Integrand size = 221, antiderivative size = 26 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\log \left (x \left (-e^{e^x-x^2 \log (-1+x+\log (4+x))}+x\right )\right ) \]
Time = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\log (x)-x^2 \log (-1+x+\log (4+x))+\log \left (e^{e^x}-x (-1+x+\log (4+x))^{x^2}\right ) \]
Integrate[(8*x - 6*x^2 - 2*x^3 + (-8*x - 2*x^2)*Log[4 + x] + E^(E^x - x^2* Log[-1 + x + Log[4 + x]])*(-4 + 3*x + x^2 - 5*x^3 - x^4 + E^x*(-4*x + 3*x^ 2 + x^3) + (4 + x + E^x*(4*x + x^2))*Log[4 + x] + (8*x^2 - 6*x^3 - 2*x^4 + (-8*x^2 - 2*x^3)*Log[4 + x])*Log[-1 + x + Log[4 + x]]))/(4*x^2 - 3*x^3 - x^4 + (-4*x^2 - x^3)*Log[4 + x] + E^(E^x - x^2*Log[-1 + x + Log[4 + x]])*( -4*x + 3*x^2 + x^3 + (4*x + x^2)*Log[4 + 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.
\(\displaystyle \int \frac {-2 x^3-6 x^2+\left (-2 x^2-8 x\right ) \log (x+4)+e^{e^x-x^2 \log (x+\log (x+4)-1)} \left (-x^4-5 x^3+x^2+\left (e^x \left (x^2+4 x\right )+x+4\right ) \log (x+4)+e^x \left (x^3+3 x^2-4 x\right )+\left (-2 x^4-6 x^3+8 x^2+\left (-2 x^3-8 x^2\right ) \log (x+4)\right ) \log (x+\log (x+4)-1)+3 x-4\right )+8 x}{-x^4-3 x^3+4 x^2+\left (-x^3-4 x^2\right ) \log (x+4)+e^{e^x-x^2 \log (x+\log (x+4)-1)} \left (x^3+3 x^2+\left (x^2+4 x\right ) \log (x+4)-4 x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {(x+\log (x+4)-1)^{x^2-1} \left (-2 x^3-6 x^2+\left (-2 x^2-8 x\right ) \log (x+4)+e^{e^x-x^2 \log (x+\log (x+4)-1)} \left (-x^4-5 x^3+x^2+\left (e^x \left (x^2+4 x\right )+x+4\right ) \log (x+4)+e^x \left (x^3+3 x^2-4 x\right )+\left (-2 x^4-6 x^3+8 x^2+\left (-2 x^3-8 x^2\right ) \log (x+4)\right ) \log (x+\log (x+4)-1)+3 x-4\right )+8 x\right )}{x (x+4) \left (e^{e^x}-x (x+\log (x+4)-1)^{x^2}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{e^x} x^4+2 e^{e^x} x^4 \log (x+\log (x+4)-1)+5 e^{e^x} x^3-e^{x+e^x} x^3+6 e^{e^x} x^3 \log (x+\log (x+4)-1)+2 e^{e^x} x^3 \log (x+4) \log (x+\log (x+4)-1)-e^{e^x} x^2-3 e^{x+e^x} x^2+6 x^2 (x+\log (x+4)-1)^{x^2}-8 x (x+\log (x+4)-1)^{x^2}+2 x^2 \log (x+4) (x+\log (x+4)-1)^{x^2}+8 x \log (x+4) (x+\log (x+4)-1)^{x^2}-e^{x+e^x} x^2 \log (x+4)-8 e^{e^x} x^2 \log (x+\log (x+4)-1)+8 e^{e^x} x^2 \log (x+4) \log (x+\log (x+4)-1)+2 x^3 (x+\log (x+4)-1)^{x^2}+4 e^{e^x}-3 e^{e^x} x+4 e^{x+e^x} x-4 e^{e^x} \log (x+4)-e^{e^x} x \log (x+4)-4 e^{x+e^x} x \log (x+4)}{4 x (x+\log (x+4)-1) \left (x (x+\log (x+4)-1)^{x^2}-e^{e^x}\right )}-\frac {e^{e^x} x^4+2 e^{e^x} x^4 \log (x+\log (x+4)-1)+5 e^{e^x} x^3-e^{x+e^x} x^3+6 e^{e^x} x^3 \log (x+\log (x+4)-1)+2 e^{e^x} x^3 \log (x+4) \log (x+\log (x+4)-1)-e^{e^x} x^2-3 e^{x+e^x} x^2+6 x^2 (x+\log (x+4)-1)^{x^2}-8 x (x+\log (x+4)-1)^{x^2}+2 x^2 \log (x+4) (x+\log (x+4)-1)^{x^2}+8 x \log (x+4) (x+\log (x+4)-1)^{x^2}-e^{x+e^x} x^2 \log (x+4)-8 e^{e^x} x^2 \log (x+\log (x+4)-1)+8 e^{e^x} x^2 \log (x+4) \log (x+\log (x+4)-1)+2 x^3 (x+\log (x+4)-1)^{x^2}+4 e^{e^x}-3 e^{e^x} x+4 e^{x+e^x} x-4 e^{e^x} \log (x+4)-e^{e^x} x \log (x+4)-4 e^{x+e^x} x \log (x+4)}{4 (x+4) (x+\log (x+4)-1) \left (x (x+\log (x+4)-1)^{x^2}-e^{e^x}\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{e^x} x^4+5 e^{e^x} x^3-e^{x+e^x} x^3-e^{e^x} x^2-3 e^{x+e^x} x^2+6 x^2 (x+\log (x+4)-1)^{x^2}-8 x (x+\log (x+4)-1)^{x^2}+2 e^{e^x} x^2 \left (x^2+3 x-4\right ) \log (x+\log (x+4)-1)+(x+4) \log (x+4) \left (2 x (x+\log (x+4)-1)^{x^2}+2 e^{e^x} x^2 \log (x+\log (x+4)-1)-e^{e^x}-e^{x+e^x} x\right )+2 x^3 (x+\log (x+4)-1)^{x^2}+4 e^{e^x}-3 e^{e^x} x+4 e^{x+e^x} x}{x (x+4) (-x-\log (x+4)+1) \left (e^{e^x}-x (x+\log (x+4)-1)^{x^2}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2}{x}-\frac {e^{e^x} \left (x^4+2 x^4 \log (x+\log (x+4)-1)-e^x x^3+5 x^3+2 x^3 \log (x+4) \log (x+\log (x+4)-1)+6 x^3 \log (x+\log (x+4)-1)-3 e^x x^2+x^2-e^x x^2 \log (x+4)+8 x^2 \log (x+4) \log (x+\log (x+4)-1)-8 x^2 \log (x+\log (x+4)-1)+4 e^x x+3 x-4 e^x x \log (x+4)+x \log (x+4)+4 \log (x+4)-4\right )}{x (x+4) (x+\log (x+4)-1) \left (e^{e^x}-x (x+\log (x+4)-1)^{x^2}\right )}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {2}{x}-\frac {e^{e^x} \left (x^4+2 x^4 \log (x+\log (x+4)-1)-e^x x^3+5 x^3+2 x^3 \log (x+4) \log (x+\log (x+4)-1)+6 x^3 \log (x+\log (x+4)-1)-3 e^x x^2+x^2-e^x x^2 \log (x+4)+8 x^2 \log (x+4) \log (x+\log (x+4)-1)-8 x^2 \log (x+\log (x+4)-1)+4 e^x x+3 x-4 e^x x \log (x+4)+x \log (x+4)+4 \log (x+4)-4\right )}{x (x+4) (x+\log (x+4)-1) \left (e^{e^x}-x (x+\log (x+4)-1)^{x^2}\right )}\right )dx\) |
Int[(8*x - 6*x^2 - 2*x^3 + (-8*x - 2*x^2)*Log[4 + x] + E^(E^x - x^2*Log[-1 + x + Log[4 + x]])*(-4 + 3*x + x^2 - 5*x^3 - x^4 + E^x*(-4*x + 3*x^2 + x^ 3) + (4 + x + E^x*(4*x + x^2))*Log[4 + x] + (8*x^2 - 6*x^3 - 2*x^4 + (-8*x ^2 - 2*x^3)*Log[4 + x])*Log[-1 + x + Log[4 + x]]))/(4*x^2 - 3*x^3 - x^4 + (-4*x^2 - x^3)*Log[4 + x] + E^(E^x - x^2*Log[-1 + x + Log[4 + x]])*(-4*x + 3*x^2 + x^3 + (4*x + x^2)*Log[4 + x])),x]
3.29.12.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 242.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\ln \left (x \right )+\ln \left (\left (\ln \left (4+x \right )+x -1\right )^{-x^{2}} {\mathrm e}^{{\mathrm e}^{x}}-x \right )\) | \(26\) |
parallelrisch | \(\ln \left (x \right )+\ln \left (x -{\mathrm e}^{-x^{2} \ln \left (\ln \left (4+x \right )+x -1\right )+{\mathrm e}^{x}}\right )\) | \(26\) |
int(((((-2*x^3-8*x^2)*ln(4+x)-2*x^4-6*x^3+8*x^2)*ln(ln(4+x)+x-1)+((x^2+4*x )*exp(x)+4+x)*ln(4+x)+(x^3+3*x^2-4*x)*exp(x)-x^4-5*x^3+x^2+3*x-4)*exp(-x^2 *ln(ln(4+x)+x-1)+exp(x))+(-2*x^2-8*x)*ln(4+x)-2*x^3-6*x^2+8*x)/(((x^2+4*x) *ln(4+x)+x^3+3*x^2-4*x)*exp(-x^2*ln(ln(4+x)+x-1)+exp(x))+(-x^3-4*x^2)*ln(4 +x)-x^4-3*x^3+4*x^2),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\log \left (x\right ) + \log \left (-x + e^{\left (-x^{2} \log \left (x + \log \left (x + 4\right ) - 1\right ) + e^{x}\right )}\right ) \]
integrate(((((-2*x^3-8*x^2)*log(4+x)-2*x^4-6*x^3+8*x^2)*log(log(4+x)+x-1)+ ((x^2+4*x)*exp(x)+4+x)*log(4+x)+(x^3+3*x^2-4*x)*exp(x)-x^4-5*x^3+x^2+3*x-4 )*exp(-x^2*log(log(4+x)+x-1)+exp(x))+(-2*x^2-8*x)*log(4+x)-2*x^3-6*x^2+8*x )/(((x^2+4*x)*log(4+x)+x^3+3*x^2-4*x)*exp(-x^2*log(log(4+x)+x-1)+exp(x))+( -x^3-4*x^2)*log(4+x)-x^4-3*x^3+4*x^2),x, algorithm=\
Time = 2.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\log {\left (x \right )} + \log {\left (- x + e^{- x^{2} \log {\left (x + \log {\left (x + 4 \right )} - 1 \right )} + e^{x}} \right )} \]
integrate(((((-2*x**3-8*x**2)*ln(4+x)-2*x**4-6*x**3+8*x**2)*ln(ln(4+x)+x-1 )+((x**2+4*x)*exp(x)+4+x)*ln(4+x)+(x**3+3*x**2-4*x)*exp(x)-x**4-5*x**3+x** 2+3*x-4)*exp(-x**2*ln(ln(4+x)+x-1)+exp(x))+(-2*x**2-8*x)*ln(4+x)-2*x**3-6* x**2+8*x)/(((x**2+4*x)*ln(4+x)+x**3+3*x**2-4*x)*exp(-x**2*ln(ln(4+x)+x-1)+ exp(x))+(-x**3-4*x**2)*ln(4+x)-x**4-3*x**3+4*x**2),x)
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=-x^{2} \log \left (x + \log \left (x + 4\right ) - 1\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {{\left (x + \log \left (x + 4\right ) - 1\right )}^{\left (x^{2}\right )} x - e^{\left (e^{x}\right )}}{x}\right ) \]
integrate(((((-2*x^3-8*x^2)*log(4+x)-2*x^4-6*x^3+8*x^2)*log(log(4+x)+x-1)+ ((x^2+4*x)*exp(x)+4+x)*log(4+x)+(x^3+3*x^2-4*x)*exp(x)-x^4-5*x^3+x^2+3*x-4 )*exp(-x^2*log(log(4+x)+x-1)+exp(x))+(-2*x^2-8*x)*log(4+x)-2*x^3-6*x^2+8*x )/(((x^2+4*x)*log(4+x)+x^3+3*x^2-4*x)*exp(-x^2*log(log(4+x)+x-1)+exp(x))+( -x^3-4*x^2)*log(4+x)-x^4-3*x^3+4*x^2),x, algorithm=\
Time = 9.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\log \left (x - e^{\left (-x^{2} \log \left (x + \log \left (x + 4\right ) - 1\right ) + e^{x}\right )}\right ) + \log \left (x\right ) \]
integrate(((((-2*x^3-8*x^2)*log(4+x)-2*x^4-6*x^3+8*x^2)*log(log(4+x)+x-1)+ ((x^2+4*x)*exp(x)+4+x)*log(4+x)+(x^3+3*x^2-4*x)*exp(x)-x^4-5*x^3+x^2+3*x-4 )*exp(-x^2*log(log(4+x)+x-1)+exp(x))+(-2*x^2-8*x)*log(4+x)-2*x^3-6*x^2+8*x )/(((x^2+4*x)*log(4+x)+x^3+3*x^2-4*x)*exp(-x^2*log(log(4+x)+x-1)+exp(x))+( -x^3-4*x^2)*log(4+x)-x^4-3*x^3+4*x^2),x, algorithm=\
Time = 14.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {8 x-6 x^2-2 x^3+\left (-8 x-2 x^2\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4+3 x+x^2-5 x^3-x^4+e^x \left (-4 x+3 x^2+x^3\right )+\left (4+x+e^x \left (4 x+x^2\right )\right ) \log (4+x)+\left (8 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3\right ) \log (4+x)\right ) \log (-1+x+\log (4+x))\right )}{4 x^2-3 x^3-x^4+\left (-4 x^2-x^3\right ) \log (4+x)+e^{e^x-x^2 \log (-1+x+\log (4+x))} \left (-4 x+3 x^2+x^3+\left (4 x+x^2\right ) \log (4+x)\right )} \, dx=\ln \left (x\right )+\ln \left (\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{{\left (x+\ln \left (x+4\right )-1\right )}^{x^2}}-x\right ) \]
int((log(x + 4)*(8*x + 2*x^2) - 8*x + 6*x^2 + 2*x^3 - exp(exp(x) - x^2*log (x + log(x + 4) - 1))*(3*x - log(x + log(x + 4) - 1)*(log(x + 4)*(8*x^2 + 2*x^3) - 8*x^2 + 6*x^3 + 2*x^4) + exp(x)*(3*x^2 - 4*x + x^3) + log(x + 4)* (x + exp(x)*(4*x + x^2) + 4) + x^2 - 5*x^3 - x^4 - 4))/(log(x + 4)*(4*x^2 + x^3) - exp(exp(x) - x^2*log(x + log(x + 4) - 1))*(log(x + 4)*(4*x + x^2) - 4*x + 3*x^2 + x^3) - 4*x^2 + 3*x^3 + x^4),x)