Integrand size = 138, antiderivative size = 30 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {e \left (x+e^{4/x} x\right )}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \] Output:
1/2*(x+x*exp(4/x))/ln(4-ln(x)^2/x)*exp(1)
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {e \left (1+e^{4/x}\right ) x}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \] Input:
Integrate[((-2*E*x - 2*E^(1 + 4/x)*x)*Log[x] + (E*x + E^(1 + 4/x)*x)*Log[x ]^2 + (-4*E*x^2 + E^(1 + 4/x)*(16*x - 4*x^2) + (E^(1 + 4/x)*(-4 + x) + E*x )*Log[x]^2)*Log[(4*x - Log[x]^2)/x])/((-8*x^2 + 2*x*Log[x]^2)*Log[(4*x - L og[x]^2)/x]^2),x]
Output:
(E*(1 + E^(4/x))*x)/(2*Log[4 - Log[x]^2/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.
| \(\displaystyle \int \frac {\left (-4 e x^2+e^{\frac {4}{x}+1} \left (16 x-4 x^2\right )+\left (e^{\frac {4}{x}+1} (x-4)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )+\left (e^{\frac {4}{x}+1} x+e x\right ) \log ^2(x)+\left (-2 e^{\frac {4}{x}+1} x-2 e x\right ) \log (x)}{\left (2 x \log ^2(x)-8 x^2\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {\left (-4 e x^2+e^{\frac {4}{x}+1} \left (16 x-4 x^2\right )+\left (e^{\frac {4}{x}+1} (x-4)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )+\left (e^{\frac {4}{x}+1} x+e x\right ) \log ^2(x)+\left (-2 e^{\frac {4}{x}+1} x-2 e x\right ) \log (x)}{x \left (2 \log ^2(x)-8 x\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (-4 e x^2+e^{\frac {4}{x}+1} \left (16 x-4 x^2\right )+\left (e^{\frac {4}{x}+1} (x-4)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )-\left (\left (e^{\frac {4}{x}+1} x+e x\right ) \log ^2(x)\right )-\left (-2 e^{\frac {4}{x}+1} x-2 e x\right ) \log (x)}{2 x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {-\left (\left (e^{1+\frac {4}{x}} x+e x\right ) \log ^2(x)\right )+2 \left (e^{1+\frac {4}{x}} x+e x\right ) \log (x)+\left (4 e x^2+\left (e^{1+\frac {4}{x}} (4-x)-e x\right ) \log ^2(x)-4 e^{1+\frac {4}{x}} \left (4 x-x^2\right )\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {e \left (-\log \left (4-\frac {\log ^2(x)}{x}\right ) \log ^2(x)-\log ^2(x)+2 \log (x)+4 x \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {e^{1+\frac {4}{x}} \left (4 \log \left (4-\frac {\log ^2(x)}{x}\right ) x^2-\log ^2(x) x+2 \log (x) x-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right ) x-16 \log \left (4-\frac {\log ^2(x)}{x}\right ) x+4 \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-2 e \int \frac {\log (x)}{\left (\log ^2(x)-4 x\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}dx+e \int \frac {\log ^2(x)}{\left (\log ^2(x)-4 x\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}dx+e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )}dx+\frac {e^{\frac {4}{x}+1} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right )\) |
Input:
Int[((-2*E*x - 2*E^(1 + 4/x)*x)*Log[x] + (E*x + E^(1 + 4/x)*x)*Log[x]^2 + (-4*E*x^2 + E^(1 + 4/x)*(16*x - 4*x^2) + (E^(1 + 4/x)*(-4 + x) + E*x)*Log[ x]^2)*Log[(4*x - Log[x]^2)/x])/((-8*x^2 + 2*x*Log[x]^2)*Log[(4*x - Log[x]^ 2)/x]^2),x]
Output:
$Aborted
Time = 22.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {x \,{\mathrm e} \,{\mathrm e}^{\frac {4}{x}}+x \,{\mathrm e}}{2 \ln \left (-\frac {\ln \left (x \right )^{2}-4 x}{x}\right )}\) | \(34\) |
| risch | \(\frac {i \left ({\mathrm e}^{\frac {4}{x}}+1\right ) x \,{\mathrm e}}{\pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )}^{3}-\pi {\operatorname {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+4 i \ln \left (2\right )-2 i \ln \left (x \right )+2 i \ln \left (-\frac {\ln \left (x \right )^{2}}{4}+x \right )}\) | \(166\) |
Input:
int(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*ln(x)^2+(-4*x^2+16*x)*exp(1)*exp(4/ x)-4*x^2*exp(1))*ln((-ln(x)^2+4*x)/x)+(x*exp(1)*exp(4/x)+x*exp(1))*ln(x)^2 +(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*ln(x))/(2*x*ln(x)^2-8*x^2)/ln((-ln(x)^2 +4*x)/x)^2,x,method=_RETURNVERBOSE)
Output:
1/2*(x*exp(1)*exp(4/x)+x*exp(1))/ln(-(ln(x)^2-4*x)/x)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, \log \left (-\frac {\log \left (x\right )^{2} - 4 \, x}{x}\right )} \] Input:
integrate(((((-4+x)*exp(1)*exp(4/x)+exp(1)*x)*log(x)^2+(-4*x^2+16*x)*exp(1 )*exp(4/x)-4*x^2*exp(1))*log((-log(x)^2+4*x)/x)+(x*exp(1)*exp(4/x)+exp(1)* x)*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*exp(1)*x)*log(x))/(2*x*log(x)^2-8*x^2) /log((-log(x)^2+4*x)/x)^2,x, algorithm="fricas")
Output:
1/2*(x*e + x*e^((x + 4)/x))/log(-(log(x)^2 - 4*x)/x)
Time = 0.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {e x e^{\frac {4}{x}}}{2 \log {\left (\frac {4 x - \log {\left (x \right )}^{2}}{x} \right )}} + \frac {e x}{2 \log {\left (\frac {4 x - \log {\left (x \right )}^{2}}{x} \right )}} \] Input:
integrate(((((-4+x)*exp(1)*exp(4/x)+exp(1)*x)*ln(x)**2+(-4*x**2+16*x)*exp( 1)*exp(4/x)-4*x**2*exp(1))*ln((-ln(x)**2+4*x)/x)+(x*exp(1)*exp(4/x)+exp(1) *x)*ln(x)**2+(-2*x*exp(1)*exp(4/x)-2*exp(1)*x)*ln(x))/(2*x*ln(x)**2-8*x**2 )/ln((-ln(x)**2+4*x)/x)**2,x)
Output:
E*x*exp(4/x)/(2*log((4*x - log(x)**2)/x)) + E*x/(2*log((4*x - log(x)**2)/x ))
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {x e + x e^{\left (\frac {4}{x} + 1\right )}}{2 \, {\left (\log \left (-\log \left (x\right )^{2} + 4 \, x\right ) - \log \left (x\right )\right )}} \] Input:
integrate(((((-4+x)*exp(1)*exp(4/x)+exp(1)*x)*log(x)^2+(-4*x^2+16*x)*exp(1 )*exp(4/x)-4*x^2*exp(1))*log((-log(x)^2+4*x)/x)+(x*exp(1)*exp(4/x)+exp(1)* x)*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*exp(1)*x)*log(x))/(2*x*log(x)^2-8*x^2) /log((-log(x)^2+4*x)/x)^2,x, algorithm="maxima")
Output:
1/2*(x*e + x*e^(4/x + 1))/(log(-log(x)^2 + 4*x) - log(x))
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, {\left (\log \left (-\log \left (x\right )^{2} + 4 \, x\right ) - \log \left (x\right )\right )}} \] Input:
integrate(((((-4+x)*exp(1)*exp(4/x)+exp(1)*x)*log(x)^2+(-4*x^2+16*x)*exp(1 )*exp(4/x)-4*x^2*exp(1))*log((-log(x)^2+4*x)/x)+(x*exp(1)*exp(4/x)+exp(1)* x)*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*exp(1)*x)*log(x))/(2*x*log(x)^2-8*x^2) /log((-log(x)^2+4*x)/x)^2,x, algorithm="giac")
Output:
1/2*(x*e + x*e^((x + 4)/x))/(log(-log(x)^2 + 4*x) - log(x))
Timed out. \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left (x\,\mathrm {e}+x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )+\ln \left (\frac {4\,x-{\ln \left (x\right )}^2}{x}\right )\,\left ({\ln \left (x\right )}^2\,\left (x\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (x-4\right )\right )-4\,x^2\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (16\,x-4\,x^2\right )\right )-\ln \left (x\right )\,\left (2\,x\,\mathrm {e}+2\,x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )}{{\ln \left (\frac {4\,x-{\ln \left (x\right )}^2}{x}\right )}^2\,\left (2\,x\,{\ln \left (x\right )}^2-8\,x^2\right )} \,d x \] Input:
int((log(x)^2*(x*exp(1) + x*exp(1)*exp(4/x)) + log((4*x - log(x)^2)/x)*(lo g(x)^2*(x*exp(1) + exp(1)*exp(4/x)*(x - 4)) - 4*x^2*exp(1) + exp(1)*exp(4/ x)*(16*x - 4*x^2)) - log(x)*(2*x*exp(1) + 2*x*exp(1)*exp(4/x)))/(log((4*x - log(x)^2)/x)^2*(2*x*log(x)^2 - 8*x^2)),x)
Output:
int((log(x)^2*(x*exp(1) + x*exp(1)*exp(4/x)) + log((4*x - log(x)^2)/x)*(lo g(x)^2*(x*exp(1) + exp(1)*exp(4/x)*(x - 4)) - 4*x^2*exp(1) + exp(1)*exp(4/ x)*(16*x - 4*x^2)) - log(x)*(2*x*exp(1) + 2*x*exp(1)*exp(4/x)))/(log((4*x - log(x)^2)/x)^2*(2*x*log(x)^2 - 8*x^2)), x)
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx=\frac {e x \left (e^{\frac {4}{x}}+1\right )}{2 \,\mathrm {log}\left (\frac {-\mathrm {log}\left (x \right )^{2}+4 x}{x}\right )} \] Input:
int(((((-4+x)*exp(1)*exp(4/x)+exp(1)*x)*log(x)^2+(-4*x^2+16*x)*exp(1)*exp( 4/x)-4*x^2*exp(1))*log((-log(x)^2+4*x)/x)+(x*exp(1)*exp(4/x)+exp(1)*x)*log (x)^2+(-2*x*exp(1)*exp(4/x)-2*exp(1)*x)*log(x))/(2*x*log(x)^2-8*x^2)/log(( -log(x)^2+4*x)/x)^2,x)
Output:
(e*x*(e**(4/x) + 1))/(2*log(( - log(x)**2 + 4*x)/x))