Integrand size = 166, antiderivative size = 25 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=-\frac {e}{2+\log (5)+\frac {9 e^x (3+\log (5))}{x}}+\log (\log (x)) \]
Time = 0.82 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=-\frac {e x}{27 e^x+2 x+9 e^x \log (5)+x \log (5)}+\log (\log (x)) \]
Integrate[(4*x^2 + 4*x^2*Log[5] + x^2*Log[5]^2 + E^(2*x)*(729 + 486*Log[5] + 81*Log[5]^2) + E^x*(108*x + 90*x*Log[5] + 18*x*Log[5]^2) + E^x*(E*(-27* x + 27*x^2) + E*(-9*x + 9*x^2)*Log[5])*Log[x])/((4*x^3 + 4*x^3*Log[5] + x^ 3*Log[5]^2 + E^(2*x)*(729*x + 486*x*Log[5] + 81*x*Log[5]^2) + E^x*(108*x^2 + 90*x^2*Log[5] + 18*x^2*Log[5]^2))*Log[x]),x]
Time = 0.92 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 6, 6, 6, 7239, 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.
| \(\displaystyle \int \frac {4 x^2+x^2 \log ^2(5)+4 x^2 \log (5)+e^x \left (e \left (27 x^2-27 x\right )+e \left (9 x^2-9 x\right ) \log (5)\right ) \log (x)+e^x \left (108 x+18 x \log ^2(5)+90 x \log (5)\right )+e^{2 x} \left (729+81 \log ^2(5)+486 \log (5)\right )}{\left (4 x^3+x^3 \log ^2(5)+4 x^3 \log (5)+e^x \left (108 x^2+18 x^2 \log ^2(5)+90 x^2 \log (5)\right )+e^{2 x} \left (729 x+81 x \log ^2(5)+486 x \log (5)\right )\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 x^2+x^2 \log ^2(5)+4 x^2 \log (5)+e^x \left (e \left (27 x^2-27 x\right )+e \left (9 x^2-9 x\right ) \log (5)\right ) \log (x)+e^x \left (108 x+18 x \log ^2(5)+90 x \log (5)\right )+e^{2 x} \left (729+81 \log ^2(5)+486 \log (5)\right )}{\left (x^3 \log ^2(5)+x^3 (4+4 \log (5))+e^x \left (108 x^2+18 x^2 \log ^2(5)+90 x^2 \log (5)\right )+e^{2 x} \left (729 x+81 x \log ^2(5)+486 x \log (5)\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {4 x^2+x^2 \log ^2(5)+4 x^2 \log (5)+e^x \left (e \left (27 x^2-27 x\right )+e \left (9 x^2-9 x\right ) \log (5)\right ) \log (x)+e^x \left (108 x+18 x \log ^2(5)+90 x \log (5)\right )+e^{2 x} \left (729+81 \log ^2(5)+486 \log (5)\right )}{\left (x^3 \left (4+\log ^2(5)+4 \log (5)\right )+e^x \left (108 x^2+18 x^2 \log ^2(5)+90 x^2 \log (5)\right )+e^{2 x} \left (729 x+81 x \log ^2(5)+486 x \log (5)\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^2 \log ^2(5)+x^2 (4+4 \log (5))+e^x \left (e \left (27 x^2-27 x\right )+e \left (9 x^2-9 x\right ) \log (5)\right ) \log (x)+e^x \left (108 x+18 x \log ^2(5)+90 x \log (5)\right )+e^{2 x} \left (729+81 \log ^2(5)+486 \log (5)\right )}{\left (x^3 \left (4+\log ^2(5)+4 \log (5)\right )+e^x \left (108 x^2+18 x^2 \log ^2(5)+90 x^2 \log (5)\right )+e^{2 x} \left (729 x+81 x \log ^2(5)+486 x \log (5)\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^2 \left (4+\log ^2(5)+4 \log (5)\right )+e^x \left (e \left (27 x^2-27 x\right )+e \left (9 x^2-9 x\right ) \log (5)\right ) \log (x)+e^x \left (108 x+18 x \log ^2(5)+90 x \log (5)\right )+e^{2 x} \left (729+81 \log ^2(5)+486 \log (5)\right )}{\left (x^3 \left (4+\log ^2(5)+4 \log (5)\right )+e^x \left (108 x^2+18 x^2 \log ^2(5)+90 x^2 \log (5)\right )+e^{2 x} \left (729 x+81 x \log ^2(5)+486 x \log (5)\right )\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (\frac {9 e^{x+1} (x-1) (3+\log (5))}{\left (x (2+\log (5))+9 e^x (3+\log (5))\right )^2}+\frac {1}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \log (\log (x))+\frac {9 e (3+\log (5))}{(2+\log (5)) \left (e^{-x} x (2+\log (5))+9 (3+\log (5))\right )}\) |
Int[(4*x^2 + 4*x^2*Log[5] + x^2*Log[5]^2 + E^(2*x)*(729 + 486*Log[5] + 81* Log[5]^2) + E^x*(108*x + 90*x*Log[5] + 18*x*Log[5]^2) + E^x*(E*(-27*x + 27 *x^2) + E*(-9*x + 9*x^2)*Log[5])*Log[x])/((4*x^3 + 4*x^3*Log[5] + x^3*Log[ 5]^2 + E^(2*x)*(729*x + 486*x*Log[5] + 81*x*Log[5]^2) + E^x*(108*x^2 + 90* x^2*Log[5] + 18*x^2*Log[5]^2))*Log[x]),x]
3.10.96.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(-\frac {x \,{\mathrm e}}{9 \,{\mathrm e}^{x} \ln \left (5\right )+x \ln \left (5\right )+27 \,{\mathrm e}^{x}+2 x}+\ln \left (\ln \left (x \right )\right )\) | \(30\) |
| parallelrisch | \(\frac {\ln \left (5\right )^{2} \ln \left (\ln \left (x \right )\right ) x +9 \ln \left (5\right )^{2} \ln \left (\ln \left (x \right )\right ) {\mathrm e}^{x}+9 \,{\mathrm e} \ln \left (5\right ) {\mathrm e}^{x}+4 \ln \left (5\right ) \ln \left (\ln \left (x \right )\right ) x +45 \ln \left (5\right ) {\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )+27 \,{\mathrm e} \,{\mathrm e}^{x}+4 x \ln \left (\ln \left (x \right )\right )+54 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )}{\left (2+\ln \left (5\right )\right ) \left (9 \,{\mathrm e}^{x} \ln \left (5\right )+x \ln \left (5\right )+27 \,{\mathrm e}^{x}+2 x \right )}\) | \(93\) |
int((((9*x^2-9*x)*exp(1)*ln(5)+(27*x^2-27*x)*exp(1))*exp(x)*ln(x)+(81*ln(5 )^2+486*ln(5)+729)*exp(x)^2+(18*x*ln(5)^2+90*x*ln(5)+108*x)*exp(x)+x^2*ln( 5)^2+4*x^2*ln(5)+4*x^2)/((81*x*ln(5)^2+486*x*ln(5)+729*x)*exp(x)^2+(18*x^2 *ln(5)^2+90*x^2*ln(5)+108*x^2)*exp(x)+x^3*ln(5)^2+4*x^3*ln(5)+4*x^3)/ln(x) ,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=-\frac {x e - {\left (9 \, {\left (\log \left (5\right ) + 3\right )} e^{x} + x \log \left (5\right ) + 2 \, x\right )} \log \left (\log \left (x\right )\right )}{9 \, {\left (\log \left (5\right ) + 3\right )} e^{x} + x \log \left (5\right ) + 2 \, x} \]
integrate((((9*x^2-9*x)*exp(1)*log(5)+(27*x^2-27*x)*exp(1))*exp(x)*log(x)+ (81*log(5)^2+486*log(5)+729)*exp(x)^2+(18*x*log(5)^2+90*x*log(5)+108*x)*ex p(x)+x^2*log(5)^2+4*x^2*log(5)+4*x^2)/((81*x*log(5)^2+486*x*log(5)+729*x)* exp(x)^2+(18*x^2*log(5)^2+90*x^2*log(5)+108*x^2)*exp(x)+x^3*log(5)^2+4*x^3 *log(5)+4*x^3)/log(x),x, algorithm=\
-(x*e - (9*(log(5) + 3)*e^x + x*log(5) + 2*x)*log(log(x)))/(9*(log(5) + 3) *e^x + x*log(5) + 2*x)
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=- \frac {e x}{x \log {\left (5 \right )} + 2 x + \left (9 \log {\left (5 \right )} + 27\right ) e^{x}} + \log {\left (\log {\left (x \right )} \right )} \]
integrate((((9*x**2-9*x)*exp(1)*ln(5)+(27*x**2-27*x)*exp(1))*exp(x)*ln(x)+ (81*ln(5)**2+486*ln(5)+729)*exp(x)**2+(18*x*ln(5)**2+90*x*ln(5)+108*x)*exp (x)+x**2*ln(5)**2+4*x**2*ln(5)+4*x**2)/((81*x*ln(5)**2+486*x*ln(5)+729*x)* exp(x)**2+(18*x**2*ln(5)**2+90*x**2*ln(5)+108*x**2)*exp(x)+x**3*ln(5)**2+4 *x**3*ln(5)+4*x**3)/ln(x),x)
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=-\frac {x e}{x {\left (\log \left (5\right ) + 2\right )} + 9 \, {\left (\log \left (5\right ) + 3\right )} e^{x}} + \log \left (\log \left (x\right )\right ) \]
integrate((((9*x^2-9*x)*exp(1)*log(5)+(27*x^2-27*x)*exp(1))*exp(x)*log(x)+ (81*log(5)^2+486*log(5)+729)*exp(x)^2+(18*x*log(5)^2+90*x*log(5)+108*x)*ex p(x)+x^2*log(5)^2+4*x^2*log(5)+4*x^2)/((81*x*log(5)^2+486*x*log(5)+729*x)* exp(x)^2+(18*x^2*log(5)^2+90*x^2*log(5)+108*x^2)*exp(x)+x^3*log(5)^2+4*x^3 *log(5)+4*x^3)/log(x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).
Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=\frac {x \log \left (5\right ) \log \left (\log \left (x\right )\right ) + 9 \, e^{x} \log \left (5\right ) \log \left (\log \left (x\right )\right ) - x e + 2 \, x \log \left (\log \left (x\right )\right ) + 27 \, e^{x} \log \left (\log \left (x\right )\right )}{x \log \left (5\right ) + 9 \, e^{x} \log \left (5\right ) + 2 \, x + 27 \, e^{x}} \]
integrate((((9*x^2-9*x)*exp(1)*log(5)+(27*x^2-27*x)*exp(1))*exp(x)*log(x)+ (81*log(5)^2+486*log(5)+729)*exp(x)^2+(18*x*log(5)^2+90*x*log(5)+108*x)*ex p(x)+x^2*log(5)^2+4*x^2*log(5)+4*x^2)/((81*x*log(5)^2+486*x*log(5)+729*x)* exp(x)^2+(18*x^2*log(5)^2+90*x^2*log(5)+108*x^2)*exp(x)+x^3*log(5)^2+4*x^3 *log(5)+4*x^3)/log(x),x, algorithm=\
(x*log(5)*log(log(x)) + 9*e^x*log(5)*log(log(x)) - x*e + 2*x*log(log(x)) + 27*e^x*log(log(x)))/(x*log(5) + 9*e^x*log(5) + 2*x + 27*e^x)
Timed out. \[ \int \frac {4 x^2+4 x^2 \log (5)+x^2 \log ^2(5)+e^{2 x} \left (729+486 \log (5)+81 \log ^2(5)\right )+e^x \left (108 x+90 x \log (5)+18 x \log ^2(5)\right )+e^x \left (e \left (-27 x+27 x^2\right )+e \left (-9 x+9 x^2\right ) \log (5)\right ) \log (x)}{\left (4 x^3+4 x^3 \log (5)+x^3 \log ^2(5)+e^{2 x} \left (729 x+486 x \log (5)+81 x \log ^2(5)\right )+e^x \left (108 x^2+90 x^2 \log (5)+18 x^2 \log ^2(5)\right )\right ) \log (x)} \, dx=\int \frac {x^2\,{\ln \left (5\right )}^2+{\mathrm {e}}^x\,\left (108\,x+90\,x\,\ln \left (5\right )+18\,x\,{\ln \left (5\right )}^2\right )+4\,x^2\,\ln \left (5\right )+4\,x^2+{\mathrm {e}}^{2\,x}\,\left (486\,\ln \left (5\right )+81\,{\ln \left (5\right )}^2+729\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (\mathrm {e}\,\left (27\,x-27\,x^2\right )+\mathrm {e}\,\ln \left (5\right )\,\left (9\,x-9\,x^2\right )\right )}{\ln \left (x\right )\,\left (x^3\,{\ln \left (5\right )}^2+{\mathrm {e}}^x\,\left (18\,x^2\,{\ln \left (5\right )}^2+90\,x^2\,\ln \left (5\right )+108\,x^2\right )+4\,x^3\,\ln \left (5\right )+{\mathrm {e}}^{2\,x}\,\left (729\,x+486\,x\,\ln \left (5\right )+81\,x\,{\ln \left (5\right )}^2\right )+4\,x^3\right )} \,d x \]
int((x^2*log(5)^2 + exp(x)*(108*x + 90*x*log(5) + 18*x*log(5)^2) + 4*x^2*l og(5) + 4*x^2 + exp(2*x)*(486*log(5) + 81*log(5)^2 + 729) - exp(x)*log(x)* (exp(1)*(27*x - 27*x^2) + exp(1)*log(5)*(9*x - 9*x^2)))/(log(x)*(x^3*log(5 )^2 + exp(x)*(18*x^2*log(5)^2 + 90*x^2*log(5) + 108*x^2) + 4*x^3*log(5) + exp(2*x)*(729*x + 486*x*log(5) + 81*x*log(5)^2) + 4*x^3)),x)
int((x^2*log(5)^2 + exp(x)*(108*x + 90*x*log(5) + 18*x*log(5)^2) + 4*x^2*l og(5) + 4*x^2 + exp(2*x)*(486*log(5) + 81*log(5)^2 + 729) - exp(x)*log(x)* (exp(1)*(27*x - 27*x^2) + exp(1)*log(5)*(9*x - 9*x^2)))/(log(x)*(x^3*log(5 )^2 + exp(x)*(18*x^2*log(5)^2 + 90*x^2*log(5) + 108*x^2) + 4*x^3*log(5) + exp(2*x)*(729*x + 486*x*log(5) + 81*x*log(5)^2) + 4*x^3)), x)