\(\int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} (-8 x^2+8 x^3)}{e^{2 x} x^2+x^3}} (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} (-16-8 x+16 x^2+16 x^4))}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx\) [6050]

Optimal result

Integrand size = 121, antiderivative size = 24 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=e^{4 (2-2 x) \left (-1+\frac {1}{x^2 \left (e^{2 x}+x\right )}\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=\int \frac {\exp \left (\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}\right ) \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx \]

[In]

Int[(E^((8 - 8*x - 8*x^3 + 8*x^4 + E^(2*x)*(-8*x^2 + 8*x^3))/(E^(2*x)*x^2 + x^3))*(-24*x + 16*x^2 + 8*E^(4*x)*
x^3 + 8*x^5 + E^(2*x)*(-16 - 8*x + 16*x^2 + 16*x^4)))/(E^(4*x)*x^3 + 2*E^(2*x)*x^4 + x^5),x]

[Out]

8*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)*x^2 + x^3))/(x^2*(E^(2*x) + x))), x] - 16*Defer[Int][E^((8*(-1 + x)*
(-1 + E^(2*x)*x^2 + x^3))/(x^2*(E^(2*x) + x)))/(E^(2*x) + x)^2, x] - 8*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)
*x^2 + x^3))/(x^2*(E^(2*x) + x)))/(x^2*(E^(2*x) + x)^2), x] + 24*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)*x^2 +
x^3))/(x^2*(E^(2*x) + x)))/(x*(E^(2*x) + x)^2), x] - 16*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)*x^2 + x^3))/(
x^2*(E^(2*x) + x)))/(x^3*(E^(2*x) + x)), x] - 8*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)*x^2 + x^3))/(x^2*(E^(2
*x) + x)))/(x^2*(E^(2*x) + x)), x] + 16*Defer[Int][E^((8*(-1 + x)*(-1 + E^(2*x)*x^2 + x^3))/(x^2*(E^(2*x) + x)
))/(x*(E^(2*x) + x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (e^{4 x} x^3+x \left (-3+2 x+x^4\right )+e^{2 x} \left (-2-x+2 x^2+2 x^4\right )\right )}{x^3 \left (e^{2 x}+x\right )^2} \, dx \\ & = 8 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (e^{4 x} x^3+x \left (-3+2 x+x^4\right )+e^{2 x} \left (-2-x+2 x^2+2 x^4\right )\right )}{x^3 \left (e^{2 x}+x\right )^2} \, dx \\ & = 8 \int \left (\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )-\frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (1-3 x+2 x^2\right )}{x^2 \left (e^{2 x}+x\right )^2}+\frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (-2-x+2 x^2\right )}{x^3 \left (e^{2 x}+x\right )}\right ) \, dx \\ & = 8 \int \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \, dx-8 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (1-3 x+2 x^2\right )}{x^2 \left (e^{2 x}+x\right )^2} \, dx+8 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \left (-2-x+2 x^2\right )}{x^3 \left (e^{2 x}+x\right )} \, dx \\ & = 8 \int \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \, dx-8 \int \left (\frac {2 \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{\left (e^{2 x}+x\right )^2}+\frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^2 \left (e^{2 x}+x\right )^2}-\frac {3 \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x \left (e^{2 x}+x\right )^2}\right ) \, dx+8 \int \left (-\frac {2 \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^3 \left (e^{2 x}+x\right )}-\frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^2 \left (e^{2 x}+x\right )}+\frac {2 \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x \left (e^{2 x}+x\right )}\right ) \, dx \\ & = 8 \int \exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right ) \, dx-8 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^2 \left (e^{2 x}+x\right )^2} \, dx-8 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^2 \left (e^{2 x}+x\right )} \, dx-16 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{\left (e^{2 x}+x\right )^2} \, dx-16 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x^3 \left (e^{2 x}+x\right )} \, dx+16 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x \left (e^{2 x}+x\right )} \, dx+24 \int \frac {\exp \left (\frac {8 (-1+x) \left (-1+e^{2 x} x^2+x^3\right )}{x^2 \left (e^{2 x}+x\right )}\right )}{x \left (e^{2 x}+x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=e^{-8+8 x-\frac {8 (-1+x)}{x^2 \left (e^{2 x}+x\right )}} \]

[In]

Integrate[(E^((8 - 8*x - 8*x^3 + 8*x^4 + E^(2*x)*(-8*x^2 + 8*x^3))/(E^(2*x)*x^2 + x^3))*(-24*x + 16*x^2 + 8*E^
(4*x)*x^3 + 8*x^5 + E^(2*x)*(-16 - 8*x + 16*x^2 + 16*x^4)))/(E^(4*x)*x^3 + 2*E^(2*x)*x^4 + x^5),x]

[Out]

E^(-8 + 8*x - (8*(-1 + x))/(x^2*(E^(2*x) + x)))

Maple [A] (verified)

Time = 3.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29

method	result	size
risch	\({\mathrm e}^{\frac {8 \left (-1+x \right ) \left ({\mathrm e}^{2 x} x^{2}+x^{3}-1\right )}{x^{2} \left ({\mathrm e}^{2 x}+x \right )}}\)	\(31\)
parallelrisch	\({\mathrm e}^{\frac {\left (8 x^{3}-8 x^{2}\right ) {\mathrm e}^{2 x}+8 x^{4}-8 x^{3}-8 x +8}{x^{2} \left ({\mathrm e}^{2 x}+x \right )}}\)	\(45\)

[In]

int((8*x^3*exp(x)^4+(16*x^4+16*x^2-8*x-16)*exp(x)^2+8*x^5+16*x^2-24*x)*exp(((8*x^3-8*x^2)*exp(x)^2+8*x^4-8*x^3
-8*x+8)/(exp(x)^2*x^2+x^3))/(x^3*exp(x)^4+2*exp(x)^2*x^4+x^5),x,method=_RETURNVERBOSE)

[Out]

exp(8*(-1+x)*(exp(2*x)*x^2+x^3-1)/x^2/(exp(2*x)+x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=e^{\left (\frac {8 \, {\left (x^{4} - x^{3} + {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x\right )} - x + 1\right )}}{x^{3} + x^{2} e^{\left (2 \, x\right )}}\right )} \]

[In]

integrate((8*x^3*exp(x)^4+(16*x^4+16*x^2-8*x-16)*exp(x)^2+8*x^5+16*x^2-24*x)*exp(((8*x^3-8*x^2)*exp(x)^2+8*x^4
-8*x^3-8*x+8)/(exp(x)^2*x^2+x^3))/(x^3*exp(x)^4+2*exp(x)^2*x^4+x^5),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=e^{\frac {8 x^{4} - 8 x^{3} - 8 x + \left (8 x^{3} - 8 x^{2}\right ) e^{2 x} + 8}{x^{3} + x^{2} e^{2 x}}} \]

[In]

integrate((8*x**3*exp(x)**4+(16*x**4+16*x**2-8*x-16)*exp(x)**2+8*x**5+16*x**2-24*x)*exp(((8*x**3-8*x**2)*exp(x
)**2+8*x**4-8*x**3-8*x+8)/(exp(x)**2*x**2+x**3))/(x**3*exp(x)**4+2*exp(x)**2*x**4+x**5),x)

[Out]

exp((8*x**4 - 8*x**3 - 8*x + (8*x**3 - 8*x**2)*exp(2*x) + 8)/(x**3 + x**2*exp(2*x)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).

Time = 0.44 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=e^{\left (8 \, x - \frac {8 \, e^{\left (-2 \, x\right )}}{x} - \frac {8 \, e^{\left (-4 \, x\right )}}{x} + \frac {8}{x e^{\left (4 \, x\right )} + e^{\left (6 \, x\right )}} + \frac {8}{x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} + \frac {8 \, e^{\left (-2 \, x\right )}}{x^{2}} - 8\right )} \]

[In]

integrate((8*x^3*exp(x)^4+(16*x^4+16*x^2-8*x-16)*exp(x)^2+8*x^5+16*x^2-24*x)*exp(((8*x^3-8*x^2)*exp(x)^2+8*x^4
-8*x^3-8*x+8)/(exp(x)^2*x^2+x^3))/(x^3*exp(x)^4+2*exp(x)^2*x^4+x^5),x, algorithm="maxima")

[Out]

e^(8*x - 8*e^(-2*x)/x - 8*e^(-4*x)/x + 8/(x*e^(4*x) + e^(6*x)) + 8/(x*e^(2*x) + e^(4*x)) + 8*e^(-2*x)/x^2 - 8)

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((8*x^3*exp(x)^4+(16*x^4+16*x^2-8*x-16)*exp(x)^2+8*x^5+16*x^2-24*x)*exp(((8*x^3-8*x^2)*exp(x)^2+8*x^4
-8*x^3-8*x+8)/(exp(x)^2*x^2+x^3))/(x^3*exp(x)^4+2*exp(x)^2*x^4+x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{512,[1,17]%%%}+%%%{-2048,[1,16]%%%}+%%%{2816,[1,15]%%%}+
%%%{-6752,[

Mupad [B] (verification not implemented)

Time = 11.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75 \[ \int \frac {e^{\frac {8-8 x-8 x^3+8 x^4+e^{2 x} \left (-8 x^2+8 x^3\right )}{e^{2 x} x^2+x^3}} \left (-24 x+16 x^2+8 e^{4 x} x^3+8 x^5+e^{2 x} \left (-16-8 x+16 x^2+16 x^4\right )\right )}{e^{4 x} x^3+2 e^{2 x} x^4+x^5} \, dx={\mathrm {e}}^{\frac {8\,x^2}{x+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {8}{x\,{\mathrm {e}}^{2\,x}+x^2}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^{2\,x}}{x+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{-\frac {8\,x}{x+{\mathrm {e}}^{2\,x}}}\,{\mathrm {e}}^{\frac {8}{x^2\,{\mathrm {e}}^{2\,x}+x^3}}\,{\mathrm {e}}^{\frac {8\,x\,{\mathrm {e}}^{2\,x}}{x+{\mathrm {e}}^{2\,x}}} \]

[In]

int((exp(-(8*x + exp(2*x)*(8*x^2 - 8*x^3) + 8*x^3 - 8*x^4 - 8)/(x^2*exp(2*x) + x^3))*(8*x^3*exp(4*x) - exp(2*x
)*(8*x - 16*x^2 - 16*x^4 + 16) - 24*x + 16*x^2 + 8*x^5))/(2*x^4*exp(2*x) + x^3*exp(4*x) + x^5),x)

[Out]

exp((8*x^2)/(x + exp(2*x)))*exp(-8/(x*exp(2*x) + x^2))*exp(-(8*exp(2*x))/(x + exp(2*x)))*exp(-(8*x)/(x + exp(2
*x)))*exp(8/(x^2*exp(2*x) + x^3))*exp((8*x*exp(2*x))/(x + exp(2*x)))