Integrand size = 135, antiderivative size = 30 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\left (2+\frac {1}{25} x^2 \left (x+\frac {x-\log \left (\frac {5}{x+\log (x)}\right )}{x}\right )\right )^2 \end {dmath*}
Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {1}{625} \left (50+x^2+x^3-x \log \left (\frac {5}{x+\log (x)}\right )\right )^2 \end {dmath*}
Integrate[(100 + 100*x + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + (200 *x + 300*x^2 + 4*x^3 + 10*x^4 + 6*x^5)*Log[x] + (-102*x - 2*x^2 - 6*x^3 - 8*x^4 + (-100 - 6*x^2 - 8*x^3)*Log[x])*Log[5/(x + Log[x])] + (2*x^2 + 2*x* Log[x])*Log[5/(x + Log[x])]^2)/(625*x + 625*Log[x]),x]
Time = 0.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {6 x^6+10 x^5+6 x^4+304 x^3+202 x^2+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )+\left (-8 x^4-6 x^3-2 x^2+\left (-8 x^3-6 x^2-100\right ) \log (x)-102 x\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (6 x^5+10 x^4+4 x^3+300 x^2+200 x\right ) \log (x)+100 x+100}{625 x+625 \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (x^3+x^2-x \log \left (\frac {5}{x+\log (x)}\right )+50\right ) \left (3 x^3+2 x^2+x-x \log \left (\frac {5}{x+\log (x)}\right )+\log (x) \left (x (3 x+2)-\log \left (\frac {5}{x+\log (x)}\right )\right )+1\right )}{625 (x+\log (x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{625} \int \frac {\left (x^3+x^2-\log \left (\frac {5}{x+\log (x)}\right ) x+50\right ) \left (3 x^3+2 x^2-\log \left (\frac {5}{x+\log (x)}\right ) x+x+\log (x) \left (x (3 x+2)-\log \left (\frac {5}{x+\log (x)}\right )\right )+1\right )}{x+\log (x)}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {1}{625} \left (x^3+x^2-x \log \left (\frac {5}{x+\log (x)}\right )+50\right )^2\) |
Int[(100 + 100*x + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + (200*x + 3 00*x^2 + 4*x^3 + 10*x^4 + 6*x^5)*Log[x] + (-102*x - 2*x^2 - 6*x^3 - 8*x^4 + (-100 - 6*x^2 - 8*x^3)*Log[x])*Log[5/(x + Log[x])] + (2*x^2 + 2*x*Log[x] )*Log[5/(x + Log[x])]^2)/(625*x + 625*Log[x]),x]
3.7.51.3.1 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(82\) vs. \(2(28)=56\).
Time = 5.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77
method | result | size |
parallelrisch | \(\frac {x^{6}}{625}+\frac {2 x^{5}}{625}-\frac {2 \ln \left (\frac {5}{x +\ln \left (x \right )}\right ) x^{4}}{625}+\frac {x^{4}}{625}-\frac {2 \ln \left (\frac {5}{x +\ln \left (x \right )}\right ) x^{3}}{625}+\frac {\ln \left (\frac {5}{x +\ln \left (x \right )}\right )^{2} x^{2}}{625}+\frac {4 x^{3}}{25}+\frac {4 x^{2}}{25}-\frac {4 x \ln \left (\frac {5}{x +\ln \left (x \right )}\right )}{25}\) | \(83\) |
risch | \(\frac {x^{2} \ln \left (x +\ln \left (x \right )\right )^{2}}{625}+\frac {x \left (100+2 x^{3}-2 x \ln \left (5\right )+2 x^{2}\right ) \ln \left (x +\ln \left (x \right )\right )}{625}+\frac {x^{6}}{625}-\frac {2 x^{4} \ln \left (5\right )}{625}+\frac {2 x^{5}}{625}+\frac {x^{2} \ln \left (5\right )^{2}}{625}-\frac {2 x^{3} \ln \left (5\right )}{625}+\frac {x^{4}}{625}+\frac {4 x^{3}}{25}-\frac {4 x \ln \left (5\right )}{25}+\frac {4 x^{2}}{25}+4\) | \(93\) |
int(((2*x*ln(x)+2*x^2)*ln(5/(x+ln(x)))^2+((-8*x^3-6*x^2-100)*ln(x)-8*x^4-6 *x^3-2*x^2-102*x)*ln(5/(x+ln(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+200*x)*ln(x) +6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*ln(x)+625*x),x,method= _RETURNVERBOSE)
1/625*x^6+2/625*x^5-2/625*ln(5/(x+ln(x)))*x^4+1/625*x^4-2/625*ln(5/(x+ln(x )))*x^3+1/625*ln(5/(x+ln(x)))^2*x^2+4/25*x^3+4/25*x^2-4/25*x*ln(5/(x+ln(x) ))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} + \frac {1}{625} \, x^{4} + \frac {1}{625} \, x^{2} \log \left (\frac {5}{x + \log \left (x\right )}\right )^{2} + \frac {4}{25} \, x^{3} + \frac {4}{25} \, x^{2} - \frac {2}{625} \, {\left (x^{4} + x^{3} + 50 \, x\right )} \log \left (\frac {5}{x + \log \left (x\right )}\right ) \end {dmath*}
integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log( x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+ 200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+62 5*x),x, algorithm=\
1/625*x^6 + 2/625*x^5 + 1/625*x^4 + 1/625*x^2*log(5/(x + log(x)))^2 + 4/25 *x^3 + 4/25*x^2 - 2/625*(x^4 + x^3 + 50*x)*log(5/(x + log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {x^{6}}{625} + \frac {2 x^{5}}{625} + \frac {x^{4}}{625} + \frac {4 x^{3}}{25} + \frac {x^{2} \log {\left (\frac {5}{x + \log {\left (x \right )}} \right )}^{2}}{625} + \frac {4 x^{2}}{25} + \left (- \frac {2 x^{4}}{625} - \frac {2 x^{3}}{625} - \frac {4 x}{25}\right ) \log {\left (\frac {5}{x + \log {\left (x \right )}} \right )} \end {dmath*}
integrate(((2*x*ln(x)+2*x**2)*ln(5/(x+ln(x)))**2+((-8*x**3-6*x**2-100)*ln( x)-8*x**4-6*x**3-2*x**2-102*x)*ln(5/(x+ln(x)))+(6*x**5+10*x**4+4*x**3+300* x**2+200*x)*ln(x)+6*x**6+10*x**5+6*x**4+304*x**3+202*x**2+100*x+100)/(625* ln(x)+625*x),x)
x**6/625 + 2*x**5/625 + x**4/625 + 4*x**3/25 + x**2*log(5/(x + log(x)))**2 /625 + 4*x**2/25 + (-2*x**4/625 - 2*x**3/625 - 4*x/25)*log(5/(x + log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} - \frac {1}{625} \, x^{4} {\left (2 \, \log \left (5\right ) - 1\right )} - \frac {2}{625} \, x^{3} {\left (\log \left (5\right ) - 50\right )} + \frac {1}{625} \, x^{2} \log \left (x + \log \left (x\right )\right )^{2} + \frac {1}{625} \, {\left (\log \left (5\right )^{2} + 100\right )} x^{2} - \frac {4}{25} \, x \log \left (5\right ) + \frac {2}{625} \, {\left (x^{4} + x^{3} - x^{2} \log \left (5\right ) + 50 \, x\right )} \log \left (x + \log \left (x\right )\right ) \end {dmath*}
integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log( x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+ 200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+62 5*x),x, algorithm=\
1/625*x^6 + 2/625*x^5 - 1/625*x^4*(2*log(5) - 1) - 2/625*x^3*(log(5) - 50) + 1/625*x^2*log(x + log(x))^2 + 1/625*(log(5)^2 + 100)*x^2 - 4/25*x*log(5 ) + 2/625*(x^4 + x^3 - x^2*log(5) + 50*x)*log(x + log(x))
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {1}{625} \, x^{6} + \frac {2}{625} \, x^{5} - \frac {1}{625} \, x^{4} {\left (2 \, \log \left (5\right ) - 1\right )} - \frac {2}{625} \, x^{3} {\left (\log \left (5\right ) - 50\right )} + \frac {1}{625} \, x^{2} \log \left (x + \log \left (x\right )\right )^{2} + \frac {1}{625} \, {\left (\log \left (5\right )^{2} + 100\right )} x^{2} - \frac {4}{25} \, x \log \left (5\right ) + \frac {2}{625} \, {\left (x^{4} + x^{3} - x^{2} \log \left (5\right ) + 50 \, x\right )} \log \left (x + \log \left (x\right )\right ) \end {dmath*}
integrate(((2*x*log(x)+2*x^2)*log(5/(x+log(x)))^2+((-8*x^3-6*x^2-100)*log( x)-8*x^4-6*x^3-2*x^2-102*x)*log(5/(x+log(x)))+(6*x^5+10*x^4+4*x^3+300*x^2+ 200*x)*log(x)+6*x^6+10*x^5+6*x^4+304*x^3+202*x^2+100*x+100)/(625*log(x)+62 5*x),x, algorithm=\
1/625*x^6 + 2/625*x^5 - 1/625*x^4*(2*log(5) - 1) - 2/625*x^3*(log(5) - 50) + 1/625*x^2*log(x + log(x))^2 + 1/625*(log(5)^2 + 100)*x^2 - 4/25*x*log(5 ) + 2/625*(x^4 + x^3 - x^2*log(5) + 50*x)*log(x + log(x))
Time = 13.75 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \begin {dmath*} \int \frac {100+100 x+202 x^2+304 x^3+6 x^4+10 x^5+6 x^6+\left (200 x+300 x^2+4 x^3+10 x^4+6 x^5\right ) \log (x)+\left (-102 x-2 x^2-6 x^3-8 x^4+\left (-100-6 x^2-8 x^3\right ) \log (x)\right ) \log \left (\frac {5}{x+\log (x)}\right )+\left (2 x^2+2 x \log (x)\right ) \log ^2\left (\frac {5}{x+\log (x)}\right )}{625 x+625 \log (x)} \, dx=\frac {x^2\,{\ln \left (\frac {5}{x+\ln \left (x\right )}\right )}^2}{625}+\frac {4\,x^2}{25}+\frac {4\,x^3}{25}+\frac {x^4}{625}+\frac {2\,x^5}{625}+\frac {x^6}{625}-\ln \left (\frac {5}{x+\ln \left (x\right )}\right )\,\left (\frac {2\,x^4}{625}+\frac {2\,x^3}{625}+\frac {4\,x}{25}\right ) \end {dmath*}
int((100*x - log(5/(x + log(x)))*(102*x + log(x)*(6*x^2 + 8*x^3 + 100) + 2 *x^2 + 6*x^3 + 8*x^4) + log(x)*(200*x + 300*x^2 + 4*x^3 + 10*x^4 + 6*x^5) + 202*x^2 + 304*x^3 + 6*x^4 + 10*x^5 + 6*x^6 + log(5/(x + log(x)))^2*(2*x* log(x) + 2*x^2) + 100)/(625*x + 625*log(x)),x)