Integrand size = 57, antiderivative size = 21 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=-4+x+x^{4 x \left (x+\frac {e^8 \log (3 x)}{x}\right )} \] Output:
exp(ln(x)*x*(4*exp(4)^2*ln(3*x)/x+4*x))+x-4
\[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=\int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx \] Input:
Integrate[(x + E^(4*x^2*Log[x] + 4*E^8*Log[x]*Log[3*x])*(4*x^2 + (4*E^8 + 8*x^2)*Log[x] + 4*E^8*Log[3*x]))/x,x]
Output:
Integrate[(x + E^(4*x^2*Log[x] + 4*E^8*Log[x]*Log[3*x])*(4*x^2 + (4*E^8 + 8*x^2)*Log[x] + 4*E^8*Log[3*x]))/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.
\(\displaystyle \int \frac {e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (8 x^2+4 e^8\right ) \log (x)+4 e^8 \log (3 x)\right )+x}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (4 \left (x^2+2 x^2 \log (x)+2 e^8 \log (x)+e^8 \log (3)\right ) x^{4 x^2+4 e^8 \log (3 x)-1}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 e^8 \log (3) \int x^{4 x^2+4 e^8 \log (3 x)-1}dx+4 \int x^{4 x^2+4 e^8 \log (3 x)+1}dx+8 e^8 \int x^{4 x^2+4 e^8 \log (3 x)-1} \log (x)dx+8 \int x^{4 x^2+4 e^8 \log (3 x)+1} \log (x)dx+x\) |
Input:
Int[(x + E^(4*x^2*Log[x] + 4*E^8*Log[x]*Log[3*x])*(4*x^2 + (4*E^8 + 8*x^2) *Log[x] + 4*E^8*Log[3*x]))/x,x]
Output:
$Aborted
Time = 1.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(x +{\mathrm e}^{4 \ln \left (x \right ) \left ({\mathrm e}^{8} \ln \left (3 x \right )+x^{2}\right )}\) | \(21\) |
risch | \(x +x^{4 \,{\mathrm e}^{8} \left (\ln \left (x \right )+\ln \left (3\right )\right )} x^{4 x^{2}}\) | \(22\) |
Input:
int(((4*exp(4)^2*ln(3*x)+(4*exp(4)^2+8*x^2)*ln(x)+4*x^2)*exp(4*exp(4)^2*ln (x)*ln(3*x)+4*x^2*ln(x))+x)/x,x,method=_RETURNVERBOSE)
Output:
x+exp(4*ln(x)*(exp(4)^2*ln(3*x)+x^2))
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x + e^{\left (4 \, e^{8} \log \left (x\right )^{2} + 4 \, {\left (x^{2} + e^{8} \log \left (3\right )\right )} \log \left (x\right )\right )} \] Input:
integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp (4)^2*log(x)*log(3*x)+4*x^2*log(x))+x)/x,x, algorithm="fricas")
Output:
x + e^(4*e^8*log(x)^2 + 4*(x^2 + e^8*log(3))*log(x))
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x + e^{4 x^{2} \log {\left (x \right )} + 4 \left (\log {\left (x \right )} + \log {\left (3 \right )}\right ) e^{8} \log {\left (x \right )}} \] Input:
integrate(((4*exp(4)**2*ln(3*x)+(4*exp(4)**2+8*x**2)*ln(x)+4*x**2)*exp(4*e xp(4)**2*ln(x)*ln(3*x)+4*x**2*ln(x))+x)/x,x)
Output:
x + exp(4*x**2*log(x) + 4*(log(x) + log(3))*exp(8)*log(x))
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x + e^{\left (4 \, x^{2} \log \left (x\right ) + 4 \, e^{8} \log \left (3\right ) \log \left (x\right ) + 4 \, e^{8} \log \left (x\right )^{2}\right )} \] Input:
integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp (4)^2*log(x)*log(3*x)+4*x^2*log(x))+x)/x,x, algorithm="maxima")
Output:
x + e^(4*x^2*log(x) + 4*e^8*log(3)*log(x) + 4*e^8*log(x)^2)
Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x + e^{\left (4 \, x^{2} \log \left (x\right ) + 4 \, e^{8} \log \left (3 \, x\right ) \log \left (x\right )\right )} \] Input:
integrate(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp (4)^2*log(x)*log(3*x)+4*x^2*log(x))+x)/x,x, algorithm="giac")
Output:
x + e^(4*x^2*log(x) + 4*e^8*log(3*x)*log(x))
Time = 2.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x+x^{4\,{\mathrm {e}}^8\,\ln \left (3\right )}\,x^{4\,x^2}\,{\mathrm {e}}^{4\,{\mathrm {e}}^8\,{\ln \left (x\right )}^2} \] Input:
int((x + exp(4*x^2*log(x) + 4*log(3*x)*exp(8)*log(x))*(4*log(3*x)*exp(8) + log(x)*(4*exp(8) + 8*x^2) + 4*x^2))/x,x)
Output:
x + x^(4*exp(8)*log(3))*x^(4*x^2)*exp(4*exp(8)*log(x)^2)
\[ \int \frac {x+e^{4 x^2 \log (x)+4 e^8 \log (x) \log (3 x)} \left (4 x^2+\left (4 e^8+8 x^2\right ) \log (x)+4 e^8 \log (3 x)\right )}{x} \, dx=x^{4 \,\mathrm {log}\left (x \right ) e^{8}+4 x^{2}} 3^{4 \,\mathrm {log}\left (x \right ) e^{8}}+4 \left (\int \frac {x^{4 \,\mathrm {log}\left (x \right ) e^{8}+4 x^{2}} 3^{4 \,\mathrm {log}\left (x \right ) e^{8}} \mathrm {log}\left (3 x \right )}{x}d x \right ) e^{8}-4 \left (\int \frac {x^{4 \,\mathrm {log}\left (x \right ) e^{8}+4 x^{2}} 3^{4 \,\mathrm {log}\left (x \right ) e^{8}} \mathrm {log}\left (x \right )}{x}d x \right ) e^{8}-4 \left (\int \frac {x^{4 \,\mathrm {log}\left (x \right ) e^{8}+4 x^{2}} 3^{4 \,\mathrm {log}\left (x \right ) e^{8}}}{x}d x \right ) \mathrm {log}\left (3\right ) e^{8}+x \] Input:
int(((4*exp(4)^2*log(3*x)+(4*exp(4)^2+8*x^2)*log(x)+4*x^2)*exp(4*exp(4)^2* log(x)*log(3*x)+4*x^2*log(x))+x)/x,x)
Output:
x**(4*log(x)*e**8 + 4*x**2)*3**(4*log(x)*e**8) + 4*int((x**(4*log(x)*e**8 + 4*x**2)*3**(4*log(x)*e**8)*log(3*x))/x,x)*e**8 - 4*int((x**(4*log(x)*e** 8 + 4*x**2)*3**(4*log(x)*e**8)*log(x))/x,x)*e**8 - 4*int((x**(4*log(x)*e** 8 + 4*x**2)*3**(4*log(x)*e**8))/x,x)*log(3)*e**8 + x