3.40.42 \(\int \frac {e^{e^{\frac {-8-12 x-28 x^2-13 x^3+e^x (4+4 x+12 x^2)}{4 x^3}}+\frac {-8-12 x-28 x^2-13 x^3+e^x (4+4 x+12 x^2)}{4 x^3}} (e^4 (6+6 x+7 x^2)+e^{4+x} (-3-x-2 x^2+3 x^3))}{x^4} \, dx\)

Optimal. Leaf size=34 \[ e^{4+e^{\frac {-\frac {x}{4}+\left (-2+e^x-x\right ) \left (3+\frac {1+x}{x^2}\right )}{x}}} \]

________________________________________________________________________________________

Rubi [F]  time = 25.17, 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 {\exp \left (\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right ) \left (e^4 \left (6+6 x+7 x^2\right )+e^{4+x} \left (-3-x-2 x^2+3 x^3\right )\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 - 13*x^3 + E^
x*(4 + 4*x + 12*x^2))/(4*x^3))*(E^4*(6 + 6*x + 7*x^2) + E^(4 + x)*(-3 - x - 2*x^2 + 3*x^3)))/x^4,x]

[Out]

6*Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 -
13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^4, x] - 3*Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x
*(4 + 4*x + 12*x^2))/(4*x^3)) + x + (-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^4, x] +
6*Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 -
13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^3, x] - Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(
4 + 4*x + 12*x^2))/(4*x^3)) + x + (-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^3, x] + 7*
Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 - 1
3*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^2, x] - 2*Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(
4 + 4*x + 12*x^2))/(4*x^3)) + x + (-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x^2, x] + 3*
Defer[Int][E^(4 + E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + x + (-8 - 12*x - 28*x^2
- 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right ) \left (6+6 x+7 x^2\right )}{x^4}+\frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right ) \left (-3-x-2 x^2+3 x^3\right )}{x^4}\right ) \, dx\\ &=\int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right ) \left (6+6 x+7 x^2\right )}{x^4} \, dx+\int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right ) \left (-3-x-2 x^2+3 x^3\right )}{x^4} \, dx\\ &=\int \left (\frac {6 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^4}+\frac {6 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^3}+\frac {7 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^2}\right ) \, dx+\int \left (-\frac {3 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^4}-\frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^3}-\frac {2 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^2}+\frac {3 \exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^2} \, dx\right )-3 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^4} \, dx+3 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x} \, dx+6 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^4} \, dx+6 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^3} \, dx+7 \int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^2} \, dx-\int \frac {\exp \left (4+\exp \left (\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )+x+\frac {-8-12 x-28 x^2-13 x^3+e^x \left (4+4 x+12 x^2\right )}{4 x^3}\right )}{x^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.48, size = 40, normalized size = 1.18 \begin {gather*} e^{4+e^{-\frac {13}{4}-\frac {2}{x^3}-\frac {3}{x^2}-\frac {7}{x}+\frac {e^x \left (1+x+3 x^2\right )}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-8 - 12*x - 28*x^2 - 13*x^3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3)) + (-8 - 12*x - 28*x^2 - 13*x^
3 + E^x*(4 + 4*x + 12*x^2))/(4*x^3))*(E^4*(6 + 6*x + 7*x^2) + E^(4 + x)*(-3 - x - 2*x^2 + 3*x^3)))/x^4,x]

[Out]

E^(4 + E^(-13/4 - 2/x^3 - 3/x^2 - 7/x + (E^x*(1 + x + 3*x^2))/x^3))

________________________________________________________________________________________

fricas [B]  time = 0.93, size = 132, normalized size = 3.88 \begin {gather*} e^{\left (\frac {{\left (4 \, x^{3} e^{\left (-\frac {{\left ({\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + 4\right )} - {\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} + 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + \frac {{\left ({\left (13 \, x^{3} + 28 \, x^{2} + 12 \, x + 8\right )} e^{4} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{4 \, x^{3}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-
12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="fricas")

[Out]

e^(1/4*(4*x^3*e^(-1/4*((13*x^3 + 28*x^2 + 12*x + 8)*e^4 - 4*(3*x^2 + x + 1)*e^(x + 4))*e^(-4)/x^3 + 4) - (13*x
^3 + 28*x^2 + 12*x + 8)*e^4 + 4*(3*x^2 + x + 1)*e^(x + 4))*e^(-4)/x^3 + 1/4*((13*x^3 + 28*x^2 + 12*x + 8)*e^4
- 4*(3*x^2 + x + 1)*e^(x + 4))*e^(-4)/x^3 + 4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (7 \, x^{2} + 6 \, x + 6\right )} e^{4} + {\left (3 \, x^{3} - 2 \, x^{2} - x - 3\right )} e^{\left (x + 4\right )}\right )} e^{\left (-\frac {13 \, x^{3} + 28 \, x^{2} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{x} + 12 \, x + 8}{4 \, x^{3}} + e^{\left (-\frac {13 \, x^{3} + 28 \, x^{2} - 4 \, {\left (3 \, x^{2} + x + 1\right )} e^{x} + 12 \, x + 8}{4 \, x^{3}}\right )}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-
12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="giac")

[Out]

integrate(((7*x^2 + 6*x + 6)*e^4 + (3*x^3 - 2*x^2 - x - 3)*e^(x + 4))*e^(-1/4*(13*x^3 + 28*x^2 - 4*(3*x^2 + x
+ 1)*e^x + 12*x + 8)/x^3 + e^(-1/4*(13*x^3 + 28*x^2 - 4*(3*x^2 + x + 1)*e^x + 12*x + 8)/x^3))/x^4, x)

________________________________________________________________________________________

maple [A]  time = 0.27, size = 41, normalized size = 1.21

method	result	size
risch	\({\mathrm e}^{4+{\mathrm e}^{\frac {12 \,{\mathrm e}^{x} x^{2}-13 x^{3}+4 \,{\mathrm e}^{x} x -28 x^{2}+4 \,{\mathrm e}^{x}-12 x -8}{4 x^{3}}}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8
)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x,method=_RETURNVERBOSE)

[Out]

exp(4+exp(1/4*(12*exp(x)*x^2-13*x^3+4*exp(x)*x-28*x^2+4*exp(x)-12*x-8)/x^3))

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 40, normalized size = 1.18 \begin {gather*} e^{\left (e^{\left (\frac {3 \, e^{x}}{x} - \frac {7}{x} + \frac {e^{x}}{x^{2}} - \frac {3}{x^{2}} + \frac {e^{x}}{x^{3}} - \frac {2}{x^{3}} - \frac {13}{4}\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^3-2*x^2-x-3)*exp(4)*exp(x)+(7*x^2+6*x+6)*exp(4))*exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-
12*x-8)/x^3)*exp(exp(1/4*((12*x^2+4*x+4)*exp(x)-13*x^3-28*x^2-12*x-8)/x^3))/x^4,x, algorithm="maxima")

[Out]

e^(e^(3*e^x/x - 7/x + e^x/x^2 - 3/x^2 + e^x/x^3 - 2/x^3 - 13/4) + 4)

________________________________________________________________________________________

mupad [B]  time = 2.72, size = 47, normalized size = 1.38 \begin {gather*} {\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{-\frac {13}{4}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^2}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^3}}\,{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-\frac {3}{x^2}}\,{\mathrm {e}}^{-\frac {2}{x^3}}\,{\mathrm {e}}^{-\frac {7}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(-(3*x - (exp(x)*(4*x + 12*x^2 + 4))/4 + 7*x^2 + (13*x^3)/4 + 2)/x^3))*exp(-(3*x - (exp(x)*(4*x +
12*x^2 + 4))/4 + 7*x^2 + (13*x^3)/4 + 2)/x^3)*(exp(4)*(6*x + 7*x^2 + 6) - exp(4)*exp(x)*(x + 2*x^2 - 3*x^3 + 3
)))/x^4,x)

[Out]

exp(4)*exp(exp(-13/4)*exp(exp(x)/x^2)*exp(exp(x)/x^3)*exp((3*exp(x))/x)*exp(-3/x^2)*exp(-2/x^3)*exp(-7/x))

________________________________________________________________________________________

sympy [A]  time = 1.95, size = 41, normalized size = 1.21 \begin {gather*} e^{4} e^{e^{\frac {- \frac {13 x^{3}}{4} - 7 x^{2} - 3 x + \frac {\left (12 x^{2} + 4 x + 4\right ) e^{x}}{4} - 2}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**3-2*x**2-x-3)*exp(4)*exp(x)+(7*x**2+6*x+6)*exp(4))*exp(1/4*((12*x**2+4*x+4)*exp(x)-13*x**3-28
*x**2-12*x-8)/x**3)*exp(exp(1/4*((12*x**2+4*x+4)*exp(x)-13*x**3-28*x**2-12*x-8)/x**3))/x**4,x)

[Out]

exp(4)*exp(exp((-13*x**3/4 - 7*x**2 - 3*x + (12*x**2 + 4*x + 4)*exp(x)/4 - 2)/x**3))

________________________________________________________________________________________