3.15.77 \(\int \frac {x^3+e^{\frac {4-e^{1+e^{2 e^5}+4 x+4 x^2+e^{e^5} (2+4 x)}+3 x^2+x^3}{x^2}} (-8+x^3+e^{1+e^{2 e^5}+4 x+4 x^2+e^{e^5} (2+4 x)} (2-4 x-4 e^{e^5} x-8 x^2))}{x^3} \, dx\)

Optimal. Leaf size=29 \[ e^{3+\frac {4-e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x}+x \]

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Rubi [F]  time = 16.42, 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 {x^3+\exp \left (\frac {4-e^{1+e^{2 e^5}+4 x+4 x^2+e^{e^5} (2+4 x)}+3 x^2+x^3}{x^2}\right ) \left (-8+x^3+e^{1+e^{2 e^5}+4 x+4 x^2+e^{e^5} (2+4 x)} \left (2-4 x-4 e^{e^5} x-8 x^2\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3 + E^((4 - E^(1 + E^(2*E^5) + 4*x + 4*x^2 + E^E^5*(2 + 4*x)) + 3*x^2 + x^3)/x^2)*(-8 + x^3 + E^(1 + E^
(2*E^5) + 4*x + 4*x^2 + E^E^5*(2 + 4*x))*(2 - 4*x - 4*E^E^5*x - 8*x^2)))/x^3,x]

[Out]

x + Defer[Int][E^(3 - (-4 + E^(1 + E^E^5 + 2*x)^2)/x^2 + x), x] - 8*Defer[Int][E^(3 - (-4 + E^(1 + E^E^5 + 2*x
)^2)/x^2 + x)/x^3, x] + 2*Defer[Int][E^(4*(1 + E^(2*E^5)/4) + 4/x^2 - E^(1 + E^E^5 + 2*x)^2/x^2 + 5*x + 4*x^2
+ E^E^5*(2 + 4*x))/x^3, x] - 4*(1 + E^E^5)*Defer[Int][E^(4*(1 + E^(2*E^5)/4) + 4/x^2 - E^(1 + E^E^5 + 2*x)^2/x
^2 + 5*x + 4*x^2 + E^E^5*(2 + 4*x))/x^2, x] - 8*Defer[Int][E^(4*(1 + E^(2*E^5)/4) + 4/x^2 - E^(1 + E^E^5 + 2*x
)^2/x^2 + 5*x + 4*x^2 + E^E^5*(2 + 4*x))/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right ) \left (1-2 \left (1+e^{e^5}\right ) x-4 x^2\right )}{x^3}+\frac {\exp \left (-\frac {e^{\left (1+e^{e^5}\right )^2+4 \left (1+e^{e^5}\right ) x+4 x^2}}{x^2}\right ) \left (-8 e^{3+\frac {4}{x^2}+x}+e^{\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}} x^3+e^{3+\frac {4}{x^2}+x} x^3\right )}{x^3}\right ) \, dx\\ &=2 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right ) \left (1-2 \left (1+e^{e^5}\right ) x-4 x^2\right )}{x^3} \, dx+\int \frac {\exp \left (-\frac {e^{\left (1+e^{e^5}\right )^2+4 \left (1+e^{e^5}\right ) x+4 x^2}}{x^2}\right ) \left (-8 e^{3+\frac {4}{x^2}+x}+e^{\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}} x^3+e^{3+\frac {4}{x^2}+x} x^3\right )}{x^3} \, dx\\ &=2 \int \left (\frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^3}-\frac {2 \exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right ) \left (1+e^{e^5}\right )}{x^2}-\frac {4 \exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x}\right ) \, dx+\int \left (1+\frac {e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x} \left (-8+x^3\right )}{x^3}\right ) \, dx\\ &=x+2 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^3} \, dx-8 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x} \, dx-\left (4 \left (1+e^{e^5}\right )\right ) \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^2} \, dx+\int \frac {e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x} \left (-8+x^3\right )}{x^3} \, dx\\ &=x+2 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^3} \, dx-8 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x} \, dx-\left (4 \left (1+e^{e^5}\right )\right ) \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^2} \, dx+\int \left (e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x}-\frac {8 e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x}}{x^3}\right ) \, dx\\ &=x+2 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^3} \, dx-8 \int \frac {e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x}}{x^3} \, dx-8 \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x} \, dx-\left (4 \left (1+e^{e^5}\right )\right ) \int \frac {\exp \left (4 \left (1+\frac {e^{2 e^5}}{4}\right )+\frac {4}{x^2}-\frac {e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+5 x+4 x^2+e^{e^5} (2+4 x)\right )}{x^2} \, dx+\int e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.58, size = 28, normalized size = 0.97 \begin {gather*} e^{3-\frac {-4+e^{\left (1+e^{e^5}+2 x\right )^2}}{x^2}+x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3 + E^((4 - E^(1 + E^(2*E^5) + 4*x + 4*x^2 + E^E^5*(2 + 4*x)) + 3*x^2 + x^3)/x^2)*(-8 + x^3 + E^(
1 + E^(2*E^5) + 4*x + 4*x^2 + E^E^5*(2 + 4*x))*(2 - 4*x - 4*E^E^5*x - 8*x^2)))/x^3,x]

[Out]

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

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fricas [A]  time = 0.84, size = 45, normalized size = 1.55 \begin {gather*} x + e^{\left (\frac {x^{3} + 3 \, x^{2} - e^{\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (e^{5}\right )} + 4 \, x + e^{\left (2 \, e^{5}\right )} + 1\right )} + 4}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*exp(exp(5))-8*x^2-4*x+2)*exp(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3-8)*exp((-exp
(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3+3*x^2+4)/x^2)+x^3)/x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} + {\left (x^{3} - 2 \, {\left (4 \, x^{2} + 2 \, x e^{\left (e^{5}\right )} + 2 \, x - 1\right )} e^{\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (e^{5}\right )} + 4 \, x + e^{\left (2 \, e^{5}\right )} + 1\right )} - 8\right )} e^{\left (\frac {x^{3} + 3 \, x^{2} - e^{\left (4 \, x^{2} + 2 \, {\left (2 \, x + 1\right )} e^{\left (e^{5}\right )} + 4 \, x + e^{\left (2 \, e^{5}\right )} + 1\right )} + 4}{x^{2}}\right )}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*exp(exp(5))-8*x^2-4*x+2)*exp(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3-8)*exp((-exp
(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3+3*x^2+4)/x^2)+x^3)/x^3,x, algorithm="giac")

[Out]

integrate((x^3 + (x^3 - 2*(4*x^2 + 2*x*e^(e^5) + 2*x - 1)*e^(4*x^2 + 2*(2*x + 1)*e^(e^5) + 4*x + e^(2*e^5) + 1
) - 8)*e^((x^3 + 3*x^2 - e^(4*x^2 + 2*(2*x + 1)*e^(e^5) + 4*x + e^(2*e^5) + 1) + 4)/x^2))/x^3, x)

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maple [A]  time = 0.14, size = 47, normalized size = 1.62

method	result	size
risch	\(x +{\mathrm e}^{\frac {-{\mathrm e}^{4 x \,{\mathrm e}^{{\mathrm e}^{5}}+4 x^{2}+{\mathrm e}^{2 \,{\mathrm e}^{5}}+2 \,{\mathrm e}^{{\mathrm e}^{5}}+4 x +1}+x^{3}+3 x^{2}+4}{x^{2}}}\)	\(47\)
norman	\(\frac {x^{3}+x^{2} {\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{5}}+\left (4 x +2\right ) {\mathrm e}^{{\mathrm e}^{5}}+4 x^{2}+4 x +1}+x^{3}+3 x^{2}+4}{x^{2}}}}{x^{2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x*exp(exp(5))-8*x^2-4*x+2)*exp(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3-8)*exp((-exp(exp(e
xp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3+3*x^2+4)/x^2)+x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.67, size = 43, normalized size = 1.48 \begin {gather*} x + e^{\left (x - \frac {e^{\left (4 \, x^{2} + 4 \, x e^{\left (e^{5}\right )} + 4 \, x + e^{\left (2 \, e^{5}\right )} + 2 \, e^{\left (e^{5}\right )} + 1\right )}}{x^{2}} + \frac {4}{x^{2}} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*exp(exp(5))-8*x^2-4*x+2)*exp(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3-8)*exp((-exp
(exp(exp(5))^2+(4*x+2)*exp(exp(5))+4*x^2+4*x+1)+x^3+3*x^2+4)/x^2)+x^3)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.30, size = 50, normalized size = 1.72 \begin {gather*} x+{\mathrm {e}}^{-\frac {{\mathrm {e}}^{4\,x\,{\mathrm {e}}^{{\mathrm {e}}^5}}\,{\mathrm {e}}^{4\,x}\,\mathrm {e}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^5}}\,{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^5}}\,{\mathrm {e}}^{4\,x^2}}{x^2}}\,{\mathrm {e}}^3\,{\mathrm {e}}^{\frac {4}{x^2}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((3*x^2 - exp(4*x + exp(2*exp(5)) + exp(exp(5))*(4*x + 2) + 4*x^2 + 1) + x^3 + 4)/x^2)*(exp(4*x + exp
(2*exp(5)) + exp(exp(5))*(4*x + 2) + 4*x^2 + 1)*(4*x + 4*x*exp(exp(5)) + 8*x^2 - 2) - x^3 + 8) - x^3)/x^3,x)

[Out]

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

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sympy [A]  time = 0.57, size = 44, normalized size = 1.52 \begin {gather*} x + e^{\frac {x^{3} + 3 x^{2} - e^{4 x^{2} + 4 x + \left (4 x + 2\right ) e^{e^{5}} + 1 + e^{2 e^{5}}} + 4}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x*exp(exp(5))-8*x**2-4*x+2)*exp(exp(exp(5))**2+(4*x+2)*exp(exp(5))+4*x**2+4*x+1)+x**3-8)*exp((
-exp(exp(exp(5))**2+(4*x+2)*exp(exp(5))+4*x**2+4*x+1)+x**3+3*x**2+4)/x**2)+x**3)/x**3,x)

[Out]

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

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