3.57.8 \(\int \frac {2+8 x+8 x^2+e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} (2+8 x+8 x^2+e^{-e+e^x+\frac {e^{-e+e^x}}{2+4 x}} (-2+e^x (1+2 x)))}{2+8 x+8 x^2} \, dx\)

Optimal. Leaf size=28 \[ e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}+x \]

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Rubi [F]  time = 21.48, 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 {2+8 x+8 x^2+e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} \left (2+8 x+8 x^2+e^{-e+e^x+\frac {e^{-e+e^x}}{2+4 x}} \left (-2+e^x (1+2 x)\right )\right )}{2+8 x+8 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 8*x + 8*x^2 + E^(E^25 + E^(E^(-E + E^x)/(2 + 4*x)) + x)*(2 + 8*x + 8*x^2 + E^(-E + E^x + E^(-E + E^x)
/(2 + 4*x))*(-2 + E^x*(1 + 2*x))))/(2 + 8*x + 8*x^2),x]

[Out]

x + Defer[Int][E^(E^25 + E^(E^(-E + E^x)/(2 + 4*x)) + x), x] - Defer[Int][E^(E^x + E^(E^(-E + E^x)/(2 + 4*x))
- E*(1 - E^24) + x + E^(-E + E^x)/(2 + 4*x))/(1 + 2*x)^2, x] + Defer[Int][E^(E^x + E^(E^(-E + E^x)/(2 + 4*x))
- E*(1 - E^24) + 2*x + E^(-E + E^x)/(2 + 4*x))/(1 + 2*x), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+8 x+8 x^2+e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} \left (2+8 x+8 x^2+e^{-e+e^x+\frac {e^{-e+e^x}}{2+4 x}} \left (-2+e^x (1+2 x)\right )\right )}{2 (1+2 x)^2} \, dx\\ &=\frac {1}{2} \int \frac {2+8 x+8 x^2+e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} \left (2+8 x+8 x^2+e^{-e+e^x+\frac {e^{-e+e^x}}{2+4 x}} \left (-2+e^x (1+2 x)\right )\right )}{(1+2 x)^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {2}{(1+2 x)^2}+\frac {2 e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{(1+2 x)^2}+\frac {8 x}{(1+2 x)^2}+\frac {8 e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} x}{(1+2 x)^2}+\frac {8 x^2}{(1+2 x)^2}+\frac {8 e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} x^2}{(1+2 x)^2}+\frac {\exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+x+\frac {e^{-e+e^x}}{2+4 x}\right ) \left (-2+e^x+2 e^x x\right )}{(1+2 x)^2}\right ) \, dx\\ &=-\frac {1}{2 (1+2 x)}+\frac {1}{2} \int \frac {\exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+x+\frac {e^{-e+e^x}}{2+4 x}\right ) \left (-2+e^x+2 e^x x\right )}{(1+2 x)^2} \, dx+4 \int \frac {x}{(1+2 x)^2} \, dx+4 \int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} x}{(1+2 x)^2} \, dx+4 \int \frac {x^2}{(1+2 x)^2} \, dx+4 \int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} x^2}{(1+2 x)^2} \, dx+\int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{(1+2 x)^2} \, dx\\ &=-\frac {1}{2 (1+2 x)}+\frac {1}{2} \int \left (-\frac {2 \exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+x+\frac {e^{-e+e^x}}{2+4 x}\right )}{(1+2 x)^2}+\frac {\exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+2 x+\frac {e^{-e+e^x}}{2+4 x}\right )}{1+2 x}\right ) \, dx+4 \int \left (\frac {1}{4}+\frac {1}{4 (1+2 x)^2}-\frac {1}{2 (1+2 x)}\right ) \, dx+4 \int \left (-\frac {1}{2 (1+2 x)^2}+\frac {1}{2 (1+2 x)}\right ) \, dx+4 \int \left (\frac {1}{4} e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}+\frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{4 (1+2 x)^2}-\frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{2 (1+2 x)}\right ) \, dx+4 \int \left (-\frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{2 (1+2 x)^2}+\frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{2 (1+2 x)}\right ) \, dx+\int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{(1+2 x)^2} \, dx\\ &=x+\frac {1}{2} \int \frac {\exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+2 x+\frac {e^{-e+e^x}}{2+4 x}\right )}{1+2 x} \, dx-2 \int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{(1+2 x)^2} \, dx+\int e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x} \, dx+2 \int \frac {e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}}{(1+2 x)^2} \, dx-\int \frac {\exp \left (e^x+e^{\frac {e^{-e+e^x}}{2+4 x}}-e \left (1-e^{24}\right )+x+\frac {e^{-e+e^x}}{2+4 x}\right )}{(1+2 x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 28, normalized size = 1.00 \begin {gather*} e^{e^{25}+e^{\frac {e^{-e+e^x}}{2+4 x}}+x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 8*x + 8*x^2 + E^(E^25 + E^(E^(-E + E^x)/(2 + 4*x)) + x)*(2 + 8*x + 8*x^2 + E^(-E + E^x + E^(-E
+ E^x)/(2 + 4*x))*(-2 + E^x*(1 + 2*x))))/(2 + 8*x + 8*x^2),x]

[Out]

E^(E^25 + E^(E^(-E + E^x)/(2 + 4*x)) + x) + x

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fricas [B]  time = 0.90, size = 65, normalized size = 2.32 \begin {gather*} x + e^{\left ({\left ({\left (x + e^{25}\right )} e^{\left (-e + e^{x}\right )} + e^{\left (-\frac {2 \, {\left (2 \, x + 1\right )} e - 2 \, {\left (2 \, x + 1\right )} e^{x} - e^{\left (-e + e^{x}\right )}}{2 \, {\left (2 \, x + 1\right )}}\right )}\right )} e^{\left (e - e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x+1)*exp(x)-2)*exp(exp(x)-exp(1))*exp(exp(exp(x)-exp(1))/(4*x+2))+8*x^2+8*x+2)*exp(exp(exp(exp
(x)-exp(1))/(4*x+2))+exp(25)+x)+8*x^2+8*x+2)/(8*x^2+8*x+2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {8 \, x^{2} + {\left (8 \, x^{2} + {\left ({\left (2 \, x + 1\right )} e^{x} - 2\right )} e^{\left (\frac {e^{\left (-e + e^{x}\right )}}{2 \, {\left (2 \, x + 1\right )}} - e + e^{x}\right )} + 8 \, x + 2\right )} e^{\left (x + e^{25} + e^{\left (\frac {e^{\left (-e + e^{x}\right )}}{2 \, {\left (2 \, x + 1\right )}}\right )}\right )} + 8 \, x + 2}{2 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x+1)*exp(x)-2)*exp(exp(x)-exp(1))*exp(exp(exp(x)-exp(1))/(4*x+2))+8*x^2+8*x+2)*exp(exp(exp(exp
(x)-exp(1))/(4*x+2))+exp(25)+x)+8*x^2+8*x+2)/(8*x^2+8*x+2),x, algorithm="giac")

[Out]

integrate(1/2*(8*x^2 + (8*x^2 + ((2*x + 1)*e^x - 2)*e^(1/2*e^(-e + e^x)/(2*x + 1) - e + e^x) + 8*x + 2)*e^(x +
e^25 + e^(1/2*e^(-e + e^x)/(2*x + 1))) + 8*x + 2)/(4*x^2 + 4*x + 1), x)

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maple [A]  time = 0.32, size = 26, normalized size = 0.93

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{x}-{\mathrm e}}}{4 x +2}}+{\mathrm e}^{25}+x}+x\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((2*x+1)*exp(x)-2)*exp(exp(x)-exp(1))*exp(exp(exp(x)-exp(1))/(4*x+2))+8*x^2+8*x+2)*exp(exp(exp(exp(x)-ex
p(1))/(4*x+2))+exp(25)+x)+8*x^2+8*x+2)/(8*x^2+8*x+2),x,method=_RETURNVERBOSE)

[Out]

x+exp(exp(1/2*exp(exp(x)-exp(1))/(2*x+1))+exp(25)+x)

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maxima [A]  time = 0.61, size = 25, normalized size = 0.89 \begin {gather*} x + e^{\left (x + e^{25} + e^{\left (\frac {e^{\left (e^{x}\right )}}{2 \, {\left (2 \, x e^{e} + e^{e}\right )}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x+1)*exp(x)-2)*exp(exp(x)-exp(1))*exp(exp(exp(x)-exp(1))/(4*x+2))+8*x^2+8*x+2)*exp(exp(exp(exp
(x)-exp(1))/(4*x+2))+exp(25)+x)+8*x^2+8*x+2)/(8*x^2+8*x+2),x, algorithm="maxima")

[Out]

x + e^(x + e^25 + e^(1/2*e^(e^x)/(2*x*e^e + e^e)))

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mupad [B]  time = 4.08, size = 26, normalized size = 0.93 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{{\mathrm {e}}^x}}{4\,x+2}}}\,{\mathrm {e}}^{{\mathrm {e}}^{25}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + exp(x + exp(25) + exp(exp(exp(x) - exp(1))/(4*x + 2)))*(8*x + 8*x^2 + exp(exp(x) - exp(1))*exp(exp(
exp(x) - exp(1))/(4*x + 2))*(exp(x)*(2*x + 1) - 2) + 2) + 8*x^2 + 2)/(8*x + 8*x^2 + 2),x)

[Out]

x + exp(exp((exp(-exp(1))*exp(exp(x)))/(4*x + 2)))*exp(exp(25))*exp(x)

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sympy [A]  time = 6.40, size = 22, normalized size = 0.79 \begin {gather*} x + e^{x + e^{\frac {e^{e^{x} - e}}{4 x + 2}} + e^{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((2*x+1)*exp(x)-2)*exp(exp(x)-exp(1))*exp(exp(exp(x)-exp(1))/(4*x+2))+8*x**2+8*x+2)*exp(exp(exp(ex
p(x)-exp(1))/(4*x+2))+exp(25)+x)+8*x**2+8*x+2)/(8*x**2+8*x+2),x)

[Out]

x + exp(x + exp(exp(exp(x) - E)/(4*x + 2)) + exp(25))

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