Integrand size = 148, antiderivative size = 25 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log (5)}{\log (2+x)-\frac {5 x}{\log \left (x-\frac {x^2}{81}\right )}} \]
Time = 5.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log (5) \log \left (x-\frac {x^2}{81}\right )}{-5 x+\log (2+x) \log \left (x-\frac {x^2}{81}\right )} \]
Integrate[((810 + 385*x - 10*x^2)*Log[5] + (-810 - 395*x + 5*x^2)*Log[5]*L og[(81*x - x^2)/81] + (81 - x)*Log[5]*Log[(81*x - x^2)/81]^2)/(-4050*x^2 - 1975*x^3 + 25*x^4 + (1620*x + 790*x^2 - 10*x^3)*Log[2 + x]*Log[(81*x - x^ 2)/81] + (-162 - 79*x + x^2)*Log[2 + x]^2*Log[(81*x - x^2)/81]^2),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.
| \(\displaystyle \int \frac {(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (5 x^2-395 x-810\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-10 x^2+385 x+810\right ) \log (5)}{25 x^4-1975 x^3-4050 x^2+\left (x^2-79 x-162\right ) \log ^2(x+2) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-10 x^3+790 x^2+1620 x\right ) \log (x+2) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log (5) \left (10 x^2+(x-81) \log ^2\left (x-\frac {x^2}{81}\right )-5 \left (x^2-79 x-162\right ) \log \left (x-\frac {x^2}{81}\right )-385 x-810\right )}{\left (-x^2+79 x+162\right ) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \log (5) \int -\frac {-10 x^2+385 x+(81-x) \log ^2\left (x-\frac {x^2}{81}\right )-5 \left (-x^2+79 x+162\right ) \log \left (x-\frac {x^2}{81}\right )+810}{\left (-x^2+79 x+162\right ) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\log (5) \int \frac {-10 x^2+385 x+(81-x) \log ^2\left (x-\frac {x^2}{81}\right )-5 \left (-x^2+79 x+162\right ) \log \left (x-\frac {x^2}{81}\right )+810}{\left (-x^2+79 x+162\right ) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\log (5) \int \left (\frac {5 (\log (x+2) x-2 x+2 \log (x+2))}{(x+2) \log ^2(x+2) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )}+\frac {1}{(x+2) \log ^2(x+2)}-\frac {5 \left (5 \log (x+2) x^3-5 x^3-2 \log ^2(x+2) x^2-395 \log (x+2) x^2+405 x^2+77 \log ^2(x+2) x-810 \log (x+2) x+162 \log ^2(x+2)\right )}{(x-81) (x+2) \log ^2(x+2) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\log (5) \left (100 \int \frac {1}{(x+2) \log ^2(x+2) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx+20 \int \frac {1}{(x+2) \log ^2(x+2) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )}dx-50 \int \frac {1}{\log ^2(x+2) \left (\log (x+2) \log \left (x-\frac {x^2}{81}\right )-5 x\right )^2}dx+25 \int \frac {x}{\log ^2(x+2) \left (\log (x+2) \log \left (x-\frac {x^2}{81}\right )-5 x\right )^2}dx+10 \int \frac {1}{\log ^2(x+2) \left (\log (x+2) \log \left (x-\frac {x^2}{81}\right )-5 x\right )}dx+10 \int \frac {1}{\left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx+405 \int \frac {1}{(x-81) \left (5 x-\log (x+2) \log \left (x-\frac {x^2}{81}\right )\right )^2}dx-25 \int \frac {x}{\log (x+2) \left (\log (x+2) \log \left (x-\frac {x^2}{81}\right )-5 x\right )^2}dx-5 \int \frac {1}{\log (x+2) \left (\log (x+2) \log \left (x-\frac {x^2}{81}\right )-5 x\right )}dx-\frac {1}{\log (x+2)}\right )\) |
Int[((810 + 385*x - 10*x^2)*Log[5] + (-810 - 395*x + 5*x^2)*Log[5]*Log[(81 *x - x^2)/81] + (81 - x)*Log[5]*Log[(81*x - x^2)/81]^2)/(-4050*x^2 - 1975* x^3 + 25*x^4 + (1620*x + 790*x^2 - 10*x^3)*Log[2 + x]*Log[(81*x - x^2)/81] + (-162 - 79*x + x^2)*Log[2 + x]^2*Log[(81*x - x^2)/81]^2),x]
3.12.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 12.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32
| method | result | size |
| parallelrisch | \(-\frac {\ln \left (5\right ) \ln \left (-\frac {1}{81} x^{2}+x \right )}{-\ln \left (-\frac {1}{81} x^{2}+x \right ) \ln \left (2+x \right )+5 x}\) | \(33\) |
| 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 | \(\frac {\ln \left (5\right )}{\ln \left (2+x \right )}-\frac {10 x \ln \left (5\right )}{\ln \left (2+x \right ) \left (i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x -81\right )\right ) \operatorname {csgn}\left (i x \left (x -81\right )\right ) \ln \left (2+x \right )-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )+2 i \pi \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \,\operatorname {csgn}\left (i \left (x -81\right )\right ) \operatorname {csgn}\left (i x \left (x -81\right )\right )^{2} \ln \left (2+x \right )-i \pi \operatorname {csgn}\left (i x \left (x -81\right )\right )^{3} \ln \left (2+x \right )-2 i \pi \ln \left (2+x \right )+8 \ln \left (2+x \right ) \ln \left (3\right )-2 \ln \left (x \right ) \ln \left (2+x \right )-2 \ln \left (2+x \right ) \ln \left (x -81\right )+10 x \right )}\) | \(174\) |
int(((81-x)*ln(5)*ln(-1/81*x^2+x)^2+(5*x^2-395*x-810)*ln(5)*ln(-1/81*x^2+x )+(-10*x^2+385*x+810)*ln(5))/((x^2-79*x-162)*ln(-1/81*x^2+x)^2*ln(2+x)^2+( -10*x^3+790*x^2+1620*x)*ln(-1/81*x^2+x)*ln(2+x)+25*x^4-1975*x^3-4050*x^2), x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {\log \left (5\right ) \log \left (-\frac {1}{81} \, x^{2} + x\right )}{\log \left (-\frac {1}{81} \, x^{2} + x\right ) \log \left (x + 2\right ) - 5 \, x} \]
integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(- 1/81*x^2+x)+(-10*x^2+385*x+810)*log(5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2 *log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-1975 *x^3-4050*x^2),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {5 x \log {\left (5 \right )}}{- 5 x \log {\left (x + 2 \right )} + \log {\left (x + 2 \right )}^{2} \log {\left (- \frac {x^{2}}{81} + x \right )}} + \frac {\log {\left (5 \right )}}{\log {\left (x + 2 \right )}} \]
integrate(((81-x)*ln(5)*ln(-1/81*x**2+x)**2+(5*x**2-395*x-810)*ln(5)*ln(-1 /81*x**2+x)+(-10*x**2+385*x+810)*ln(5))/((x**2-79*x-162)*ln(-1/81*x**2+x)* *2*ln(2+x)**2+(-10*x**3+790*x**2+1620*x)*ln(-1/81*x**2+x)*ln(2+x)+25*x**4- 1975*x**3-4050*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=\frac {4 \, \log \left (5\right ) \log \left (3\right ) - \log \left (5\right ) \log \left (x\right ) - \log \left (5\right ) \log \left (-x + 81\right )}{{\left (4 \, \log \left (3\right ) - \log \left (x\right )\right )} \log \left (x + 2\right ) - \log \left (x + 2\right ) \log \left (-x + 81\right ) + 5 \, x} \]
integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(- 1/81*x^2+x)+(-10*x^2+385*x+810)*log(5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2 *log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-1975 *x^3-4050*x^2),x, algorithm=\
(4*log(5)*log(3) - log(5)*log(x) - log(5)*log(-x + 81))/((4*log(3) - log(x ))*log(x + 2) - log(x + 2)*log(-x + 81) + 5*x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=-\frac {5 \, x \log \left (5\right )}{4 \, \log \left (3\right ) \log \left (x + 2\right )^{2} - \log \left (-x^{2} + 81 \, x\right ) \log \left (x + 2\right )^{2} + 5 \, x \log \left (x + 2\right )} + \frac {\log \left (5\right )}{\log \left (x + 2\right )} \]
integrate(((81-x)*log(5)*log(-1/81*x^2+x)^2+(5*x^2-395*x-810)*log(5)*log(- 1/81*x^2+x)+(-10*x^2+385*x+810)*log(5))/((x^2-79*x-162)*log(-1/81*x^2+x)^2 *log(2+x)^2+(-10*x^3+790*x^2+1620*x)*log(-1/81*x^2+x)*log(2+x)+25*x^4-1975 *x^3-4050*x^2),x, algorithm=\
-5*x*log(5)/(4*log(3)*log(x + 2)^2 - log(-x^2 + 81*x)*log(x + 2)^2 + 5*x*l og(x + 2)) + log(5)/log(x + 2)
Time = 10.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {\left (810+385 x-10 x^2\right ) \log (5)+\left (-810-395 x+5 x^2\right ) \log (5) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+(81-x) \log (5) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )}{-4050 x^2-1975 x^3+25 x^4+\left (1620 x+790 x^2-10 x^3\right ) \log (2+x) \log \left (\frac {1}{81} \left (81 x-x^2\right )\right )+\left (-162-79 x+x^2\right ) \log ^2(2+x) \log ^2\left (\frac {1}{81} \left (81 x-x^2\right )\right )} \, dx=-\frac {\ln \left (5\right )\,\ln \left (x-\frac {x^2}{81}\right )}{5\,x-\ln \left (x+2\right )\,\ln \left (x-\frac {x^2}{81}\right )} \]