}\right )}\) | \(85\) |
default | \(-\frac {-\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \ln \left (\frac {5}{{\mathrm e}+4}\right ) x}{4}+\frac {5 \ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right )}{4}}{2 \ln \left (\frac {5}{{\mathrm e}+4}\right )}\) | \(85\) |
parts | \(-\frac {5 \ln \left (4 x \ln \left (\frac {5}{{\mathrm e}+4}\right )+5\right )}{8 \ln \left (\frac {5}{{\mathrm e}+4}\right )}-\frac {5 \left (\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}\right )}{4}-\frac {\ln \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \left (\frac {5}{x \ln \left (\frac {5}{{\mathrm e}+4}\right )}+4\right ) \ln \left (\frac {5}{{\mathrm e}+4}\right ) x}{20}\right )}{2 \ln \left (\frac {5}{{\mathrm e}+4}\right )}\) | \(112\) |
Input:
int(((4*x*ln(5/(exp(1)+4))+5)*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(exp(1)+4 )))-5)/(8*x*ln(5/(exp(1)+4))+10),x,method=_RETURNVERBOSE)
Output:
1/2*x*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(exp(1)+4)))
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=\frac {1}{2} \, x \log \left (\frac {4 \, x \log \left (\frac {5}{e + 4}\right ) + 5}{x \log \left (\frac {5}{e + 4}\right )}\right ) \] Input:
integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5 /(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+4))+10),x, algorithm="fricas")
Output:
1/2*x*log((4*x*log(5/(e + 4)) + 5)/(x*log(5/(e + 4))))
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=\frac {x \log {\left (\frac {4 x \log {\left (\frac {5}{e + 4} \right )} + 5}{x \log {\left (\frac {5}{e + 4} \right )}} \right )}}{2} \] Input:
integrate(((4*x*ln(5/(exp(1)+4))+5)*ln((4*x*ln(5/(exp(1)+4))+5)/x/ln(5/(ex p(1)+4)))-5)/(8*x*ln(5/(exp(1)+4))+10),x)
Output:
x*log((4*x*log(5/(E + 4)) + 5)/(x*log(5/(E + 4))))/2
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 5.57 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=-\frac {4 \, x {\left (\log \left (5\right ) - \log \left (e + 4\right )\right )} \log \left (x\right ) + 4 \, {\left (\log \left (5\right ) \log \left (-\log \left (5\right ) + \log \left (e + 4\right )\right ) - \log \left (e + 4\right ) \log \left (-\log \left (5\right ) + \log \left (e + 4\right )\right )\right )} x - {\left (4 \, x {\left (\log \left (5\right ) - \log \left (e + 4\right )\right )} + 5\right )} \log \left (-4 \, x {\left (\log \left (5\right ) - \log \left (e + 4\right )\right )} - 5\right )}{8 \, {\left (\log \left (5\right ) - \log \left (e + 4\right )\right )}} - \frac {5 \, \log \left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )}{8 \, \log \left (\frac {5}{e + 4}\right )} \] Input:
integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5 /(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+4))+10),x, algorithm="maxima")
Output:
-1/8*(4*x*(log(5) - log(e + 4))*log(x) + 4*(log(5)*log(-log(5) + log(e + 4 )) - log(e + 4)*log(-log(5) + log(e + 4)))*x - (4*x*(log(5) - log(e + 4)) + 5)*log(-4*x*(log(5) - log(e + 4)) - 5))/(log(5) - log(e + 4)) - 5/8*log( 4*x*log(5/(e + 4)) + 5)/log(5/(e + 4))
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (22) = 44\).
Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 7.65 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=-\frac {5 \, \log \left (\frac {5}{e + 4}\right ) \log \left (\frac {4 \, x \log \left (\frac {5}{e + 4}\right ) + 5}{x \log \left (\frac {5}{e + 4}\right )}\right )}{2 \, {\left (4 \, \log \left (5\right )^{2} - 8 \, \log \left (5\right ) \log \left (e + 4\right ) + 4 \, \log \left (e + 4\right )^{2} - \frac {{\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \left (5\right )^{2}}{x \log \left (\frac {5}{e + 4}\right )} + \frac {2 \, {\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \left (5\right ) \log \left (e + 4\right )}{x \log \left (\frac {5}{e + 4}\right )} - \frac {{\left (4 \, x \log \left (\frac {5}{e + 4}\right ) + 5\right )} \log \left (e + 4\right )^{2}}{x \log \left (\frac {5}{e + 4}\right )}\right )}} \] Input:
integrate(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5 /(exp(1)+4)))-5)/(8*x*log(5/(exp(1)+4))+10),x, algorithm="giac")
Output:
-5/2*log(5/(e + 4))*log((4*x*log(5/(e + 4)) + 5)/(x*log(5/(e + 4))))/(4*lo g(5)^2 - 8*log(5)*log(e + 4) + 4*log(e + 4)^2 - (4*x*log(5/(e + 4)) + 5)*l og(5)^2/(x*log(5/(e + 4))) + 2*(4*x*log(5/(e + 4)) + 5)*log(5)*log(e + 4)/ (x*log(5/(e + 4))) - (4*x*log(5/(e + 4)) + 5)*log(e + 4)^2/(x*log(5/(e + 4 ))))
Time = 5.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=-\frac {x\,\left (\ln \left (-\ln \left (\frac {5}{\mathrm {e}+4}\right )\right )-\ln \left (-\frac {4\,x\,\ln \left (\frac {5}{\mathrm {e}+4}\right )+5}{x}\right )\right )}{2} \] Input:
int((log((4*x*log(5/(exp(1) + 4)) + 5)/(x*log(5/(exp(1) + 4))))*(4*x*log(5 /(exp(1) + 4)) + 5) - 5)/(8*x*log(5/(exp(1) + 4)) + 10),x)
Output:
-(x*(log(-log(5/(exp(1) + 4))) - log(-(4*x*log(5/(exp(1) + 4)) + 5)/x)))/2
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.43 \[ \int \frac {-5+\left (5+4 x \log \left (\frac {5}{4+e}\right )\right ) \log \left (\frac {5+4 x \log \left (\frac {5}{4+e}\right )}{x \log \left (\frac {5}{4+e}\right )}\right )}{10+8 x \log \left (\frac {5}{4+e}\right )} \, dx=\frac {-5 \,\mathrm {log}\left (4 \,\mathrm {log}\left (\frac {5}{e +4}\right ) x +5\right )+4 \,\mathrm {log}\left (\frac {4 \,\mathrm {log}\left (\frac {5}{e +4}\right ) x +5}{\mathrm {log}\left (\frac {5}{e +4}\right ) x}\right ) \mathrm {log}\left (\frac {5}{e +4}\right ) x +5 \,\mathrm {log}\left (\frac {4 \,\mathrm {log}\left (\frac {5}{e +4}\right ) x +5}{\mathrm {log}\left (\frac {5}{e +4}\right ) x}\right )+5 \,\mathrm {log}\left (x \right )}{8 \,\mathrm {log}\left (\frac {5}{e +4}\right )} \] Input:
int(((4*x*log(5/(exp(1)+4))+5)*log((4*x*log(5/(exp(1)+4))+5)/x/log(5/(exp( 1)+4)))-5)/(8*x*log(5/(exp(1)+4))+10),x)
Output:
( - 5*log(4*log(5/(e + 4))*x + 5) + 4*log((4*log(5/(e + 4))*x + 5)/(log(5/ (e + 4))*x))*log(5/(e + 4))*x + 5*log((4*log(5/(e + 4))*x + 5)/(log(5/(e + 4))*x)) + 5*log(x))/(8*log(5/(e + 4)))