Integrand size = 148, antiderivative size = 26 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\left (1+x+\frac {49}{e^x+x}\right ) \left (3+x \left (-\frac {1}{x}+x\right ) \log (x)\right ) \] Output:
(ln(x)*x*(x-1/x)+3)*(x+49/(exp(x)+x)+1)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \left (49+e^x x+x^2\right )+\left (-1+x^2\right ) \left (49+x+x^2+e^x (1+x)\right ) \log (x)}{e^x+x} \] Input:
Integrate[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x ^3) + E^x*(-49 - 149*x + 53*x^2 + 2*x^3 + 2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6*x ^4))*Log[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]
Output:
(3*(49 + E^x*x + x^2) + (-1 + x^2)*(49 + x + x^2 + E^x*(1 + x))*Log[x])/(E ^x + 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.
\(\displaystyle \int \frac {x^5+x^4+51 x^3-x^2+e^{2 x} \left (x^3+x^2+2 x-1\right )+e^x \left (2 x^4+2 x^3+53 x^2-149 x-49\right )+\left (3 x^5+2 x^4+48 x^3+e^{2 x} \left (3 x^3+2 x^2-x\right )+e^x \left (6 x^4-45 x^3+96 x^2+49 x\right )+49 x\right ) \log (x)-196 x}{x^3+2 e^x x^2+e^{2 x} x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^5+x^4+51 x^3-x^2+e^{2 x} \left (x^3+x^2+2 x-1\right )+e^x \left (2 x^4+2 x^3+53 x^2-149 x-49\right )+\left (3 x^5+2 x^4+48 x^3+e^{2 x} \left (3 x^3+2 x^2-x\right )+e^x \left (6 x^4-45 x^3+96 x^2+49 x\right )+49 x\right ) \log (x)-196 x}{x \left (x+e^x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {49 (x-1) \left (x^2 \log (x)-\log (x)+3\right )}{\left (x+e^x\right )^2}-\frac {49 \left (x^3 \log (x)-x^2-2 x^2 \log (x)+3 x-x \log (x)+1\right )}{x \left (x+e^x\right )}+\frac {x^3+3 x^3 \log (x)+x^2+2 x^2 \log (x)+2 x-x \log (x)-1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -49 \int \frac {\int \frac {x^3}{\left (x+e^x\right )^2}dx}{x}dx+49 \log (x) \int \frac {x^3}{\left (x+e^x\right )^2}dx+49 \int \frac {\int \frac {x^2}{\left (x+e^x\right )^2}dx}{x}dx+49 \int \frac {\int \frac {x^2}{x+e^x}dx}{x}dx-49 \log (x) \int \frac {x^2}{\left (x+e^x\right )^2}dx-49 \log (x) \int \frac {x^2}{x+e^x}dx+49 \int \frac {x}{x+e^x}dx-98 \int \frac {\int \frac {x}{x+e^x}dx}{x}dx+98 \log (x) \int \frac {x}{x+e^x}dx+x^3 \log (x)+x^2 \log (x)+3 x+\frac {147}{x+e^x}-x \log (x)-\frac {49 \log (x)}{x+e^x}-\log (x)\) |
Input:
Int[(-196*x - x^2 + 51*x^3 + x^4 + x^5 + E^(2*x)*(-1 + 2*x + x^2 + x^3) + E^x*(-49 - 149*x + 53*x^2 + 2*x^3 + 2*x^4) + (49*x + 48*x^3 + 2*x^4 + 3*x^ 5 + E^(2*x)*(-x + 2*x^2 + 3*x^3) + E^x*(49*x + 96*x^2 - 45*x^3 + 6*x^4))*L og[x])/(E^(2*x)*x + 2*E^x*x^2 + x^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.69
\[\frac {\left (x^{4}+{\mathrm e}^{x} x^{3}+x^{3}+{\mathrm e}^{x} x^{2}+48 x^{2}-{\mathrm e}^{x} x -49\right ) \ln \left (x \right )}{{\mathrm e}^{x}+x}-\frac {x \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )-3 x^{2}-3 \,{\mathrm e}^{x} x -147}{{\mathrm e}^{x}+x}\]
Input:
int((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x ^4+48*x^3+49*x)*ln(x)+(x^3+x^2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x-4 9)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x)
Output:
(x^4+exp(x)*x^3+x^3+exp(x)*x^2+48*x^2-exp(x)*x-49)/(exp(x)+x)*ln(x)-(x*ln( x)+exp(x)*ln(x)-3*x^2-3*exp(x)*x-147)/(exp(x)+x)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \, x^{2} + 3 \, x e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} + {\left (x^{3} + x^{2} - x - 1\right )} e^{x} - x - 49\right )} \log \left (x\right ) + 147}{x + e^{x}} \] Input:
integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x ^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2- 149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x , algorithm="fricas")
Output:
(3*x^2 + 3*x*e^x + (x^4 + x^3 + 48*x^2 + (x^3 + x^2 - x - 1)*e^x - x - 49) *log(x) + 147)/(x + e^x)
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=3 x + \left (x^{3} + x^{2} - x\right ) \log {\left (x \right )} - \log {\left (x \right )} + \frac {49 x^{2} \log {\left (x \right )} - 49 \log {\left (x \right )} + 147}{x + e^{x}} \] Input:
integrate((((3*x**3+2*x**2-x)*exp(x)**2+(6*x**4-45*x**3+96*x**2+49*x)*exp( x)+3*x**5+2*x**4+48*x**3+49*x)*ln(x)+(x**3+x**2+2*x-1)*exp(x)**2+(2*x**4+2 *x**3+53*x**2-149*x-49)*exp(x)+x**5+x**4+51*x**3-x**2-196*x)/(x*exp(x)**2+ 2*exp(x)*x**2+x**3),x)
Output:
3*x + (x**3 + x**2 - x)*log(x) - log(x) + (49*x**2*log(x) - 49*log(x) + 14 7)/(x + exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {3 \, x^{2} + {\left ({\left (x^{3} + x^{2} - x - 1\right )} \log \left (x\right ) + 3 \, x\right )} e^{x} + {\left (x^{4} + x^{3} + 48 \, x^{2} - x - 49\right )} \log \left (x\right ) + 147}{x + e^{x}} \] Input:
integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x ^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2- 149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x , algorithm="maxima")
Output:
(3*x^2 + ((x^3 + x^2 - x - 1)*log(x) + 3*x)*e^x + (x^4 + x^3 + 48*x^2 - x - 49)*log(x) + 147)/(x + e^x)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (25) = 50\).
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {x^{4} \log \left (x\right ) + x^{3} e^{x} \log \left (x\right ) + x^{3} \log \left (x\right ) + x^{2} e^{x} \log \left (x\right ) + 48 \, x^{2} \log \left (x\right ) - x e^{x} \log \left (x\right ) + 3 \, x^{2} + 3 \, x e^{x} - x \log \left (x\right ) - e^{x} \log \left (x\right ) - 49 \, \log \left (x\right ) + 147}{x + e^{x}} \] Input:
integrate((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x ^5+2*x^4+48*x^3+49*x)*log(x)+(x^3+x^2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2- 149*x-49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x , algorithm="giac")
Output:
(x^4*log(x) + x^3*e^x*log(x) + x^3*log(x) + x^2*e^x*log(x) + 48*x^2*log(x) - x*e^x*log(x) + 3*x^2 + 3*x*e^x - x*log(x) - e^x*log(x) - 49*log(x) + 14 7)/(x + e^x)
Time = 4.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=3\,x-\ln \left (x\right )+\frac {147}{x+{\mathrm {e}}^x}+\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (x^3+x^2-x\right )+x\,\left (x^3+x^2-x\right )+49\,x^2-49\right )}{x+{\mathrm {e}}^x} \] Input:
int((exp(x)*(53*x^2 - 149*x + 2*x^3 + 2*x^4 - 49) - 196*x + log(x)*(49*x + exp(x)*(49*x + 96*x^2 - 45*x^3 + 6*x^4) + exp(2*x)*(2*x^2 - x + 3*x^3) + 48*x^3 + 2*x^4 + 3*x^5) + exp(2*x)*(2*x + x^2 + x^3 - 1) - x^2 + 51*x^3 + x^4 + x^5)/(x*exp(2*x) + 2*x^2*exp(x) + x^3),x)
Output:
3*x - log(x) + 147/(x + exp(x)) + (log(x)*(exp(x)*(x^2 - x + x^3) + x*(x^2 - x + x^3) + 49*x^2 - 49))/(x + exp(x))
Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {-196 x-x^2+51 x^3+x^4+x^5+e^{2 x} \left (-1+2 x+x^2+x^3\right )+e^x \left (-49-149 x+53 x^2+2 x^3+2 x^4\right )+\left (49 x+48 x^3+2 x^4+3 x^5+e^{2 x} \left (-x+2 x^2+3 x^3\right )+e^x \left (49 x+96 x^2-45 x^3+6 x^4\right )\right ) \log (x)}{e^{2 x} x+2 e^x x^2+x^3} \, dx=\frac {e^{x} \mathrm {log}\left (x \right ) x^{3}+e^{x} \mathrm {log}\left (x \right ) x^{2}-e^{x} \mathrm {log}\left (x \right ) x -e^{x} \mathrm {log}\left (x \right )+3 e^{x} x +\mathrm {log}\left (x \right ) x^{4}+\mathrm {log}\left (x \right ) x^{3}+48 \,\mathrm {log}\left (x \right ) x^{2}-\mathrm {log}\left (x \right ) x -49 \,\mathrm {log}\left (x \right )+3 x^{2}+147}{e^{x}+x} \] Input:
int((((3*x^3+2*x^2-x)*exp(x)^2+(6*x^4-45*x^3+96*x^2+49*x)*exp(x)+3*x^5+2*x ^4+48*x^3+49*x)*log(x)+(x^3+x^2+2*x-1)*exp(x)^2+(2*x^4+2*x^3+53*x^2-149*x- 49)*exp(x)+x^5+x^4+51*x^3-x^2-196*x)/(x*exp(x)^2+2*exp(x)*x^2+x^3),x)
Output:
(e**x*log(x)*x**3 + e**x*log(x)*x**2 - e**x*log(x)*x - e**x*log(x) + 3*e** x*x + log(x)*x**4 + log(x)*x**3 + 48*log(x)*x**2 - log(x)*x - 49*log(x) + 3*x**2 + 147)/(e**x + x)