3.58.59 \(\int \frac {e^{2+e^{-e^{2 x}+x}+x} (12+84 x-28 x^2+160 x^3+320 x^4+e^{-e^{2 x}+x} (-12 x-28 x^2+160 x^3+320 x^4+e^{2 x} (24 x+56 x^2-320 x^3-640 x^4)))}{x^2+8 x^3+16 x^4} \, dx\)

Optimal. Leaf size=32 \[ 4 e^{2+e^{-e^{2 x}+x}+x} \left (5-\frac {3}{x (1+4 x)}\right ) \]

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Rubi [F]  time = 8.05, 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 {e^{2+e^{-e^{2 x}+x}+x} \left (12+84 x-28 x^2+160 x^3+320 x^4+e^{-e^{2 x}+x} \left (-12 x-28 x^2+160 x^3+320 x^4+e^{2 x} \left (24 x+56 x^2-320 x^3-640 x^4\right )\right )\right )}{x^2+8 x^3+16 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2 + E^(-E^(2*x) + x) + x)*(12 + 84*x - 28*x^2 + 160*x^3 + 320*x^4 + E^(-E^(2*x) + x)*(-12*x - 28*x^2 +
160*x^3 + 320*x^4 + E^(2*x)*(24*x + 56*x^2 - 320*x^3 - 640*x^4))))/(x^2 + 8*x^3 + 16*x^4),x]

[Out]

12*Defer[Int][E^(2 + E^(-E^(2*x) + x) + x)/x^2, x] - 12*Defer[Int][E^(2 + E^(-E^(2*x) + x) + x)/x, x] - 12*Def
er[Int][E^(2 - E^(2*x) + E^(-E^(2*x) + x) + 2*x)/x, x] + 24*Defer[Int][E^(2 - E^(2*x) + E^(-E^(2*x) + x) + 4*x
)/x, x] - 192*Defer[Int][E^(2 + E^(-E^(2*x) + x) + x)/(1 + 4*x)^2, x] + 48*Defer[Int][E^(2 + E^(-E^(2*x) + x)
+ x)/(1 + 4*x), x] + 48*Defer[Int][E^(2 - E^(2*x) + E^(-E^(2*x) + x) + 2*x)/(1 + 4*x), x] - 96*Defer[Int][E^(2
- E^(2*x) + E^(-E^(2*x) + x) + 4*x)/(1 + 4*x), x] + 20*Defer[Subst][Defer[Int][E^(2 + x/E^x^2), x], x, E^x] +
20*Defer[Subst][Defer[Int][E^(2 + x/E^x^2 - x^2)*x, x], x, E^x] - 40*Defer[Subst][Defer[Int][E^(2 + x/E^x^2 -
x^2)*x^3, x], x, E^x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2+e^{-e^{2 x}+x}+x} \left (12+84 x-28 x^2+160 x^3+320 x^4+e^{-e^{2 x}+x} \left (-12 x-28 x^2+160 x^3+320 x^4+e^{2 x} \left (24 x+56 x^2-320 x^3-640 x^4\right )\right )\right )}{x^2 \left (1+8 x+16 x^2\right )} \, dx\\ &=\int \frac {e^{2+e^{-e^{2 x}+x}+x} \left (12+84 x-28 x^2+160 x^3+320 x^4+e^{-e^{2 x}+x} \left (-12 x-28 x^2+160 x^3+320 x^4+e^{2 x} \left (24 x+56 x^2-320 x^3-640 x^4\right )\right )\right )}{x^2 (1+4 x)^2} \, dx\\ &=\int \left (-\frac {28 e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2}+\frac {12 e^{2+e^{-e^{2 x}+x}+x}}{x^2 (1+4 x)^2}+\frac {84 e^{2+e^{-e^{2 x}+x}+x}}{x (1+4 x)^2}+\frac {160 e^{2+e^{-e^{2 x}+x}+x} x}{(1+4 x)^2}+\frac {320 e^{2+e^{-e^{2 x}+x}+x} x^2}{(1+4 x)^2}+\frac {4 e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x} \left (-3+5 x+20 x^2\right )}{x (1+4 x)}-\frac {8 e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x} \left (-3+5 x+20 x^2\right )}{x (1+4 x)}\right ) \, dx\\ &=4 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x} \left (-3+5 x+20 x^2\right )}{x (1+4 x)} \, dx-8 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x} \left (-3+5 x+20 x^2\right )}{x (1+4 x)} \, dx+12 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x^2 (1+4 x)^2} \, dx-28 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx+84 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x (1+4 x)^2} \, dx+160 \int \frac {e^{2+e^{-e^{2 x}+x}+x} x}{(1+4 x)^2} \, dx+320 \int \frac {e^{2+e^{-e^{2 x}+x}+x} x^2}{(1+4 x)^2} \, dx\\ &=4 \int \left (5 e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}-\frac {3 e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{x}+\frac {12 e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{1+4 x}\right ) \, dx-8 \int \left (5 e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}-\frac {3 e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{x}+\frac {12 e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{1+4 x}\right ) \, dx+12 \int \left (\frac {e^{2+e^{-e^{2 x}+x}+x}}{x^2}-\frac {8 e^{2+e^{-e^{2 x}+x}+x}}{x}+\frac {16 e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2}+\frac {32 e^{2+e^{-e^{2 x}+x}+x}}{1+4 x}\right ) \, dx-28 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx+84 \int \left (\frac {e^{2+e^{-e^{2 x}+x}+x}}{x}-\frac {4 e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2}-\frac {4 e^{2+e^{-e^{2 x}+x}+x}}{1+4 x}\right ) \, dx+160 \int \left (-\frac {e^{2+e^{-e^{2 x}+x}+x}}{4 (1+4 x)^2}+\frac {e^{2+e^{-e^{2 x}+x}+x}}{4 (1+4 x)}\right ) \, dx+320 \int \left (\frac {1}{16} e^{2+e^{-e^{2 x}+x}+x}+\frac {e^{2+e^{-e^{2 x}+x}+x}}{16 (1+4 x)^2}-\frac {e^{2+e^{-e^{2 x}+x}+x}}{8 (1+4 x)}\right ) \, dx\\ &=12 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x^2} \, dx-12 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{x} \, dx+20 \int e^{2+e^{-e^{2 x}+x}+x} \, dx+20 \int e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x} \, dx+20 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx+24 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{x} \, dx-28 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-40 \int e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x} \, dx-40 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx+48 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{1+4 x} \, dx+84 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x} \, dx-96 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x} \, dx-96 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{1+4 x} \, dx+192 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-336 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-336 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{1+4 x} \, dx+384 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{1+4 x} \, dx\\ &=12 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x^2} \, dx-12 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{x} \, dx+20 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx+20 \operatorname {Subst}\left (\int e^{2+e^{-x^2} x} \, dx,x,e^x\right )+20 \operatorname {Subst}\left (\int e^{2+e^{-x^2} x-x^2} x \, dx,x,e^x\right )+24 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{x} \, dx-28 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-40 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-40 \operatorname {Subst}\left (\int e^{2+e^{-x^2} x-x^2} x^3 \, dx,x,e^x\right )+48 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+2 x}}{1+4 x} \, dx+84 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x} \, dx-96 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{x} \, dx-96 \int \frac {e^{2-e^{2 x}+e^{-e^{2 x}+x}+4 x}}{1+4 x} \, dx+192 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-336 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{(1+4 x)^2} \, dx-336 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{1+4 x} \, dx+384 \int \frac {e^{2+e^{-e^{2 x}+x}+x}}{1+4 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 38, normalized size = 1.19 \begin {gather*} -\frac {4 e^{2+e^{-e^{2 x}+x}+x} \left (3-5 x-20 x^2\right )}{x (1+4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 + E^(-E^(2*x) + x) + x)*(12 + 84*x - 28*x^2 + 160*x^3 + 320*x^4 + E^(-E^(2*x) + x)*(-12*x - 28
*x^2 + 160*x^3 + 320*x^4 + E^(2*x)*(24*x + 56*x^2 - 320*x^3 - 640*x^4))))/(x^2 + 8*x^3 + 16*x^4),x]

[Out]

(-4*E^(2 + E^(-E^(2*x) + x) + x)*(3 - 5*x - 20*x^2))/(x*(1 + 4*x))

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fricas [A]  time = 0.49, size = 34, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (20 \, x^{2} + 5 \, x - 3\right )} e^{\left (x + e^{\left (x - e^{\left (2 \, x\right )}\right )} + 2\right )}}{4 \, x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-640*x^4-320*x^3+56*x^2+24*x)*exp(x)^2+320*x^4+160*x^3-28*x^2-12*x)*exp(-exp(x)^2+x)+320*x^4+160*
x^3-28*x^2+84*x+12)*exp(exp(-exp(x)^2+x)+2+x)/(16*x^4+8*x^3+x^2),x, algorithm="fricas")

[Out]

4*(20*x^2 + 5*x - 3)*e^(x + e^(x - e^(2*x)) + 2)/(4*x^2 + x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left (80 \, x^{4} + 40 \, x^{3} - 7 \, x^{2} + {\left (80 \, x^{4} + 40 \, x^{3} - 7 \, x^{2} - 2 \, {\left (80 \, x^{4} + 40 \, x^{3} - 7 \, x^{2} - 3 \, x\right )} e^{\left (2 \, x\right )} - 3 \, x\right )} e^{\left (x - e^{\left (2 \, x\right )}\right )} + 21 \, x + 3\right )} e^{\left (x + e^{\left (x - e^{\left (2 \, x\right )}\right )} + 2\right )}}{16 \, x^{4} + 8 \, x^{3} + x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-640*x^4-320*x^3+56*x^2+24*x)*exp(x)^2+320*x^4+160*x^3-28*x^2-12*x)*exp(-exp(x)^2+x)+320*x^4+160*
x^3-28*x^2+84*x+12)*exp(exp(-exp(x)^2+x)+2+x)/(16*x^4+8*x^3+x^2),x, algorithm="giac")

[Out]

integrate(4*(80*x^4 + 40*x^3 - 7*x^2 + (80*x^4 + 40*x^3 - 7*x^2 - 2*(80*x^4 + 40*x^3 - 7*x^2 - 3*x)*e^(2*x) -
3*x)*e^(x - e^(2*x)) + 21*x + 3)*e^(x + e^(x - e^(2*x)) + 2)/(16*x^4 + 8*x^3 + x^2), x)

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maple [A]  time = 0.06, size = 36, normalized size = 1.12

method	result	size
risch	\(\frac {4 \left (20 x^{2}+5 x -3\right ) {\mathrm e}^{{\mathrm e}^{x -{\mathrm e}^{2 x}}+2+x}}{x \left (4 x +1\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-640*x^4-320*x^3+56*x^2+24*x)*exp(x)^2+320*x^4+160*x^3-28*x^2-12*x)*exp(-exp(x)^2+x)+320*x^4+160*x^3-28
*x^2+84*x+12)*exp(exp(-exp(x)^2+x)+2+x)/(16*x^4+8*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

4/x*(20*x^2+5*x-3)/(4*x+1)*exp(exp(x-exp(2*x))+2+x)

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maxima [A]  time = 0.55, size = 40, normalized size = 1.25 \begin {gather*} \frac {4 \, {\left (20 \, x^{2} e^{2} + 5 \, x e^{2} - 3 \, e^{2}\right )} e^{\left (x + e^{\left (x - e^{\left (2 \, x\right )}\right )}\right )}}{4 \, x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-640*x^4-320*x^3+56*x^2+24*x)*exp(x)^2+320*x^4+160*x^3-28*x^2-12*x)*exp(-exp(x)^2+x)+320*x^4+160*
x^3-28*x^2+84*x+12)*exp(exp(-exp(x)^2+x)+2+x)/(16*x^4+8*x^3+x^2),x, algorithm="maxima")

[Out]

4*(20*x^2*e^2 + 5*x*e^2 - 3*e^2)*e^(x + e^(x - e^(2*x)))/(4*x^2 + x)

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mupad [B]  time = 3.71, size = 44, normalized size = 1.38 \begin {gather*} 20\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^x-\frac {3\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^x}\,{\mathrm {e}}^x}{x^2+\frac {x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + exp(x - exp(2*x)) + 2)*(84*x + exp(x - exp(2*x))*(exp(2*x)*(24*x + 56*x^2 - 320*x^3 - 640*x^4) -
12*x - 28*x^2 + 160*x^3 + 320*x^4) - 28*x^2 + 160*x^3 + 320*x^4 + 12))/(x^2 + 8*x^3 + 16*x^4),x)

[Out]

20*exp(2)*exp(exp(-exp(2*x))*exp(x))*exp(x) - (3*exp(2)*exp(exp(-exp(2*x))*exp(x))*exp(x))/(x/4 + x^2)

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sympy [A]  time = 0.36, size = 29, normalized size = 0.91 \begin {gather*} \frac {\left (80 x^{2} + 20 x - 12\right ) e^{x + e^{x - e^{2 x}} + 2}}{4 x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-640*x**4-320*x**3+56*x**2+24*x)*exp(x)**2+320*x**4+160*x**3-28*x**2-12*x)*exp(-exp(x)**2+x)+320*
x**4+160*x**3-28*x**2+84*x+12)*exp(exp(-exp(x)**2+x)+2+x)/(16*x**4+8*x**3+x**2),x)

[Out]

(80*x**2 + 20*x - 12)*exp(x + exp(x - exp(2*x)) + 2)/(4*x**2 + x)

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