Integrand size = 130, antiderivative size = 34 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {(7-x) x \log ^2(x)}{256 \left (-2+\frac {x^2}{2 \left (5-e^x\right )}\right )} \] Output:
1/256*x*ln(x)^2/(1/2*x^2/(5-exp(x))-2)*(-x+7)
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {\left (-5+e^x\right ) (-7+x) x \log ^2(x)}{128 \left (-20+4 e^x+x^2\right )} \] Input:
Integrate[((-1400 + 200*x + 70*x^2 - 10*x^3 + E^(2*x)*(-56 + 8*x) + E^x*(5 60 - 80*x - 14*x^2 + 2*x^3))*Log[x] + (-700 + 200*x - 35*x^2 + E^(2*x)*(-2 8 + 8*x) + E^x*(280 - 80*x + 7*x^2 - 7*x^3 + x^4))*Log[x]^2)/(51200 + 2048 *E^(2*x) - 5120*x^2 + 128*x^4 + E^x*(-20480 + 1024*x^2)),x]
Output:
((-5 + E^x)*(-7 + x)*x*Log[x]^2)/(128*(-20 + 4*E^x + x^2))
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.
\(\displaystyle \int \frac {\left (-10 x^3+70 x^2+e^x \left (2 x^3-14 x^2-80 x+560\right )+200 x+e^{2 x} (8 x-56)-1400\right ) \log (x)+\left (-35 x^2+e^x \left (x^4-7 x^3+7 x^2-80 x+280\right )+200 x+e^{2 x} (8 x-28)-700\right ) \log ^2(x)}{128 x^4-5120 x^2+e^x \left (1024 x^2-20480\right )+2048 e^{2 x}+51200} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-10 x^3+70 x^2+e^x \left (2 x^3-14 x^2-80 x+560\right )+200 x+e^{2 x} (8 x-56)-1400\right ) \log (x)+\left (-35 x^2+e^x \left (x^4-7 x^3+7 x^2-80 x+280\right )+200 x+e^{2 x} (8 x-28)-700\right ) \log ^2(x)}{128 \left (-x^2-4 e^x+20\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{128} \int -\frac {\left (35 x^2-200 x+4 e^{2 x} (7-2 x)-e^x \left (x^4-7 x^3+7 x^2-80 x+280\right )+700\right ) \log ^2(x)+2 \left (5 x^3-35 x^2-100 x+4 e^{2 x} (7-x)-e^x \left (x^3-7 x^2-40 x+280\right )+700\right ) \log (x)}{\left (-x^2-4 e^x+20\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{128} \int \frac {\left (35 x^2-200 x+4 e^{2 x} (7-2 x)-e^x \left (x^4-7 x^3+7 x^2-80 x+280\right )+700\right ) \log ^2(x)+2 \left (5 x^3-35 x^2-100 x+4 e^{2 x} (7-x)-e^x \left (x^3-7 x^2-40 x+280\right )+700\right ) \log (x)}{\left (-x^2-4 e^x+20\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{128} \int \left (\frac {\left (x^3-9 x^2-6 x+140\right ) \log ^2(x) x^3}{4 \left (x^2+4 e^x-20\right )^2}-\frac {\log (x) \left (\log (x) x^2-11 \log (x) x-2 x+21 \log (x)+14\right ) x^2}{4 \left (x^2+4 e^x-20\right )}-\frac {1}{4} \log (x) (2 \log (x) x+2 x-7 \log (x)-14)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{128} \left (-\frac {7}{2} \int \frac {\int \frac {x^2}{x^2+4 e^x-20}dx}{x}dx+\frac {21}{4} \int \frac {x^2 \log ^2(x)}{x^2+4 e^x-20}dx+\frac {7}{2} \log (x) \int \frac {x^2}{x^2+4 e^x-20}dx-\frac {1}{4} \int \frac {x^6 \log ^2(x)}{\left (x^2+4 e^x-20\right )^2}dx+\frac {9}{4} \int \frac {x^5 \log ^2(x)}{\left (x^2+4 e^x-20\right )^2}dx+\frac {3}{2} \int \frac {x^4 \log ^2(x)}{\left (x^2+4 e^x-20\right )^2}dx+\frac {1}{4} \int \frac {x^4 \log ^2(x)}{x^2+4 e^x-20}dx+\frac {1}{2} \int \frac {\int \frac {x^3}{x^2+4 e^x-20}dx}{x}dx-35 \int \frac {x^3 \log ^2(x)}{\left (x^2+4 e^x-20\right )^2}dx-\frac {11}{4} \int \frac {x^3 \log ^2(x)}{x^2+4 e^x-20}dx-\frac {1}{2} \log (x) \int \frac {x^3}{x^2+4 e^x-20}dx+\frac {1}{4} x^2 \log ^2(x)-\frac {1}{4} x^2 \log (x)-\frac {7}{4} x \log ^2(x)+\frac {1}{4} (7-x)^2 \log (x)+\frac {7}{2} x \log (x)-\frac {49 \log (x)}{4}\right )\) |
Input:
Int[((-1400 + 200*x + 70*x^2 - 10*x^3 + E^(2*x)*(-56 + 8*x) + E^x*(560 - 8 0*x - 14*x^2 + 2*x^3))*Log[x] + (-700 + 200*x - 35*x^2 + E^(2*x)*(-28 + 8* x) + E^x*(280 - 80*x + 7*x^2 - 7*x^3 + x^4))*Log[x]^2)/(51200 + 2048*E^(2* x) - 5120*x^2 + 128*x^4 + E^x*(-20480 + 1024*x^2)),x]
Output:
$Aborted
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
\[\frac {x \left ({\mathrm e}^{x} x -5 x -7 \,{\mathrm e}^{x}+35\right ) \ln \left (x \right )^{2}}{128 x^{2}+512 \,{\mathrm e}^{x}-2560}\]
Input:
int((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700 )*ln(x)^2+((8*x-56)*exp(x)^2+(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2+ 200*x-1400)*ln(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4-5120*x^2 +51200),x)
Output:
1/128*x*(exp(x)*x-5*x-7*exp(x)+35)/(x^2+4*exp(x)-20)*ln(x)^2
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=-\frac {{\left (5 \, x^{2} - {\left (x^{2} - 7 \, x\right )} e^{x} - 35 \, x\right )} \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \] Input:
integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200 *x-700)*log(x)^2+((8*x-56)*exp(x)^2+(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+ 70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4- 5120*x^2+51200),x, algorithm="fricas")
Output:
-1/128*(5*x^2 - (x^2 - 7*x)*e^x - 35*x)*log(x)^2/(x^2 + 4*e^x - 20)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\left (\frac {x^{2}}{512} - \frac {7 x}{512}\right ) \log {\left (x \right )}^{2} + \frac {- x^{4} \log {\left (x \right )}^{2} + 7 x^{3} \log {\left (x \right )}^{2}}{512 x^{2} + 2048 e^{x} - 10240} \] Input:
integrate((((8*x-28)*exp(x)**2+(x**4-7*x**3+7*x**2-80*x+280)*exp(x)-35*x** 2+200*x-700)*ln(x)**2+((8*x-56)*exp(x)**2+(2*x**3-14*x**2-80*x+560)*exp(x) -10*x**3+70*x**2+200*x-1400)*ln(x))/(2048*exp(x)**2+(1024*x**2-20480)*exp( x)+128*x**4-5120*x**2+51200),x)
Output:
(x**2/512 - 7*x/512)*log(x)**2 + (-x**4*log(x)**2 + 7*x**3*log(x)**2)/(512 *x**2 + 2048*exp(x) - 10240)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {{\left (x^{2} - 7 \, x\right )} e^{x} \log \left (x\right )^{2} - 5 \, {\left (x^{2} - 7 \, x\right )} \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \] Input:
integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200 *x-700)*log(x)^2+((8*x-56)*exp(x)^2+(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+ 70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4- 5120*x^2+51200),x, algorithm="maxima")
Output:
1/128*((x^2 - 7*x)*e^x*log(x)^2 - 5*(x^2 - 7*x)*log(x)^2)/(x^2 + 4*e^x - 2 0)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {x^{2} e^{x} \log \left (x\right )^{2} - 5 \, x^{2} \log \left (x\right )^{2} - 7 \, x e^{x} \log \left (x\right )^{2} + 35 \, x \log \left (x\right )^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \] Input:
integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200 *x-700)*log(x)^2+((8*x-56)*exp(x)^2+(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+ 70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4- 5120*x^2+51200),x, algorithm="giac")
Output:
1/128*(x^2*e^x*log(x)^2 - 5*x^2*log(x)^2 - 7*x*e^x*log(x)^2 + 35*x*log(x)^ 2)/(x^2 + 4*e^x - 20)
Time = 3.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {x\,{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x-5\right )\,\left (x-7\right )}{128\,\left (4\,{\mathrm {e}}^x+x^2-20\right )} \] Input:
int((log(x)*(200*x + exp(2*x)*(8*x - 56) + 70*x^2 - 10*x^3 - exp(x)*(80*x + 14*x^2 - 2*x^3 - 560) - 1400) + log(x)^2*(200*x + exp(x)*(7*x^2 - 80*x - 7*x^3 + x^4 + 280) + exp(2*x)*(8*x - 28) - 35*x^2 - 700))/(2048*exp(2*x) + exp(x)*(1024*x^2 - 20480) - 5120*x^2 + 128*x^4 + 51200),x)
Output:
(x*log(x)^2*(exp(x) - 5)*(x - 7))/(128*(4*exp(x) + x^2 - 20))
Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx=\frac {\mathrm {log}\left (x \right )^{2} x \left (e^{x} x -7 e^{x}-5 x +35\right )}{512 e^{x}+128 x^{2}-2560} \] Input:
int((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700 )*log(x)^2+((8*x-56)*exp(x)^2+(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2 +200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4-5120*x ^2+51200),x)
Output:
(log(x)**2*x*(e**x*x - 7*e**x - 5*x + 35))/(128*(4*e**x + x**2 - 20))