Integrand size = 168, antiderivative size = 32 \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=\log \left (-2+\frac {x^4}{\left (1-x-\frac {3+e^{-5 x} x}{x}\right ) \log (x)}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(32)=64\).
Time = 0.91 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=-\log \left (3 e^{5 x}+x-e^{5 x} x+e^{5 x} x^2\right )-\log (\log (x))+\log \left (e^{5 x} x^5+6 e^{5 x} \log (x)+2 x \log (x)-2 e^{5 x} x \log (x)+2 e^{5 x} x^2 \log (x)\right ) \]
Integrate[(-(E^(5*x)*x^5) + E^(10*x)*(-3*x^4 + x^5 - x^6) + (E^(10*x)*(15* x^4 - 4*x^5 + 3*x^6) + E^(5*x)*(4*x^5 + 5*x^6))*Log[x])/((E^(5*x)*x^6 + E^ (10*x)*(3*x^5 - x^6 + x^7))*Log[x] + (2*x^2 + E^(5*x)*(12*x - 4*x^2 + 4*x^ 3) + E^(10*x)*(18 - 12*x + 14*x^2 - 4*x^3 + 2*x^4))*Log[x]^2),x]
-Log[3*E^(5*x) + x - E^(5*x)*x + E^(5*x)*x^2] - Log[Log[x]] + Log[E^(5*x)* x^5 + 6*E^(5*x)*Log[x] + 2*x*Log[x] - 2*E^(5*x)*x*Log[x] + 2*E^(5*x)*x^2*L og[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.
| \(\displaystyle \int \frac {-e^{5 x} x^5+e^{10 x} \left (-x^6+x^5-3 x^4\right )+\left (e^{5 x} \left (5 x^6+4 x^5\right )+e^{10 x} \left (3 x^6-4 x^5+15 x^4\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (x^7-x^6+3 x^5\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (4 x^3-4 x^2+12 x\right )+e^{10 x} \left (2 x^4-4 x^3+14 x^2-12 x+18\right )\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{5 x} x^5+e^{10 x} \left (-x^6+x^5-3 x^4\right )+\left (e^{5 x} \left (5 x^6+4 x^5\right )+e^{10 x} \left (3 x^6-4 x^5+15 x^4\right )\right ) \log (x)}{\left (e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}\right ) \log (x) \left (e^{5 x} x^5+2 e^{5 x} x^2 \log (x)-2 e^{5 x} x \log (x)+2 x \log (x)+6 e^{5 x} \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 x^3-4 x^2+15 x-3}{\left (x^2-x+3\right ) \left (e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}\right )}+\frac {x^4 \left (-x^2+3 x^2 \log (x)+x-4 x \log (x)+15 \log (x)-3\right )}{\left (x^2-x+3\right ) \log (x) \left (x^5+2 x^2 \log (x)-2 x \log (x)+6 \log (x)\right )}-\frac {2 \left (5 x^6 \log (x)-x^5+4 x^5 \log (x)+10 x^3 \log ^2(x)-8 x^2 \log ^2(x)+30 x \log ^2(x)-6 \log ^2(x)\right )}{\left (x^5+2 x^2 \log (x)-2 x \log (x)+6 \log (x)\right ) \left (e^{5 x} x^5+2 e^{5 x} x^2 \log (x)-2 e^{5 x} x \log (x)+2 x \log (x)+6 e^{5 x} \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{5 x} x^4 \left (e^{5 x} \left (-x^2+x-3\right )+\left (e^{5 x} \left (3 x^2-4 x+15\right )+x (5 x+4)\right ) \log (x)-x\right )}{\left (e^{5 x} \left (x^2-x+3\right )+x\right ) \log (x) \left (e^{5 x} x^5+2 \left (e^{5 x} \left (x^2-x+3\right )+x\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{5 x} \left (5 x^6 \log (x)-x^5+4 x^5 \log (x)+10 x^3 \log ^2(x)-8 x^2 \log ^2(x)+30 x \log ^2(x)-6 \log ^2(x)\right )}{x \log (x) \left (e^{5 x} x^5+2 e^{5 x} x^2 \log (x)-2 e^{5 x} x \log (x)+2 x \log (x)+6 e^{5 x} \log (x)\right )}-\frac {e^{5 x} \left (5 x^3-4 x^2+15 x-3\right )}{x \left (e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -15 \int \frac {e^{5 x}}{e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}}dx+3 \int \frac {e^{5 x}}{x \left (e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}\right )}dx+4 \int \frac {e^{5 x} x}{e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}}dx-5 \int \frac {e^{5 x} x^2}{e^{5 x} x^2-e^{5 x} x+x+3 e^{5 x}}dx+5 \int \frac {e^{5 x} x^5}{e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)}dx+30 \int \frac {e^{5 x} \log (x)}{e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)}dx-6 \int \frac {e^{5 x} \log (x)}{x \left (e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)\right )}dx-8 \int \frac {e^{5 x} x \log (x)}{e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)}dx+10 \int \frac {e^{5 x} x^2 \log (x)}{e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)}dx+4 \int \frac {e^{5 x} x^4}{e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)}dx-\int \frac {e^{5 x} x^4}{\log (x) \left (e^{5 x} x^5+2 e^{5 x} \log (x) x^2-2 e^{5 x} \log (x) x+2 \log (x) x+6 e^{5 x} \log (x)\right )}dx\) |
Int[(-(E^(5*x)*x^5) + E^(10*x)*(-3*x^4 + x^5 - x^6) + (E^(10*x)*(15*x^4 - 4*x^5 + 3*x^6) + E^(5*x)*(4*x^5 + 5*x^6))*Log[x])/((E^(5*x)*x^6 + E^(10*x) *(3*x^5 - x^6 + x^7))*Log[x] + (2*x^2 + E^(5*x)*(12*x - 4*x^2 + 4*x^3) + E ^(10*x)*(18 - 12*x + 14*x^2 - 4*x^3 + 2*x^4))*Log[x]^2),x]
3.19.45.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (x \right )+\frac {x^{5} {\mathrm e}^{5 x}}{2 x^{2} {\mathrm e}^{5 x}-2 x \,{\mathrm e}^{5 x}+2 x +6 \,{\mathrm e}^{5 x}}\right )\) | \(45\) |
| parallelrisch | \(-\ln \left (\ln \left (x \right )\right )-\ln \left (x^{2} {\mathrm e}^{5 x}-x \,{\mathrm e}^{5 x}+x +3 \,{\mathrm e}^{5 x}\right )+\ln \left (x^{5} {\mathrm e}^{5 x}+2 \,{\mathrm e}^{5 x} \ln \left (x \right ) x^{2}-2 \,{\mathrm e}^{5 x} \ln \left (x \right ) x +2 x \ln \left (x \right )+6 \,{\mathrm e}^{5 x} \ln \left (x \right )\right )\) | \(76\) |
int((((3*x^6-4*x^5+15*x^4)*exp(5*x)^2+(5*x^6+4*x^5)*exp(5*x))*ln(x)+(-x^6+ x^5-3*x^4)*exp(5*x)^2-x^5*exp(5*x))/(((2*x^4-4*x^3+14*x^2-12*x+18)*exp(5*x )^2+(4*x^3-4*x^2+12*x)*exp(5*x)+2*x^2)*ln(x)^2+((x^7-x^6+3*x^5)*exp(5*x)^2 +x^6*exp(5*x))*ln(x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66 \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=\log \left (\frac {x^{5} e^{\left (5 \, x\right )} + 2 \, {\left ({\left (x^{2} - x + 3\right )} e^{\left (5 \, x\right )} + x\right )} \log \left (x\right )}{{\left (x^{2} - x + 3\right )} e^{\left (5 \, x\right )} + x}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((3*x^6-4*x^5+15*x^4)*exp(5*x)^2+(5*x^6+4*x^5)*exp(5*x))*log(x) +(-x^6+x^5-3*x^4)*exp(5*x)^2-x^5*exp(5*x))/(((2*x^4-4*x^3+14*x^2-12*x+18)* exp(5*x)^2+(4*x^3-4*x^2+12*x)*exp(5*x)+2*x^2)*log(x)^2+((x^7-x^6+3*x^5)*ex p(5*x)^2+x^6*exp(5*x))*log(x)),x, algorithm=\
log((x^5*e^(5*x) + 2*((x^2 - x + 3)*e^(5*x) + x)*log(x))/((x^2 - x + 3)*e^ (5*x) + x)) - log(log(x))
Exception generated. \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=\text {Exception raised: PolynomialError} \]
integrate((((3*x**6-4*x**5+15*x**4)*exp(5*x)**2+(5*x**6+4*x**5)*exp(5*x))* ln(x)+(-x**6+x**5-3*x**4)*exp(5*x)**2-x**5*exp(5*x))/(((2*x**4-4*x**3+14*x **2-12*x+18)*exp(5*x)**2+(4*x**3-4*x**2+12*x)*exp(5*x)+2*x**2)*ln(x)**2+(( x**7-x**6+3*x**5)*exp(5*x)**2+x**6*exp(5*x))*ln(x)),x)
Exception raised: PolynomialError >> 1/(4*_t0**2*x**8 - 16*_t0**2*x**7 + 7 2*_t0**2*x**6 - 160*_t0**2*x**5 + 364*_t0**2*x**4 - 480*_t0**2*x**3 + 648* _t0**2*x**2 - 432*_t0**2*x + 324*_t0**2 + 4*_t0*x**11 - 12*_t0*x**10 + 48* _t0*x**9 - 76*_
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.47 \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=\log \left (\frac {x^{5} + 2 \, {\left (x^{2} - x + 3\right )} \log \left (x\right )}{2 \, {\left (x^{2} - x + 3\right )}}\right ) + \log \left (\frac {{\left (x^{5} + 2 \, {\left (x^{2} - x + 3\right )} \log \left (x\right )\right )} e^{\left (5 \, x\right )} + 2 \, x \log \left (x\right )}{x^{5} + 2 \, {\left (x^{2} - x + 3\right )} \log \left (x\right )}\right ) - \log \left (\frac {{\left (x^{2} - x + 3\right )} e^{\left (5 \, x\right )} + x}{x^{2} - x + 3}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((3*x^6-4*x^5+15*x^4)*exp(5*x)^2+(5*x^6+4*x^5)*exp(5*x))*log(x) +(-x^6+x^5-3*x^4)*exp(5*x)^2-x^5*exp(5*x))/(((2*x^4-4*x^3+14*x^2-12*x+18)* exp(5*x)^2+(4*x^3-4*x^2+12*x)*exp(5*x)+2*x^2)*log(x)^2+((x^7-x^6+3*x^5)*ex p(5*x)^2+x^6*exp(5*x))*log(x)),x, algorithm=\
log(1/2*(x^5 + 2*(x^2 - x + 3)*log(x))/(x^2 - x + 3)) + log(((x^5 + 2*(x^2 - x + 3)*log(x))*e^(5*x) + 2*x*log(x))/(x^5 + 2*(x^2 - x + 3)*log(x))) - log(((x^2 - x + 3)*e^(5*x) + x)/(x^2 - x + 3)) - log(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
Time = 0.41 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=\log \left (x^{5} e^{\left (5 \, x\right )} + 2 \, x^{2} e^{\left (5 \, x\right )} \log \left (x\right ) - 2 \, x e^{\left (5 \, x\right )} \log \left (x\right ) + 2 \, x \log \left (x\right ) + 6 \, e^{\left (5 \, x\right )} \log \left (x\right )\right ) - \log \left (x^{2} e^{\left (5 \, x\right )} - x e^{\left (5 \, x\right )} + x + 3 \, e^{\left (5 \, x\right )}\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((3*x^6-4*x^5+15*x^4)*exp(5*x)^2+(5*x^6+4*x^5)*exp(5*x))*log(x) +(-x^6+x^5-3*x^4)*exp(5*x)^2-x^5*exp(5*x))/(((2*x^4-4*x^3+14*x^2-12*x+18)* exp(5*x)^2+(4*x^3-4*x^2+12*x)*exp(5*x)+2*x^2)*log(x)^2+((x^7-x^6+3*x^5)*ex p(5*x)^2+x^6*exp(5*x))*log(x)),x, algorithm=\
log(x^5*e^(5*x) + 2*x^2*e^(5*x)*log(x) - 2*x*e^(5*x)*log(x) + 2*x*log(x) + 6*e^(5*x)*log(x)) - log(x^2*e^(5*x) - x*e^(5*x) + x + 3*e^(5*x)) - log(lo g(x))
Timed out. \[ \int \frac {-e^{5 x} x^5+e^{10 x} \left (-3 x^4+x^5-x^6\right )+\left (e^{10 x} \left (15 x^4-4 x^5+3 x^6\right )+e^{5 x} \left (4 x^5+5 x^6\right )\right ) \log (x)}{\left (e^{5 x} x^6+e^{10 x} \left (3 x^5-x^6+x^7\right )\right ) \log (x)+\left (2 x^2+e^{5 x} \left (12 x-4 x^2+4 x^3\right )+e^{10 x} \left (18-12 x+14 x^2-4 x^3+2 x^4\right )\right ) \log ^2(x)} \, dx=-\int \frac {{\mathrm {e}}^{10\,x}\,\left (x^6-x^5+3\,x^4\right )+x^5\,{\mathrm {e}}^{5\,x}-\ln \left (x\right )\,\left ({\mathrm {e}}^{5\,x}\,\left (5\,x^6+4\,x^5\right )+{\mathrm {e}}^{10\,x}\,\left (3\,x^6-4\,x^5+15\,x^4\right )\right )}{\left ({\mathrm {e}}^{5\,x}\,\left (4\,x^3-4\,x^2+12\,x\right )+{\mathrm {e}}^{10\,x}\,\left (2\,x^4-4\,x^3+14\,x^2-12\,x+18\right )+2\,x^2\right )\,{\ln \left (x\right )}^2+\left ({\mathrm {e}}^{10\,x}\,\left (x^7-x^6+3\,x^5\right )+x^6\,{\mathrm {e}}^{5\,x}\right )\,\ln \left (x\right )} \,d x \]
int(-(exp(10*x)*(3*x^4 - x^5 + x^6) + x^5*exp(5*x) - log(x)*(exp(5*x)*(4*x ^5 + 5*x^6) + exp(10*x)*(15*x^4 - 4*x^5 + 3*x^6)))/(log(x)*(exp(10*x)*(3*x ^5 - x^6 + x^7) + x^6*exp(5*x)) + log(x)^2*(exp(5*x)*(12*x - 4*x^2 + 4*x^3 ) + exp(10*x)*(14*x^2 - 12*x - 4*x^3 + 2*x^4 + 18) + 2*x^2)),x)