Integrand size = 145, antiderivative size = 25 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {10}{3}+x+\log (x)-\frac {4}{e^{2 x}-x+x^2+\log (x)} \] Output:
10/3-4/(x^2+exp(x)^2+ln(x)-x)+x+ln(x)
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=x+\log (x)-\frac {4}{e^{2 x}-x+x^2+\log (x)} \] Input:
Integrate[(4 - 4*x + 9*x^2 - x^3 - x^4 + x^5 + E^(4*x)*(1 + x) + E^(2*x)*( 6*x + 2*x^3) + (-2*x + 2*x^3 + E^(2*x)*(2 + 2*x))*Log[x] + (1 + x)*Log[x]^ 2)/(E^(4*x)*x + x^3 - 2*x^4 + x^5 + E^(2*x)*(-2*x^2 + 2*x^3) + (2*E^(2*x)* x - 2*x^2 + 2*x^3)*Log[x] + x*Log[x]^2),x]
Output:
x + Log[x] - 4/(E^(2*x) - x + 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.
\(\displaystyle \int \frac {x^5-x^4-x^3+e^{2 x} \left (2 x^3+6 x\right )+\left (2 x^3-2 x+e^{2 x} (2 x+2)\right ) \log (x)+9 x^2-4 x+e^{4 x} (x+1)+(x+1) \log ^2(x)+4}{x^5-2 x^4+x^3+e^{2 x} \left (2 x^3-2 x^2\right )+\left (2 x^3-2 x^2+2 e^{2 x} x\right ) \log (x)+e^{4 x} x+x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^5-x^4-x^3+e^{2 x} \left (2 x^3+6 x\right )+\left (2 x^3-2 x+e^{2 x} (2 x+2)\right ) \log (x)+9 x^2-4 x+e^{4 x} (x+1)+(x+1) \log ^2(x)+4}{x \left (x^2-x+e^{2 x}+\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8}{x^2-x+e^{2 x}+\log (x)}-\frac {4 \left (2 x^3-4 x^2+x+2 x \log (x)-1\right )}{x \left (x^2-x+e^{2 x}+\log (x)\right )^2}+\frac {x+1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{\left (x^2-x+e^{2 x}+\log (x)\right )^2}dx+4 \int \frac {1}{x \left (x^2-x+e^{2 x}+\log (x)\right )^2}dx+16 \int \frac {x}{\left (x^2-x+e^{2 x}+\log (x)\right )^2}dx-8 \int \frac {x^2}{\left (x^2-x+e^{2 x}+\log (x)\right )^2}dx-8 \int \frac {\log (x)}{\left (x^2-x+e^{2 x}+\log (x)\right )^2}dx+8 \int \frac {1}{x^2-x+e^{2 x}+\log (x)}dx+x+\log (x)\) |
Input:
Int[(4 - 4*x + 9*x^2 - x^3 - x^4 + x^5 + E^(4*x)*(1 + x) + E^(2*x)*(6*x + 2*x^3) + (-2*x + 2*x^3 + E^(2*x)*(2 + 2*x))*Log[x] + (1 + x)*Log[x]^2)/(E^ (4*x)*x + x^3 - 2*x^4 + x^5 + E^(2*x)*(-2*x^2 + 2*x^3) + (2*E^(2*x)*x - 2* x^2 + 2*x^3)*Log[x] + x*Log[x]^2),x]
Output:
$Aborted
Time = 22.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(x +\ln \left (x \right )-\frac {4}{x^{2}+{\mathrm e}^{2 x}+\ln \left (x \right )-x}\) | \(22\) |
parallelrisch | \(\frac {x^{2} \ln \left (x \right )+x^{3}+{\mathrm e}^{2 x} \ln \left (x \right )+x \,{\mathrm e}^{2 x}+\ln \left (x \right )^{2}-x^{2}-4}{x^{2}+{\mathrm e}^{2 x}+\ln \left (x \right )-x}\) | \(50\) |
Input:
int(((1+x)*ln(x)^2+((2+2*x)*exp(x)^2+2*x^3-2*x)*ln(x)+(1+x)*exp(x)^4+(2*x^ 3+6*x)*exp(x)^2+x^5-x^4-x^3+9*x^2-4*x+4)/(x*ln(x)^2+(2*x*exp(x)^2+2*x^3-2* x^2)*ln(x)+x*exp(x)^4+(2*x^3-2*x^2)*exp(x)^2+x^5-2*x^4+x^3),x,method=_RETU RNVERBOSE)
Output:
x+ln(x)-4/(x^2+exp(2*x)+ln(x)-x)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {x^{3} - x^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} + e^{\left (2 \, x\right )}\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} \] Input:
integrate(((1+x)*log(x)^2+((2+2*x)*exp(x)^2+2*x^3-2*x)*log(x)+(1+x)*exp(x) ^4+(2*x^3+6*x)*exp(x)^2+x^5-x^4-x^3+9*x^2-4*x+4)/(x*log(x)^2+(2*x*exp(x)^2 +2*x^3-2*x^2)*log(x)+x*exp(x)^4+(2*x^3-2*x^2)*exp(x)^2+x^5-2*x^4+x^3),x, a lgorithm="fricas")
Output:
(x^3 - x^2 + x*e^(2*x) + (x^2 + e^(2*x))*log(x) + log(x)^2 - 4)/(x^2 - x + e^(2*x) + log(x))
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=x + \log {\left (x \right )} - \frac {4}{x^{2} - x + e^{2 x} + \log {\left (x \right )}} \] Input:
integrate(((1+x)*ln(x)**2+((2+2*x)*exp(x)**2+2*x**3-2*x)*ln(x)+(1+x)*exp(x )**4+(2*x**3+6*x)*exp(x)**2+x**5-x**4-x**3+9*x**2-4*x+4)/(x*ln(x)**2+(2*x* exp(x)**2+2*x**3-2*x**2)*ln(x)+x*exp(x)**4+(2*x**3-2*x**2)*exp(x)**2+x**5- 2*x**4+x**3),x)
Output:
x + log(x) - 4/(x**2 - x + exp(2*x) + log(x))
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {x^{3} - x^{2} + x e^{\left (2 \, x\right )} + x \log \left (x\right ) - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} + \log \left (x\right ) \] Input:
integrate(((1+x)*log(x)^2+((2+2*x)*exp(x)^2+2*x^3-2*x)*log(x)+(1+x)*exp(x) ^4+(2*x^3+6*x)*exp(x)^2+x^5-x^4-x^3+9*x^2-4*x+4)/(x*log(x)^2+(2*x*exp(x)^2 +2*x^3-2*x^2)*log(x)+x*exp(x)^4+(2*x^3-2*x^2)*exp(x)^2+x^5-2*x^4+x^3),x, a lgorithm="maxima")
Output:
(x^3 - x^2 + x*e^(2*x) + x*log(x) - 4)/(x^2 - x + e^(2*x) + log(x)) + log( x)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {x^{3} + x^{2} \log \left (x\right ) - x^{2} + x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4}{x^{2} - x + e^{\left (2 \, x\right )} + \log \left (x\right )} \] Input:
integrate(((1+x)*log(x)^2+((2+2*x)*exp(x)^2+2*x^3-2*x)*log(x)+(1+x)*exp(x) ^4+(2*x^3+6*x)*exp(x)^2+x^5-x^4-x^3+9*x^2-4*x+4)/(x*log(x)^2+(2*x*exp(x)^2 +2*x^3-2*x^2)*log(x)+x*exp(x)^4+(2*x^3-2*x^2)*exp(x)^2+x^5-2*x^4+x^3),x, a lgorithm="giac")
Output:
(x^3 + x^2*log(x) - x^2 + x*e^(2*x) + e^(2*x)*log(x) + log(x)^2 - 4)/(x^2 - x + e^(2*x) + log(x))
Time = 2.71 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=x+\ln \left (x\right )-\frac {4}{{\mathrm {e}}^{2\,x}-x+\ln \left (x\right )+x^2} \] Input:
int((exp(2*x)*(6*x + 2*x^3) - 4*x + exp(4*x)*(x + 1) + log(x)^2*(x + 1) + log(x)*(exp(2*x)*(2*x + 2) - 2*x + 2*x^3) + 9*x^2 - x^3 - x^4 + x^5 + 4)/( x*exp(4*x) + x*log(x)^2 + log(x)*(2*x*exp(2*x) - 2*x^2 + 2*x^3) - exp(2*x) *(2*x^2 - 2*x^3) + x^3 - 2*x^4 + x^5),x)
Output:
x + log(x) - 4/(exp(2*x) - x + log(x) + x^2)
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {4-4 x+9 x^2-x^3-x^4+x^5+e^{4 x} (1+x)+e^{2 x} \left (6 x+2 x^3\right )+\left (-2 x+2 x^3+e^{2 x} (2+2 x)\right ) \log (x)+(1+x) \log ^2(x)}{e^{4 x} x+x^3-2 x^4+x^5+e^{2 x} \left (-2 x^2+2 x^3\right )+\left (2 e^{2 x} x-2 x^2+2 x^3\right ) \log (x)+x \log ^2(x)} \, dx=\frac {e^{2 x} \mathrm {log}\left (x \right )+e^{2 x} x +\mathrm {log}\left (x \right )^{2}+\mathrm {log}\left (x \right ) x^{2}+x^{3}-x^{2}-4}{e^{2 x}+\mathrm {log}\left (x \right )+x^{2}-x} \] Input:
int(((1+x)*log(x)^2+((2+2*x)*exp(x)^2+2*x^3-2*x)*log(x)+(1+x)*exp(x)^4+(2* x^3+6*x)*exp(x)^2+x^5-x^4-x^3+9*x^2-4*x+4)/(x*log(x)^2+(2*x*exp(x)^2+2*x^3 -2*x^2)*log(x)+x*exp(x)^4+(2*x^3-2*x^2)*exp(x)^2+x^5-2*x^4+x^3),x)
Output:
(e**(2*x)*log(x) + e**(2*x)*x + log(x)**2 + log(x)*x**2 + x**3 - x**2 - 4) /(e**(2*x) + log(x) + x**2 - x)