Optimal. Leaf size=25 \[ \frac {\log (5)}{\log (2+x)-\frac {5 x}{\log \left (x-\frac {x^2}{81}\right )}} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 13.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (5) \left (-810-385 x+10 x^2-5 \left (-162-79 x+x^2\right ) \log \left (x-\frac {x^2}{81}\right )+(-81+x) \log ^2\left (x-\frac {x^2}{81}\right )\right )}{\left (162+79 x-x^2\right ) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx\\ &=\log (5) \int \frac {-810-385 x+10 x^2-5 \left (-162-79 x+x^2\right ) \log \left (x-\frac {x^2}{81}\right )+(-81+x) \log ^2\left (x-\frac {x^2}{81}\right )}{\left (162+79 x-x^2\right ) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx\\ &=\log (5) \int \left (-\frac {1}{(2+x) \log ^2(2+x)}+\frac {5 \left (405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)\right )}{(-81+x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 (-2 x+2 \log (2+x)+x \log (2+x))}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx\\ &=-\left (\log (5) \int \frac {1}{(2+x) \log ^2(2+x)} \, dx\right )+(5 \log (5)) \int \frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{(-81+x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \frac {-2 x+2 \log (2+x)+x \log (2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx\\ &=-\left (\log (5) \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,2+x\right )\right )+(5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)-\left (162+77 x-2 x^2\right ) \log ^2(2+x)}{(81-x) (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \left (-\frac {2 x}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}+\frac {2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}+\frac {x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx\\ &=-\left (\log (5) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (2+x)\right )\right )-(5 \log (5)) \int \frac {x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx+(5 \log (5)) \int \left (\frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{83 (-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {-405 x^2+5 x^3+810 x \log (2+x)+395 x^2 \log (2+x)-5 x^3 \log (2+x)-162 \log ^2(2+x)-77 x \log ^2(2+x)+2 x^2 \log ^2(2+x)}{83 (2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+(10 \log (5)) \int \frac {x}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx\\ &=\frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \frac {405 x^2-5 x^3-810 x \log (2+x)-395 x^2 \log (2+x)+5 x^3 \log (2+x)+162 \log ^2(2+x)+77 x \log ^2(2+x)-2 x^2 \log ^2(2+x)}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (5 \log (5)) \int \frac {-405 x^2+5 x^3+810 x \log (2+x)+395 x^2 \log (2+x)-5 x^3 \log (2+x)-162 \log ^2(2+x)-77 x \log ^2(2+x)+2 x^2 \log ^2(2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx-(5 \log (5)) \int \left (-\frac {2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}-\frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx-(10 \log (5)) \int \frac {1}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx+(10 \log (5)) \int \left (-\frac {2}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}-\frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )}\right ) \, dx\\ &=\frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)-\left (162+77 x-2 x^2\right ) \log ^2(2+x)}{(81-x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+\frac {1}{83} (5 \log (5)) \int \frac {5 (-81+x) x^2-5 x \left (-162-79 x+x^2\right ) \log (2+x)+\left (-162-77 x+2 x^2\right ) \log ^2(2+x)}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2} \, dx+(5 \log (5)) \int \frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(20 \log (5)) \int \frac {1}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx\\ &=\frac {\log (5)}{\log (2+x)}+\frac {1}{83} (5 \log (5)) \int \left (\frac {162}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {77 x}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {2 x^2}{(-81+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {405 x^2}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 x^3}{(-81+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {810 x}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {395 x^2}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {5 x^3}{(-81+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+\frac {1}{83} (5 \log (5)) \int \left (-\frac {162}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {77 x}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {2 x^2}{(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {405 x^2}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {5 x^3}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {810 x}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}+\frac {395 x^2}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}-\frac {5 x^3}{(2+x) \log (2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right ) \, dx+(5 \log (5)) \int \frac {1}{\log (2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(10 \log (5)) \int \frac {1}{\log ^2(2+x) \left (-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx-(20 \log (5)) \int \frac {1}{(2+x) \log ^2(2+x) \left (5 x-\log (2+x) \log \left (x-\frac {x^2}{81}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.32, size = 34, normalized size = 1.36 \begin {gather*} \frac {\log (5) \log \left (x-\frac {x^2}{81}\right )}{-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs.
\(2(23)=46\).
time = 1.63, size = 56, normalized size = 2.24
method | result | size |
default | \(-\ln \left (5\right ) \left (-\frac {1}{\ln \left (2+x \right )}+\frac {5 x}{\ln \left (2+x \right ) \left (4 \ln \left (2+x \right ) \ln \left (3\right )-\ln \left (2+x \right ) \ln \left (-\left (2+x \right )^{2}+4+85 x \right )+5 x \right )}\right )\) | \(56\) |
risch | \(\frac {\ln \left (5\right )}{\ln \left (2+x \right )}-\frac {10 x \ln \left (5\right )}{\ln \left (2+x \right ) \left (i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x -81\right )\right ) \mathrm {csgn}\left (i x \left (x -81\right )\right ) \ln \left (2+x \right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )+2 i \pi \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \,\mathrm {csgn}\left (i \left (x -81\right )\right ) \mathrm {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \mathrm {csgn}\left (i x \left (x -81\right )\right )^{3} \ln \left (2+x \right )-2 i \pi \ln \left (2+x \right )+8 \ln \left (2+x \right ) \ln \left (3\right )-2 \ln \left (x \right ) \ln \left (2+x \right )-2 \ln \left (2+x \right ) \ln \left (x -81\right )+10 x \right )}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (26) = 52\).
time = 0.56, size = 56, normalized size = 2.24 \begin {gather*} \frac {4 \, \log \left (5\right ) \log \left (3\right ) - \log \left (5\right ) \log \left (x\right ) - \log \left (5\right ) \log \left (-x + 81\right )}{{\left (4 \, \log \left (3\right ) - \log \left (x\right )\right )} \log \left (x + 2\right ) - \log \left (x + 2\right ) \log \left (-x + 81\right ) + 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 30, normalized size = 1.20 \begin {gather*} \frac {\log \left (5\right ) \log \left (-\frac {1}{81} \, x^{2} + x\right )}{\log \left (-\frac {1}{81} \, x^{2} + x\right ) \log \left (x + 2\right ) - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.14, size = 37, normalized size = 1.48 \begin {gather*} \frac {5 x \log {\left (5 \right )}}{- 5 x \log {\left (x + 2 \right )} + \log {\left (x + 2 \right )}^{2} \log {\left (- \frac {x^{2}}{81} + x \right )}} + \frac {\log {\left (5 \right )}}{\log {\left (x + 2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (26) = 52\).
time = 0.50, size = 53, normalized size = 2.12 \begin {gather*} -\frac {5 \, x \log \left (5\right )}{4 \, \log \left (3\right ) \log \left (x + 2\right )^{2} - \log \left (-x^{2} + 81 \, x\right ) \log \left (x + 2\right )^{2} + 5 \, x \log \left (x + 2\right )} + \frac {\log \left (5\right )}{\log \left (x + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.95, size = 32, normalized size = 1.28 \begin {gather*} -\frac {\ln \left (5\right )\,\ln \left (x-\frac {x^2}{81}\right )}{5\,x-\ln \left (x+2\right )\,\ln \left (x-\frac {x^2}{81}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________