Integrand size = 137, antiderivative size = 26 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\left (-x+\frac {24}{4+\frac {e^x}{3 x^2 \log (x)}}\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=2 \left (-6 x+\frac {x^2}{2}+\frac {18 e^{2 x}}{\left (e^x+12 x^2 \log (x)\right )^2}+\frac {6 e^x (-6+x)}{e^x+12 x^2 \log (x)}\right ) \]
Integrate[(2*E^(3*x)*x - 144*E^(2*x)*x^2 + (E^(2*x)*(-432*x^2 + 216*x^3) + E^x*(10368*x^3 - 1728*x^4))*Log[x] + E^x*(20736*x^3 - 17280*x^4 + 2592*x^ 5)*Log[x]^2 + (-20736*x^6 + 3456*x^7)*Log[x]^3)/(E^(3*x) + 36*E^(2*x)*x^2* Log[x] + 432*E^x*x^4*Log[x]^2 + 1728*x^6*Log[x]^3),x]
2*(-6*x + x^2/2 + (18*E^(2*x))/(E^x + 12*x^2*Log[x])^2 + (6*E^x*(-6 + x))/ (E^x + 12*x^2*Log[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.
\(\displaystyle \int \frac {-144 e^{2 x} x^2+\left (3456 x^7-20736 x^6\right ) \log ^3(x)+e^x \left (2592 x^5-17280 x^4+20736 x^3\right ) \log ^2(x)+\left (e^x \left (10368 x^3-1728 x^4\right )+e^{2 x} \left (216 x^3-432 x^2\right )\right ) \log (x)+2 e^{3 x} x}{1728 x^6 \log ^3(x)+432 e^x x^4 \log ^2(x)+36 e^{2 x} x^2 \log (x)+e^{3 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-144 e^{2 x} x^2+\left (3456 x^7-20736 x^6\right ) \log ^3(x)+e^x \left (2592 x^5-17280 x^4+20736 x^3\right ) \log ^2(x)+\left (e^x \left (10368 x^3-1728 x^4\right )+e^{2 x} \left (216 x^3-432 x^2\right )\right ) \log (x)+2 e^{3 x} x}{\left (12 x^2 \log (x)+e^x\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {144 x^2 (x \log (x)-3 \log (x)-1)}{12 x^2 \log (x)+e^x}+\frac {124416 x^5 \log ^2(x) (x \log (x)-2 \log (x)-1)}{\left (12 x^2 \log (x)+e^x\right )^3}-\frac {1728 (x+6) x^3 \log (x) (x \log (x)-2 \log (x)-1)}{\left (12 x^2 \log (x)+e^x\right )^2}+2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -144 \int \frac {x^2}{12 \log (x) x^2+e^x}dx-432 \int \frac {x^2 \log (x)}{12 \log (x) x^2+e^x}dx+124416 \int \frac {x^6 \log ^3(x)}{\left (12 \log (x) x^2+e^x\right )^3}dx-248832 \int \frac {x^5 \log ^3(x)}{\left (12 \log (x) x^2+e^x\right )^3}dx-124416 \int \frac {x^5 \log ^2(x)}{\left (12 \log (x) x^2+e^x\right )^3}dx-1728 \int \frac {x^5 \log ^2(x)}{\left (12 \log (x) x^2+e^x\right )^2}dx-6912 \int \frac {x^4 \log ^2(x)}{\left (12 \log (x) x^2+e^x\right )^2}dx+1728 \int \frac {x^4 \log (x)}{\left (12 \log (x) x^2+e^x\right )^2}dx+20736 \int \frac {x^3 \log ^2(x)}{\left (12 \log (x) x^2+e^x\right )^2}dx+10368 \int \frac {x^3 \log (x)}{\left (12 \log (x) x^2+e^x\right )^2}dx+144 \int \frac {x^3 \log (x)}{12 \log (x) x^2+e^x}dx+x^2\) |
Int[(2*E^(3*x)*x - 144*E^(2*x)*x^2 + (E^(2*x)*(-432*x^2 + 216*x^3) + E^x*( 10368*x^3 - 1728*x^4))*Log[x] + E^x*(20736*x^3 - 17280*x^4 + 2592*x^5)*Log [x]^2 + (-20736*x^6 + 3456*x^7)*Log[x]^3)/(E^(3*x) + 36*E^(2*x)*x^2*Log[x] + 432*E^x*x^4*Log[x]^2 + 1728*x^6*Log[x]^3),x]
3.3.96.3.1 Defintions of rubi rules used
Time = 2.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81
method | result | size |
risch | \(x^{2}-12 x +\frac {12 \left (12 x^{3} \ln \left (x \right )-72 x^{2} \ln \left (x \right )+{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{x}}{\left (12 x^{2} \ln \left (x \right )+{\mathrm e}^{x}\right )^{2}}\) | \(47\) |
parallelrisch | \(-\frac {-3456 x^{6} \ln \left (x \right )^{2}+41472 x^{5} \ln \left (x \right )^{2}-124416 x^{4} \ln \left (x \right )^{2}-576 x^{4} {\mathrm e}^{x} \ln \left (x \right )+3456 x^{3} {\mathrm e}^{x} \ln \left (x \right )-24 \,{\mathrm e}^{2 x} x^{2}}{24 \left (144 x^{4} \ln \left (x \right )^{2}+24 x^{2} {\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}\right )}\) | \(83\) |
int(((3456*x^7-20736*x^6)*ln(x)^3+(2592*x^5-17280*x^4+20736*x^3)*exp(x)*ln (x)^2+((216*x^3-432*x^2)*exp(x)^2+(-1728*x^4+10368*x^3)*exp(x))*ln(x)+2*x* exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*ln(x)^3+432*x^4*exp(x)*ln(x)^2+36*x^2 *exp(x)^2*ln(x)+exp(x)^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {24 \, {\left (x^{4} - 6 \, x^{3} - 36 \, x^{2}\right )} e^{x} \log \left (x\right ) + 144 \, {\left (x^{6} - 12 \, x^{5}\right )} \log \left (x\right )^{2} + {\left (x^{2} - 36\right )} e^{\left (2 \, x\right )}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*ex p(x)*log(x)^2+((216*x^3-432*x^2)*exp(x)^2+(-1728*x^4+10368*x^3)*exp(x))*lo g(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*log( x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm=\
(24*(x^4 - 6*x^3 - 36*x^2)*e^x*log(x) + 144*(x^6 - 12*x^5)*log(x)^2 + (x^2 - 36)*e^(2*x))/(144*x^4*log(x)^2 + 24*x^2*e^x*log(x) + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=x^{2} + \frac {- 1728 x^{5} \log {\left (x \right )}^{2} + 5184 x^{4} \log {\left (x \right )}^{2} - 144 x^{3} e^{x} \log {\left (x \right )}}{144 x^{4} \log {\left (x \right )}^{2} + 24 x^{2} e^{x} \log {\left (x \right )} + e^{2 x}} \]
integrate(((3456*x**7-20736*x**6)*ln(x)**3+(2592*x**5-17280*x**4+20736*x** 3)*exp(x)*ln(x)**2+((216*x**3-432*x**2)*exp(x)**2+(-1728*x**4+10368*x**3)* exp(x))*ln(x)+2*x*exp(x)**3-144*exp(x)**2*x**2)/(1728*x**6*ln(x)**3+432*x* *4*exp(x)*ln(x)**2+36*x**2*exp(x)**2*ln(x)+exp(x)**3),x)
x**2 + (-1728*x**5*log(x)**2 + 5184*x**4*log(x)**2 - 144*x**3*exp(x)*log(x ))/(144*x**4*log(x)**2 + 24*x**2*exp(x)*log(x) + exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {x^{2} e^{\left (2 \, x\right )} + 24 \, {\left (x^{4} - 6 \, x^{3}\right )} e^{x} \log \left (x\right ) + 144 \, {\left (x^{6} - 12 \, x^{5} + 36 \, x^{4}\right )} \log \left (x\right )^{2}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*ex p(x)*log(x)^2+((216*x^3-432*x^2)*exp(x)^2+(-1728*x^4+10368*x^3)*exp(x))*lo g(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*log( x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm=\
(x^2*e^(2*x) + 24*(x^4 - 6*x^3)*e^x*log(x) + 144*(x^6 - 12*x^5 + 36*x^4)*l og(x)^2)/(144*x^4*log(x)^2 + 24*x^2*e^x*log(x) + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=\frac {144 \, x^{6} \log \left (x\right )^{2} - 1728 \, x^{5} \log \left (x\right )^{2} + 24 \, x^{4} e^{x} \log \left (x\right ) - 1872 \, x^{4} \log \left (x\right )^{2} - 144 \, x^{3} e^{x} \log \left (x\right ) - 1176 \, x^{2} e^{x} \log \left (x\right ) + x^{2} e^{\left (2 \, x\right )} - 49 \, e^{\left (2 \, x\right )}}{144 \, x^{4} \log \left (x\right )^{2} + 24 \, x^{2} e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )}} \]
integrate(((3456*x^7-20736*x^6)*log(x)^3+(2592*x^5-17280*x^4+20736*x^3)*ex p(x)*log(x)^2+((216*x^3-432*x^2)*exp(x)^2+(-1728*x^4+10368*x^3)*exp(x))*lo g(x)+2*x*exp(x)^3-144*exp(x)^2*x^2)/(1728*x^6*log(x)^3+432*x^4*exp(x)*log( x)^2+36*x^2*exp(x)^2*log(x)+exp(x)^3),x, algorithm=\
(144*x^6*log(x)^2 - 1728*x^5*log(x)^2 + 24*x^4*e^x*log(x) - 1872*x^4*log(x )^2 - 144*x^3*e^x*log(x) - 1176*x^2*e^x*log(x) + x^2*e^(2*x) - 49*e^(2*x)) /(144*x^4*log(x)^2 + 24*x^2*e^x*log(x) + e^(2*x))
Time = 9.01 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.96 \[ \int \frac {2 e^{3 x} x-144 e^{2 x} x^2+\left (e^{2 x} \left (-432 x^2+216 x^3\right )+e^x \left (10368 x^3-1728 x^4\right )\right ) \log (x)+e^x \left (20736 x^3-17280 x^4+2592 x^5\right ) \log ^2(x)+\left (-20736 x^6+3456 x^7\right ) \log ^3(x)}{e^{3 x}+36 e^{2 x} x^2 \log (x)+432 e^x x^4 \log ^2(x)+1728 x^6 \log ^3(x)} \, dx=x^2-12\,x+\frac {12\,\left (12\,x\,{\mathrm {e}}^{2\,x}-72\,x^3\,{\mathrm {e}}^x+12\,x^4\,{\mathrm {e}}^x-8\,x^2\,{\mathrm {e}}^{2\,x}+x^3\,{\mathrm {e}}^{2\,x}\right )}{\left ({\mathrm {e}}^x+12\,x^2\,\ln \left (x\right )\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )}+\frac {36\,{\mathrm {e}}^x\,\left (12\,x^5\,{\mathrm {e}}^x-2\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}\right )}{x^2\,\left ({\mathrm {e}}^{2\,x}+144\,x^4\,{\ln \left (x\right )}^2+24\,x^2\,{\mathrm {e}}^x\,\ln \left (x\right )\right )\,\left (x^2\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+12\,x^3\right )} \]
int((2*x*exp(3*x) + log(x)*(exp(x)*(10368*x^3 - 1728*x^4) - exp(2*x)*(432* x^2 - 216*x^3)) - log(x)^3*(20736*x^6 - 3456*x^7) - 144*x^2*exp(2*x) + exp (x)*log(x)^2*(20736*x^3 - 17280*x^4 + 2592*x^5))/(exp(3*x) + 1728*x^6*log( x)^3 + 36*x^2*exp(2*x)*log(x) + 432*x^4*exp(x)*log(x)^2),x)
x^2 - 12*x + (12*(12*x*exp(2*x) - 72*x^3*exp(x) + 12*x^4*exp(x) - 8*x^2*ex p(2*x) + x^3*exp(2*x)))/((exp(x) + 12*x^2*log(x))*(x^2*exp(x) - 2*x*exp(x) + 12*x^3)) + (36*exp(x)*(12*x^5*exp(x) - 2*x^3*exp(2*x) + x^4*exp(2*x)))/ (x^2*(exp(2*x) + 144*x^4*log(x)^2 + 24*x^2*exp(x)*log(x))*(x^2*exp(x) - 2* x*exp(x) + 12*x^3))