3.30.25 \(\int \frac {-400+e^{\frac {2 (e^{1+x}+2 x+e^x x)}{2+e^x}} (8+8 e^x+2 e^{2 x}+4 e^{1+x})+32 x+e^{2 x} (-100+8 x)+e^x (-400+32 x)+e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} (-184+16 x+e^{2 x} (-46+4 x)+e^{1+x} (-100+8 x)+e^x (-184+16 x))}{4+4 e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=26 \[ \left (25-e^{\frac {e^{1+x}}{2+e^x}+x}-2 x\right )^2 \]

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Rubi [F]  time = 11.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

-200/(2 + E^x) + (8*(25 - 2*x))/(2 + E^x) - (25 - 2*x)^2 - 200*x + (16*x)/(2 + E^x) + 8*x^2 - 4*(25 - 2*x)*Log
[1 + E^x/2] - 8*x*Log[1 + E^x/2] + 100*Log[2 + E^x] - 46*Defer[Int][E^((E^(1 + x) + 2*x + E^x*x)/(2 + E^x)), x
] + 2*Defer[Int][E^((2*(E^(1 + x) + 2*x + E^x*x))/(2 + E^x)), x] + 200*Defer[Int][E^((2 + E^x*(1 + E) + 2*x +
E^x*x)/(2 + E^x))/(2 + E^x)^2, x] - 8*Defer[Int][E^(1 + (2*(E^(1 + x) + 2*x + E^x*x))/(2 + E^x))/(2 + E^x)^2,
x] - 100*Defer[Int][E^((2 + E^x*(1 + E) + 2*x + E^x*x)/(2 + E^x))/(2 + E^x), x] + 4*Defer[Int][E^(1 + (2*(E^(1
+ x) + 2*x + E^x*x))/(2 + E^x))/(2 + E^x), x] + 4*Defer[Int][E^((E^(1 + x) + 2*x + E^x*x)/(2 + E^x))*x, x] -
16*Defer[Int][(E^((2 + E^x*(1 + E) + 2*x + E^x*x)/(2 + E^x))*x)/(2 + E^x)^2, x] + 8*Defer[Int][(E^((2 + E^x*(1
+ E) + 2*x + E^x*x)/(2 + E^x))*x)/(2 + E^x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\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 )}{\left (2+e^x\right )^2} \, dx\\ &=\int \left (-\frac {400}{\left (2+e^x\right )^2}+\frac {2 e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (4+4 \left (1+\frac {e}{2}\right ) e^x+e^{2 x}\right )}{\left (2+e^x\right )^2}+\frac {32 x}{\left (2+e^x\right )^2}+\frac {16 e^x (-25+2 x)}{\left (2+e^x\right )^2}+\frac {4 e^{2 x} (-25+2 x)}{\left (2+e^x\right )^2}+\frac {2 e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-92-92 \left (1+\frac {25 e}{46}\right ) e^x-23 e^{2 x}+8 x+8 \left (1+\frac {e}{2}\right ) e^x x+2 e^{2 x} x\right )}{\left (2+e^x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \left (4+4 \left (1+\frac {e}{2}\right ) e^x+e^{2 x}\right )}{\left (2+e^x\right )^2} \, dx+2 \int \frac {e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \left (-92-92 \left (1+\frac {25 e}{46}\right ) e^x-23 e^{2 x}+8 x+8 \left (1+\frac {e}{2}\right ) e^x x+2 e^{2 x} x\right )}{\left (2+e^x\right )^2} \, dx+4 \int \frac {e^{2 x} (-25+2 x)}{\left (2+e^x\right )^2} \, dx+16 \int \frac {e^x (-25+2 x)}{\left (2+e^x\right )^2} \, dx+32 \int \frac {x}{\left (2+e^x\right )^2} \, dx-400 \int \frac {1}{\left (2+e^x\right )^2} \, dx\\ &=\frac {16 (25-2 x)}{2+e^x}+2 \int \left (e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}-\frac {4 e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2}+\frac {2 e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x}\right ) \, dx+2 \int \left (-23 e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}}+2 e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x-\frac {4 e^{1+\frac {e^{1+x}+2 x+e^x x}{2+e^x}} (-25+2 x)}{\left (2+e^x\right )^2}+\frac {2 e^{1+\frac {e^{1+x}+2 x+e^x x}{2+e^x}} (-25+2 x)}{2+e^x}\right ) \, dx+4 \int \left (-25+2 x+\frac {4 (-25+2 x)}{\left (2+e^x\right )^2}-\frac {4 (-25+2 x)}{2+e^x}\right ) \, dx-16 \int \frac {e^x x}{\left (2+e^x\right )^2} \, dx+16 \int \frac {x}{2+e^x} \, dx+32 \int \frac {1}{2+e^x} \, dx-400 \operatorname {Subst}\left (\int \frac {1}{x (2+x)^2} \, dx,x,e^x\right )\\ &=\frac {16 (25-2 x)}{2+e^x}-100 x+\frac {16 x}{2+e^x}+8 x^2+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx+4 \int \frac {e^{1+\frac {e^{1+x}+2 x+e^x x}{2+e^x}} (-25+2 x)}{2+e^x} \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx-8 \int \frac {e^x x}{2+e^x} \, dx-8 \int \frac {e^{1+\frac {e^{1+x}+2 x+e^x x}{2+e^x}} (-25+2 x)}{\left (2+e^x\right )^2} \, dx-16 \int \frac {1}{2+e^x} \, dx+16 \int \frac {-25+2 x}{\left (2+e^x\right )^2} \, dx-16 \int \frac {-25+2 x}{2+e^x} \, dx+32 \operatorname {Subst}\left (\int \frac {1}{x (2+x)} \, dx,x,e^x\right )-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx-400 \operatorname {Subst}\left (\int \left (\frac {1}{4 x}-\frac {1}{2 (2+x)^2}-\frac {1}{4 (2+x)}\right ) \, dx,x,e^x\right )\\ &=-\frac {200}{2+e^x}+\frac {16 (25-2 x)}{2+e^x}-2 (25-2 x)^2-200 x+\frac {16 x}{2+e^x}+8 x^2-8 x \log \left (1+\frac {e^x}{2}\right )+100 \log \left (2+e^x\right )+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx+4 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} (-25+2 x)}{2+e^x} \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx-8 \int \frac {e^x (-25+2 x)}{\left (2+e^x\right )^2} \, dx-8 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} (-25+2 x)}{\left (2+e^x\right )^2} \, dx+8 \int \frac {-25+2 x}{2+e^x} \, dx+8 \int \frac {e^x (-25+2 x)}{2+e^x} \, dx+8 \int \log \left (1+\frac {e^x}{2}\right ) \, dx+16 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-16 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )-16 \operatorname {Subst}\left (\int \frac {1}{x (2+x)} \, dx,x,e^x\right )-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx\\ &=-\frac {200}{2+e^x}+\frac {8 (25-2 x)}{2+e^x}-(25-2 x)^2-184 x+\frac {16 x}{2+e^x}+8 x^2-8 (25-2 x) \log \left (1+\frac {e^x}{2}\right )-8 x \log \left (1+\frac {e^x}{2}\right )+84 \log \left (2+e^x\right )+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx-4 \int \frac {e^x (-25+2 x)}{2+e^x} \, dx+4 \int \left (-\frac {25 e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{2+e^x}+\frac {2 e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{2+e^x}\right ) \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx-8 \int \left (-\frac {25 e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{\left (2+e^x\right )^2}+\frac {2 e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{\left (2+e^x\right )^2}\right ) \, dx-8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+8 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )+8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right )-16 \int \frac {1}{2+e^x} \, dx-16 \int \log \left (1+\frac {e^x}{2}\right ) \, dx-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx\\ &=-\frac {200}{2+e^x}+\frac {8 (25-2 x)}{2+e^x}-(25-2 x)^2-192 x+\frac {16 x}{2+e^x}+8 x^2-4 (25-2 x) \log \left (1+\frac {e^x}{2}\right )-8 x \log \left (1+\frac {e^x}{2}\right )+92 \log \left (2+e^x\right )-8 \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx+8 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{2+e^x} \, dx+8 \int \log \left (1+\frac {e^x}{2}\right ) \, dx-16 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{\left (2+e^x\right )^2} \, dx-16 \operatorname {Subst}\left (\int \frac {1}{x (2+x)} \, dx,x,e^x\right )-16 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right )-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx-100 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{2+e^x} \, dx+200 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{\left (2+e^x\right )^2} \, dx\\ &=-\frac {200}{2+e^x}+\frac {8 (25-2 x)}{2+e^x}-(25-2 x)^2-192 x+\frac {16 x}{2+e^x}+8 x^2-4 (25-2 x) \log \left (1+\frac {e^x}{2}\right )-8 x \log \left (1+\frac {e^x}{2}\right )+92 \log \left (2+e^x\right )+8 \text {Li}_2\left (-\frac {e^x}{2}\right )+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx+8 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{2+e^x} \, dx-8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+8 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )+8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right )-16 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{\left (2+e^x\right )^2} \, dx-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx-100 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{2+e^x} \, dx+200 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{\left (2+e^x\right )^2} \, dx\\ &=-\frac {200}{2+e^x}+\frac {8 (25-2 x)}{2+e^x}-(25-2 x)^2-200 x+\frac {16 x}{2+e^x}+8 x^2-4 (25-2 x) \log \left (1+\frac {e^x}{2}\right )-8 x \log \left (1+\frac {e^x}{2}\right )+100 \log \left (2+e^x\right )+2 \int e^{\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}} \, dx+4 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{2+e^x} \, dx+4 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} x \, dx-8 \int \frac {e^{1+\frac {2 \left (e^{1+x}+2 x+e^x x\right )}{2+e^x}}}{\left (2+e^x\right )^2} \, dx+8 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{2+e^x} \, dx-16 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}} x}{\left (2+e^x\right )^2} \, dx-46 \int e^{\frac {e^{1+x}+2 x+e^x x}{2+e^x}} \, dx-100 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{2+e^x} \, dx+200 \int \frac {e^{\frac {2+e^x (1+e)+2 x+e^x x}{2+e^x}}}{\left (2+e^x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.63, size = 51, normalized size = 1.96 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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))

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fricas [B]  time = 0.72, size = 73, normalized size = 2.81 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x+1)+2*exp(x)^2+8*exp(x)+8)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))^2+((8*x-100)*exp(x+1)+(4
*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x-184)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))+(8*x-100)*exp(x)^2+(32*x-4
00)*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="fricas")

[Out]

4*x^2 + 2*(2*x - 25)*e^((2*x*e + (x + e)*e^(x + 1))/(2*e + e^(x + 1))) - 100*x + e^(2*(2*x*e + (x + e)*e^(x +
1))/(2*e + e^(x + 1)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x+1)+2*exp(x)^2+8*exp(x)+8)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))^2+((8*x-100)*exp(x+1)+(4
*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x-184)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))+(8*x-100)*exp(x)^2+(32*x-4
00)*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="giac")

[Out]

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)

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maple [B]  time = 0.35, size = 57, normalized size = 2.19

method	result	size
risch	\(4 x^{2}+{\mathrm e}^{\frac {2 \,{\mathrm e}^{x +1}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}-100 x +\left (-50+4 x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x +1}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}\)	\(57\)
norman	\(\frac {{\mathrm e}^{x} {\mathrm e}^{\frac {2 \,{\mathrm e}^{x +1}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}-200 x +8 x^{2}+2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x +1}+2 \,{\mathrm e}^{x} x +4 x}{{\mathrm e}^{x}+2}}+8 x \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}-100 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}-50 \,{\mathrm e}^{x} {\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}+4 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}-100 \,{\mathrm e}^{\frac {{\mathrm e} \,{\mathrm e}^{x}+{\mathrm e}^{x} x +2 x}{{\mathrm e}^{x}+2}}}{{\mathrm e}^{x}+2}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(x+1)+2*exp(x)^2+8*exp(x)+8)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))^2+((8*x-100)*exp(x+1)+(4*x-46)
*exp(x)^2+(16*x-184)*exp(x)+16*x-184)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))+(8*x-100)*exp(x)^2+(32*x-400)*ex
p(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x,method=_RETURNVERBOSE)

[Out]

4*x^2+exp(2*(exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))-100*x+(-50+4*x)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))

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maxima [B]  time = 0.70, size = 119, normalized size = 4.58 \begin {gather*} -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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x+1)+2*exp(x)^2+8*exp(x)+8)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))^2+((8*x-100)*exp(x+1)+(4
*x-46)*exp(x)^2+(16*x-184)*exp(x)+16*x-184)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))+(8*x-100)*exp(x)^2+(32*x-4
00)*exp(x)+32*x-400)/(exp(x)^2+4*exp(x)+4),x, algorithm="maxima")

[Out]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
*exp(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)

[Out]

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
*exp(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)

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sympy [B]  time = 0.45, size = 60, normalized size = 2.31 \begin {gather*} 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x+1)+2*exp(x)**2+8*exp(x)+8)*exp((exp(x+1)+exp(x)*x+2*x)/(exp(x)+2))**2+((8*x-100)*exp(x+1)+
(4*x-46)*exp(x)**2+(16*x-184)*exp(x)+16*x-184)*exp((exp(x+1)+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)

[Out]

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))

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