Integrand size = 100, antiderivative size = 21 \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=\log \left (e^{16-4 e^{-2 x}} x+(-2+\log (x))^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(21)=42\).
Time = 1.52 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.52 \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=-4 e^{-2 x}+\log \left (4 e^{4 e^{-2 x}}+e^{16} x-4 e^{4 e^{-2 x}} \log (x)+e^{4 e^{-2 x}} \log ^2(x)\right ) \]
Integrate[(E^(16 + 2*x)*x + 8*E^16*x^2 + E^(4/E^(2*x))*(-4*E^(2*x) + 2*E^( 2*x)*Log[x]))/(E^(16 + 2*x)*x^2 + E^(4/E^(2*x))*(4*E^(2*x)*x - 4*E^(2*x)*x *Log[x] + E^(2*x)*x*Log[x]^2)),x]
-4/E^(2*x) + Log[4*E^(4/E^(2*x)) + E^16*x - 4*E^(4/E^(2*x))*Log[x] + E^(4/ E^(2*x))*Log[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.
\(\displaystyle \int \frac {8 e^{16} x^2+e^{2 x+16} x+e^{4 e^{-2 x}} \left (2 e^{2 x} \log (x)-4 e^{2 x}\right )}{e^{2 x+16} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x+e^{2 x} x \log ^2(x)-4 e^{2 x} x \log (x)\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-2 x} \left (8 e^{16} x^2+e^{2 x+16} x+e^{4 e^{-2 x}} \left (2 e^{2 x} \log (x)-4 e^{2 x}\right )\right )}{x \left (4 e^{4 e^{-2 x}}+e^{16} x+e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8 e^{16-2 x} x}{4 e^{4 e^{-2 x}}+e^{16} x+e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)}+\frac {e^{16} x-4 e^{4 e^{-2 x}}+2 e^{4 e^{-2 x}} \log (x)}{x \left (4 e^{4 e^{-2 x}}+e^{16} x+e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \int \frac {e^{16-2 x} x}{e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)+4 e^{4 e^{-2 x}}+e^{16} x}dx-4 e^{16} \int \frac {1}{(\log (x)-2) \left (e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)+4 e^{4 e^{-2 x}}+e^{16} x\right )}dx+e^{16} \int \frac {\log (x)}{(\log (x)-2) \left (e^{4 e^{-2 x}} \log ^2(x)-4 e^{4 e^{-2 x}} \log (x)+4 e^{4 e^{-2 x}}+e^{16} x\right )}dx+2 \log (2-\log (x))\) |
Int[(E^(16 + 2*x)*x + 8*E^16*x^2 + E^(4/E^(2*x))*(-4*E^(2*x) + 2*E^(2*x)*L og[x]))/(E^(16 + 2*x)*x^2 + E^(4/E^(2*x))*(4*E^(2*x)*x - 4*E^(2*x)*x*Log[x ] + E^(2*x)*x*Log[x]^2)),x]
3.30.53.3.1 Defintions of rubi rules used
Time = 25.66 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90
method | result | size |
risch | \(2 \ln \left (\ln \left (x \right )-2\right )-4 \,{\mathrm e}^{-2 x}+\ln \left ({\mathrm e}^{4 \,{\mathrm e}^{-2 x}}+\frac {x \,{\mathrm e}^{16}}{\ln \left (x \right )^{2}-4 \ln \left (x \right )+4}\right )\) | \(40\) |
parallelrisch | \({\mathrm e}^{-2 x} \left (-4+\ln \left (\left (\ln \left (x \right )^{2} {\mathrm e}^{4 \,{\mathrm e}^{-2 x}}-4 \ln \left (x \right ) {\mathrm e}^{4 \,{\mathrm e}^{-2 x}}+x \,{\mathrm e}^{16}+4 \,{\mathrm e}^{4 \,{\mathrm e}^{-2 x}}\right ) {\mathrm e}^{-16}\right ) {\mathrm e}^{2 x}\right )\) | \(56\) |
int(((2*exp(x)^2*ln(x)-4*exp(x)^2)*exp(4/exp(x)^2)+x*exp(16)*exp(x)^2+8*x^ 2*exp(16))/((x*exp(x)^2*ln(x)^2-4*x*exp(x)^2*ln(x)+4*x*exp(x)^2)*exp(4/exp (x)^2)+x^2*exp(16)*exp(x)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.29 \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx={\left (e^{\left (2 \, x + 16\right )} \log \left (\frac {x e^{16} + {\left (\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 4\right )} e^{\left (4 \, e^{\left (-2 \, x\right )}\right )}}{\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 4}\right ) + 2 \, e^{\left (2 \, x + 16\right )} \log \left (\log \left (x\right ) - 2\right ) - 4 \, e^{16}\right )} e^{\left (-2 \, x - 16\right )} \]
integrate(((2*exp(x)^2*log(x)-4*exp(x)^2)*exp(4/exp(x)^2)+x*exp(16)*exp(x) ^2+8*x^2*exp(16))/((x*exp(x)^2*log(x)^2-4*x*exp(x)^2*log(x)+4*x*exp(x)^2)* exp(4/exp(x)^2)+x^2*exp(16)*exp(x)^2),x, algorithm=\
(e^(2*x + 16)*log((x*e^16 + (log(x)^2 - 4*log(x) + 4)*e^(4*e^(-2*x)))/(log (x)^2 - 4*log(x) + 4)) + 2*e^(2*x + 16)*log(log(x) - 2) - 4*e^16)*e^(-2*x - 16)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.49 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.95 \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=\log {\left (\frac {x e^{16}}{\log {\left (x \right )}^{2} - 4 \log {\left (x \right )} + 4} + e^{4 e^{- 2 x}} \right )} + 2 \log {\left (\log {\left (x \right )} - 2 \right )} - 4 e^{- 2 x} \]
integrate(((2*exp(x)**2*ln(x)-4*exp(x)**2)*exp(4/exp(x)**2)+x*exp(16)*exp( x)**2+8*x**2*exp(16))/((x*exp(x)**2*ln(x)**2-4*x*exp(x)**2*ln(x)+4*x*exp(x )**2)*exp(4/exp(x)**2)+x**2*exp(16)*exp(x)**2),x)
log(x*exp(16)/(log(x)**2 - 4*log(x) + 4) + exp(4*exp(-2*x))) + 2*log(log(x ) - 2) - 4*exp(-2*x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=-4 \, e^{\left (-2 \, x\right )} + \log \left (\frac {x e^{16} + {\left (\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 4\right )} e^{\left (4 \, e^{\left (-2 \, x\right )}\right )}}{\log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 4}\right ) + 2 \, \log \left (\log \left (x\right ) - 2\right ) \]
integrate(((2*exp(x)^2*log(x)-4*exp(x)^2)*exp(4/exp(x)^2)+x*exp(16)*exp(x) ^2+8*x^2*exp(16))/((x*exp(x)^2*log(x)^2-4*x*exp(x)^2*log(x)+4*x*exp(x)^2)* exp(4/exp(x)^2)+x^2*exp(16)*exp(x)^2),x, algorithm=\
-4*e^(-2*x) + log((x*e^16 + (log(x)^2 - 4*log(x) + 4)*e^(4*e^(-2*x)))/(log (x)^2 - 4*log(x) + 4)) + 2*log(log(x) - 2)
\[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=\int { \frac {8 \, x^{2} e^{16} + x e^{\left (2 \, x + 16\right )} + 2 \, {\left (e^{\left (2 \, x\right )} \log \left (x\right ) - 2 \, e^{\left (2 \, x\right )}\right )} e^{\left (4 \, e^{\left (-2 \, x\right )}\right )}}{x^{2} e^{\left (2 \, x + 16\right )} + {\left (x e^{\left (2 \, x\right )} \log \left (x\right )^{2} - 4 \, x e^{\left (2 \, x\right )} \log \left (x\right ) + 4 \, x e^{\left (2 \, x\right )}\right )} e^{\left (4 \, e^{\left (-2 \, x\right )}\right )}} \,d x } \]
integrate(((2*exp(x)^2*log(x)-4*exp(x)^2)*exp(4/exp(x)^2)+x*exp(16)*exp(x) ^2+8*x^2*exp(16))/((x*exp(x)^2*log(x)^2-4*x*exp(x)^2*log(x)+4*x*exp(x)^2)* exp(4/exp(x)^2)+x^2*exp(16)*exp(x)^2),x, algorithm=\
integrate((8*x^2*e^16 + x*e^(2*x + 16) + 2*(e^(2*x)*log(x) - 2*e^(2*x))*e^ (4*e^(-2*x)))/(x^2*e^(2*x + 16) + (x*e^(2*x)*log(x)^2 - 4*x*e^(2*x)*log(x) + 4*x*e^(2*x))*e^(4*e^(-2*x))), x)
Timed out. \[ \int \frac {e^{16+2 x} x+8 e^{16} x^2+e^{4 e^{-2 x}} \left (-4 e^{2 x}+2 e^{2 x} \log (x)\right )}{e^{16+2 x} x^2+e^{4 e^{-2 x}} \left (4 e^{2 x} x-4 e^{2 x} x \log (x)+e^{2 x} x \log ^2(x)\right )} \, dx=\int \frac {x\,{\mathrm {e}}^{2\,x+16}+8\,x^2\,{\mathrm {e}}^{16}-{\mathrm {e}}^{4\,{\mathrm {e}}^{-2\,x}}\,\left (4\,{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )\right )}{{\mathrm {e}}^{4\,{\mathrm {e}}^{-2\,x}}\,\left (x\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2-4\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+4\,x\,{\mathrm {e}}^{2\,x}\right )+x^2\,{\mathrm {e}}^{2\,x+16}} \,d x \]
int((8*x^2*exp(16) - exp(4*exp(-2*x))*(4*exp(2*x) - 2*exp(2*x)*log(x)) + x *exp(2*x)*exp(16))/(exp(4*exp(-2*x))*(4*x*exp(2*x) + x*exp(2*x)*log(x)^2 - 4*x*exp(2*x)*log(x)) + x^2*exp(2*x)*exp(16)),x)