Integrand size = 148, antiderivative size = 26 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=\left (25-e^{\frac {e^{1+x}}{2+e^x}+x}-2 x\right )^2 \] Output:
(25-exp(exp(1+x)/(2+exp(x))+x)-2*x)^2
Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=2 \left (\frac {1}{2} e^{2 \left (e-\frac {2 e}{2+e^x}+x\right )}+2 (-25+x) x+e^{e-\frac {2 e}{2+e^x}+x} (-25+2 x)\right ) \] Input:
Integrate[(-400 + E^((2*(E^(1 + x) + 2*x + E^x*x))/(2 + E^x))*(8 + 8*E^x + 2*E^(2*x) + 4*E^(1 + x)) + 32*x + E^(2*x)*(-100 + 8*x) + E^x*(-400 + 32*x ) + E^((E^(1 + x) + 2*x + E^x*x)/(2 + E^x))*(-184 + 16*x + E^(2*x)*(-46 + 4*x) + E^(1 + x)*(-100 + 8*x) + E^x*(-184 + 16*x)))/(4 + 4*E^x + E^(2*x)), x]
Output:
2*(E^(2*(E - (2*E)/(2 + E^x) + x))/2 + 2*(-25 + x)*x + E^(E - (2*E)/(2 + E ^x) + x)*(-25 + 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.
| \(\displaystyle \int \frac {e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{e^x+2}} \left (8 e^x+2 e^{2 x}+4 e^{x+1}+8\right )+32 x+e^{2 x} (8 x-100)+e^x (32 x-400)+e^{\frac {e^x x+2 x+e^{x+1}}{e^x+2}} \left (16 x+e^{2 x} (4 x-46)+e^{x+1} (8 x-100)+e^x (16 x-184)-184\right )-400}{4 e^x+e^{2 x}+4} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{e^x+2}} \left (8 e^x+2 e^{2 x}+4 e^{x+1}+8\right )+32 x+e^{2 x} (8 x-100)+e^x (32 x-400)+e^{\frac {e^x x+2 x+e^{x+1}}{e^x+2}} \left (16 x+e^{2 x} (4 x-46)+e^{x+1} (8 x-100)+e^x (16 x-184)-184\right )-400}{\left (e^x+2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{e^x+2}} \left (4 \left (1+\frac {e}{2}\right ) e^x+e^{2 x}+4\right )}{\left (e^x+2\right )^2}+\frac {32 x}{\left (e^x+2\right )^2}+\frac {16 e^x (2 x-25)}{\left (e^x+2\right )^2}+\frac {4 e^{2 x} (2 x-25)}{\left (e^x+2\right )^2}+\frac {2 e^{\frac {e^x x+2 x+e^{x+1}}{e^x+2}} \left (8 \left (1+\frac {e}{2}\right ) e^x x+2 e^{2 x} x+8 x-92 \left (1+\frac {25 e}{46}\right ) e^x-23 e^{2 x}-92\right )}{\left (e^x+2\right )^2}-\frac {400}{\left (e^x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -46 \int e^{\frac {e^x x+2 x+e^{x+1}}{2+e^x}}dx+2 \int e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{2+e^x}}dx+200 \int \frac {e^{\frac {e^x x+2 x+e^x (1+e)+2}{2+e^x}}}{\left (2+e^x\right )^2}dx-8 \int \frac {e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{2+e^x}+1}}{\left (2+e^x\right )^2}dx-100 \int \frac {e^{\frac {e^x x+2 x+e^x (1+e)+2}{2+e^x}}}{2+e^x}dx+4 \int \frac {e^{\frac {2 \left (e^x x+2 x+e^{x+1}\right )}{2+e^x}+1}}{2+e^x}dx+4 \int e^{\frac {e^x x+2 x+e^{x+1}}{2+e^x}} xdx-16 \int \frac {e^{\frac {e^x x+2 x+e^x (1+e)+2}{2+e^x}} x}{\left (2+e^x\right )^2}dx+8 \int \frac {e^{\frac {e^x x+2 x+e^x (1+e)+2}{2+e^x}} x}{2+e^x}dx+8 x^2-(25-2 x)^2+\frac {8 (25-2 x)}{e^x+2}+\frac {16 x}{e^x+2}-200 x-\frac {200}{e^x+2}-4 (25-2 x) \log \left (\frac {e^x}{2}+1\right )-8 x \log \left (\frac {e^x}{2}+1\right )+100 \log \left (e^x+2\right )\) |
Input:
Int[(-400 + E^((2*(E^(1 + x) + 2*x + E^x*x))/(2 + E^x))*(8 + 8*E^x + 2*E^( 2*x) + 4*E^(1 + x)) + 32*x + E^(2*x)*(-100 + 8*x) + E^x*(-400 + 32*x) + E^ ((E^(1 + x) + 2*x + E^x*x)/(2 + E^x))*(-184 + 16*x + E^(2*x)*(-46 + 4*x) + E^(1 + x)*(-100 + 8*x) + E^x*(-184 + 16*x)))/(4 + 4*E^x + E^(2*x)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(23)=46\).
Time = 1.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19
| method | result | size |
| risch | \(4 x^{2}+{\mathrm e}^{\frac {2 \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}-100 x +\left (-50+4 x \right ) {\mathrm e}^{\frac {{\mathrm e}^{1+x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}\) | \(57\) |
| parallelrisch | \(4 \,{\mathrm e}^{\frac {{\mathrm e}^{1+x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}} x +4 x^{2}-50 \,{\mathrm e}^{\frac {{\mathrm e}^{1+x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}+{\mathrm e}^{\frac {2 \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}-100 x\) | \(77\) |
| norman | \(\frac {{\mathrm e}^{\frac {2 \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}} {\mathrm e}^{x}-200 x +8 x^{2}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}+8 x \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}-100 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}-50 \,{\mathrm e}^{x} {\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}+4 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}-100 \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}}{{\mathrm e}^{x}+2}\) | \(178\) |
Input:
int(((4*exp(1+x)+2*exp(x)^2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x)/(exp(x )+2))^2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x-184)* exp((exp(1+x)+exp(x)*x+2*x)/(exp(x)+2))+(8*x-100)*exp(x)^2+(32*x-400)*exp( x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x,method=_RETURNVERBOSE)
Output:
4*x^2+exp(2*(exp(1+x)+exp(x)*x+2*x)/(exp(x)+2))-100*x+(-50+4*x)*exp((exp(1 +x)+exp(x)*x+2*x)/(exp(x)+2))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=4 \, x^{2} + 2 \, {\left (2 \, x - 25\right )} e^{\left (\frac {2 \, x e + {\left (x + e\right )} e^{\left (x + 1\right )}}{2 \, e + e^{\left (x + 1\right )}}\right )} - 100 \, x + e^{\left (\frac {2 \, {\left (2 \, x e + {\left (x + e\right )} e^{\left (x + 1\right )}\right )}}{2 \, e + e^{\left (x + 1\right )}}\right )} \] Input:
integrate(((4*exp(1+x)+2*exp(x)^2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x)/ (2+exp(x)))^2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x -184)*exp((exp(1+x)+exp(x)*x+2*x)/(2+exp(x)))+(8*x-100)*exp(x)^2+(32*x-400 )*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="fricas")
Output:
4*x^2 + 2*(2*x - 25)*e^((2*x*e + (x + e)*e^(x + 1))/(2*e + e^(x + 1))) - 1 00*x + e^(2*(2*x*e + (x + e)*e^(x + 1))/(2*e + e^(x + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=4 x^{2} - 100 x + \left (4 x - 50\right ) e^{\frac {x e^{x} + 2 x + e e^{x}}{e^{x} + 2}} + e^{\frac {2 \left (x e^{x} + 2 x + e e^{x}\right )}{e^{x} + 2}} \] Input:
integrate(((4*exp(1+x)+2*exp(x)**2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x) /(2+exp(x)))**2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)**2+(16*x-184)*exp(x)+1 6*x-184)*exp((exp(1+x)+exp(x)*x+2*x)/(2+exp(x)))+(8*x-100)*exp(x)**2+(32*x -400)*exp(x)+32*x-400)/(exp(x)**2+4*exp(x)+4),x)
Output:
4*x**2 - 100*x + (4*x - 50)*exp((x*exp(x) + 2*x + E*exp(x))/(exp(x) + 2)) + exp(2*(x*exp(x) + 2*x + E*exp(x))/(exp(x) + 2))
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.58 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=-100 \, x + \frac {4 \, x^{2} e^{x} + 8 \, x^{2} + {\left (e^{x} + 2\right )} e^{\left (\frac {2 \, x e^{x}}{e^{x} + 2} + \frac {4 \, x}{e^{x} + 2} + \frac {2 \, e^{\left (x + 1\right )}}{e^{x} + 2}\right )} + 2 \, {\left ({\left (2 \, x - 25\right )} e^{x} + 4 \, x - 50\right )} e^{\left (\frac {x e^{x}}{e^{x} + 2} + \frac {2 \, x}{e^{x} + 2} + \frac {e^{\left (x + 1\right )}}{e^{x} + 2}\right )} + 200}{e^{x} + 2} - \frac {200}{e^{x} + 2} \] Input:
integrate(((4*exp(1+x)+2*exp(x)^2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x)/ (2+exp(x)))^2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x -184)*exp((exp(1+x)+exp(x)*x+2*x)/(2+exp(x)))+(8*x-100)*exp(x)^2+(32*x-400 )*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="maxima")
Output:
-100*x + (4*x^2*e^x + 8*x^2 + (e^x + 2)*e^(2*x*e^x/(e^x + 2) + 4*x/(e^x + 2) + 2*e^(x + 1)/(e^x + 2)) + 2*((2*x - 25)*e^x + 4*x - 50)*e^(x*e^x/(e^x + 2) + 2*x/(e^x + 2) + e^(x + 1)/(e^x + 2)) + 200)/(e^x + 2) - 200/(e^x + 2)
\[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=\int { \frac {2 \, {\left (2 \, {\left (2 \, x - 25\right )} e^{\left (2 \, x\right )} + 8 \, {\left (2 \, x - 25\right )} e^{x} + {\left (e^{\left (2 \, x\right )} + 2 \, e^{\left (x + 1\right )} + 4 \, e^{x} + 4\right )} e^{\left (\frac {2 \, {\left (x e^{x} + 2 \, x + e^{\left (x + 1\right )}\right )}}{e^{x} + 2}\right )} + {\left ({\left (2 \, x - 23\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x - 25\right )} e^{\left (x + 1\right )} + 4 \, {\left (2 \, x - 23\right )} e^{x} + 8 \, x - 92\right )} e^{\left (\frac {x e^{x} + 2 \, x + e^{\left (x + 1\right )}}{e^{x} + 2}\right )} + 16 \, x - 200\right )}}{e^{\left (2 \, x\right )} + 4 \, e^{x} + 4} \,d x } \] Input:
integrate(((4*exp(1+x)+2*exp(x)^2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x)/ (2+exp(x)))^2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x -184)*exp((exp(1+x)+exp(x)*x+2*x)/(2+exp(x)))+(8*x-100)*exp(x)^2+(32*x-400 )*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="giac")
Output:
integrate(2*(2*(2*x - 25)*e^(2*x) + 8*(2*x - 25)*e^x + (e^(2*x) + 2*e^(x + 1) + 4*e^x + 4)*e^(2*(x*e^x + 2*x + e^(x + 1))/(e^x + 2)) + ((2*x - 23)*e ^(2*x) + 2*(2*x - 25)*e^(x + 1) + 4*(2*x - 23)*e^x + 8*x - 92)*e^((x*e^x + 2*x + e^(x + 1))/(e^x + 2)) + 16*x - 200)/(e^(2*x) + 4*e^x + 4), x)
Timed out. \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=\int \frac {32\,x+{\mathrm {e}}^{\frac {2\,x+{\mathrm {e}}^{x+1}+x\,{\mathrm {e}}^x}{{\mathrm {e}}^x+2}}\,\left (16\,x+{\mathrm {e}}^x\,\left (16\,x-184\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x-46\right )+{\mathrm {e}}^{x+1}\,\left (8\,x-100\right )-184\right )+{\mathrm {e}}^x\,\left (32\,x-400\right )+{\mathrm {e}}^{\frac {2\,\left (2\,x+{\mathrm {e}}^{x+1}+x\,{\mathrm {e}}^x\right )}{{\mathrm {e}}^x+2}}\,\left (2\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{x+1}+8\,{\mathrm {e}}^x+8\right )+{\mathrm {e}}^{2\,x}\,\left (8\,x-100\right )-400}{{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^x+4} \,d x \] Input:
int((32*x + exp((2*x + exp(x + 1) + x*exp(x))/(exp(x) + 2))*(16*x + exp(x) *(16*x - 184) + exp(2*x)*(4*x - 46) + exp(x + 1)*(8*x - 100) - 184) + exp( x)*(32*x - 400) + exp((2*(2*x + exp(x + 1) + x*exp(x)))/(exp(x) + 2))*(2*e xp(2*x) + 4*exp(x + 1) + 8*exp(x) + 8) + exp(2*x)*(8*x - 100) - 400)/(exp( 2*x) + 4*exp(x) + 4),x)
Output:
int((32*x + exp((2*x + exp(x + 1) + x*exp(x))/(exp(x) + 2))*(16*x + exp(x) *(16*x - 184) + exp(2*x)*(4*x - 46) + exp(x + 1)*(8*x - 100) - 184) + exp( x)*(32*x - 400) + exp((2*(2*x + exp(x + 1) + x*exp(x)))/(exp(x) + 2))*(2*e xp(2*x) + 4*exp(x + 1) + 8*exp(x) + 8) + exp(2*x)*(8*x - 100) - 400)/(exp( 2*x) + 4*exp(x) + 4), x)
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.46 \[ \int \frac {-400+e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (8+8 e^x+2 e^{2 x}+4 e^{1+x}\right )+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x)\right )}{4+4 e^x+e^{2 x}} \, dx=\frac {4 e^{\frac {4 e}{e^{x}+2}} x^{2}-100 e^{\frac {4 e}{e^{x}+2}} x +4 e^{\frac {e^{x} e +e^{x} x +4 e +2 x}{e^{x}+2}} x -50 e^{\frac {e^{x} e +e^{x} x +4 e +2 x}{e^{x}+2}}+e^{2 e +2 x}}{e^{\frac {4 e}{e^{x}+2}}} \] Input:
int(((4*exp(1+x)+2*exp(x)^2+8*exp(x)+8)*exp((exp(1+x)+exp(x)*x+2*x)/(2+exp (x)))^2+((8*x-100)*exp(1+x)+(4*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x-184)* exp((exp(1+x)+exp(x)*x+2*x)/(2+exp(x)))+(8*x-100)*exp(x)^2+(32*x-400)*exp( x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x)
Output:
(4*e**((4*e)/(e**x + 2))*x**2 - 100*e**((4*e)/(e**x + 2))*x + 4*e**((e**x* e + e**x*x + 4*e + 2*x)/(e**x + 2))*x - 50*e**((e**x*e + e**x*x + 4*e + 2* x)/(e**x + 2)) + e**(2*e + 2*x))/e**((4*e)/(e**x + 2))