\(\int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} (x^2+x^4)+e^{\frac {4}{e^3}} (8 x^2+8 x^4)}{1+x^2}} (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} (-16 x-32 x^3-16 x^5)+e^{\frac {8}{e^3}} (-2 x-4 x^3-2 x^5))}{1+2 x^2+x^4} \, dx\) [1803]

Optimal result

Integrand size = 127, antiderivative size = 31 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=5-e^{\left (4+e^{\frac {4}{e^3}}\right )^2 x^2-\frac {5}{1+x^2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=\int \frac {\exp \left (\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}\right ) \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx \]

[In]

Int[(E^((-5 + 16*x^2 + 16*x^4 + E^(8/E^3)*(x^2 + x^4) + E^(4/E^3)*(8*x^2 + 8*x^4))/(1 + x^2))*(-42*x - 64*x^3
- 32*x^5 + E^(4/E^3)*(-16*x - 32*x^3 - 16*x^5) + E^(8/E^3)*(-2*x - 4*x^3 - 2*x^5)))/(1 + 2*x^2 + x^4),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}\right ) \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{\left (1+x^2\right )^2} \, dx \\ & = \int \frac {\exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}\right ) \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (1+\frac {e^{\frac {4}{e^3}}}{8}\right ) \left (-16 x-32 x^3-16 x^5\right )\right )}{\left (1+x^2\right )^2} \, dx \\ & = \int \left (-2 \exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}\right ) \left (4+e^{\frac {4}{e^3}}\right )^2 x-\frac {10 \exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}\right ) x}{\left (1+x^2\right )^2}\right ) \, dx \\ & = -\left (10 \int \frac {\exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}\right ) x}{\left (1+x^2\right )^2} \, dx\right )-\left (2 \left (4+e^{\frac {4}{e^3}}\right )^2\right ) \int \exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}\right ) x \, dx \\ & = -\left (5 \text {Subst}\left (\int \frac {\exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2}{1+x}\right )}{(1+x)^2} \, dx,x,x^2\right )\right )-\left (4+e^{\frac {4}{e^3}}\right )^2 \text {Subst}\left (\int \exp \left (\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2}{1+x}\right ) \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=-e^{\frac {-5+\left (4+e^{\frac {4}{e^3}}\right )^2 x^2+\left (4+e^{\frac {4}{e^3}}\right )^2 x^4}{1+x^2}} \]

[In]

Integrate[(E^((-5 + 16*x^2 + 16*x^4 + E^(8/E^3)*(x^2 + x^4) + E^(4/E^3)*(8*x^2 + 8*x^4))/(1 + x^2))*(-42*x - 6
4*x^3 - 32*x^5 + E^(4/E^3)*(-16*x - 32*x^3 - 16*x^5) + E^(8/E^3)*(-2*x - 4*x^3 - 2*x^5)))/(1 + 2*x^2 + x^4),x]

[Out]

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

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94

method	result	size
parallelrisch	\(-{\mathrm e}^{\frac {\left (x^{4}+x^{2}\right ) {\mathrm e}^{8 \,{\mathrm e}^{-3}}+\left (8 x^{4}+8 x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{-3}}+16 x^{4}+16 x^{2}-5}{x^{2}+1}}\)	\(60\)
risch	\(-{\mathrm e}^{\frac {{\mathrm e}^{8 \,{\mathrm e}^{-3}} x^{4}+8 \,{\mathrm e}^{4 \,{\mathrm e}^{-3}} x^{4}+16 x^{4}+{\mathrm e}^{8 \,{\mathrm e}^{-3}} x^{2}+8 \,{\mathrm e}^{4 \,{\mathrm e}^{-3}} x^{2}+16 x^{2}-5}{x^{2}+1}}\)	\(62\)
gosper	\(-{\mathrm e}^{\frac {{\mathrm e}^{8 \,{\mathrm e}^{-3}} x^{4}+8 \,{\mathrm e}^{4 \,{\mathrm e}^{-3}} x^{4}+16 x^{4}+{\mathrm e}^{8 \,{\mathrm e}^{-3}} x^{2}+8 \,{\mathrm e}^{4 \,{\mathrm e}^{-3}} x^{2}+16 x^{2}-5}{x^{2}+1}}\)	\(74\)
norman	\(\frac {-x^{2} {\mathrm e}^{\frac {\left (x^{4}+x^{2}\right ) {\mathrm e}^{8 \,{\mathrm e}^{-3}}+\left (8 x^{4}+8 x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{-3}}+16 x^{4}+16 x^{2}-5}{x^{2}+1}}-{\mathrm e}^{\frac {\left (x^{4}+x^{2}\right ) {\mathrm e}^{8 \,{\mathrm e}^{-3}}+\left (8 x^{4}+8 x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{-3}}+16 x^{4}+16 x^{2}-5}{x^{2}+1}}}{x^{2}+1}\)	\(131\)

[In]

int(((-2*x^5-4*x^3-2*x)*exp(4/exp(3))^2+(-16*x^5-32*x^3-16*x)*exp(4/exp(3))-32*x^5-64*x^3-42*x)*exp(((x^4+x^2)
*exp(4/exp(3))^2+(8*x^4+8*x^2)*exp(4/exp(3))+16*x^4+16*x^2-5)/(x^2+1))/(x^4+2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

-exp(((x^4+x^2)*exp(4/exp(3))^2+(8*x^4+8*x^2)*exp(4/exp(3))+16*x^4+16*x^2-5)/(x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=-e^{\left (\frac {16 \, x^{4} + 16 \, x^{2} + {\left (x^{4} + x^{2}\right )} e^{\left (8 \, e^{\left (-3\right )}\right )} + 8 \, {\left (x^{4} + x^{2}\right )} e^{\left (4 \, e^{\left (-3\right )}\right )} - 5}{x^{2} + 1}\right )} \]

[In]

integrate(((-2*x^5-4*x^3-2*x)*exp(4/exp(3))^2+(-16*x^5-32*x^3-16*x)*exp(4/exp(3))-32*x^5-64*x^3-42*x)*exp(((x^
4+x^2)*exp(4/exp(3))^2+(8*x^4+8*x^2)*exp(4/exp(3))+16*x^4+16*x^2-5)/(x^2+1))/(x^4+2*x^2+1),x, algorithm="frica
s")

[Out]

-e^((16*x^4 + 16*x^2 + (x^4 + x^2)*e^(8*e^(-3)) + 8*(x^4 + x^2)*e^(4*e^(-3)) - 5)/(x^2 + 1))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=- e^{\frac {16 x^{4} + 16 x^{2} + \left (x^{4} + x^{2}\right ) e^{\frac {8}{e^{3}}} + \left (8 x^{4} + 8 x^{2}\right ) e^{\frac {4}{e^{3}}} - 5}{x^{2} + 1}} \]

[In]

integrate(((-2*x**5-4*x**3-2*x)*exp(4/exp(3))**2+(-16*x**5-32*x**3-16*x)*exp(4/exp(3))-32*x**5-64*x**3-42*x)*e
xp(((x**4+x**2)*exp(4/exp(3))**2+(8*x**4+8*x**2)*exp(4/exp(3))+16*x**4+16*x**2-5)/(x**2+1))/(x**4+2*x**2+1),x)

[Out]

-exp((16*x**4 + 16*x**2 + (x**4 + x**2)*exp(8*exp(-3)) + (8*x**4 + 8*x**2)*exp(4*exp(-3)) - 5)/(x**2 + 1))

Maxima [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=-e^{\left (x^{2} e^{\left (8 \, e^{\left (-3\right )}\right )} + 8 \, x^{2} e^{\left (4 \, e^{\left (-3\right )}\right )} + 16 \, x^{2} - \frac {5}{x^{2} + 1}\right )} \]

[In]

integrate(((-2*x^5-4*x^3-2*x)*exp(4/exp(3))^2+(-16*x^5-32*x^3-16*x)*exp(4/exp(3))-32*x^5-64*x^3-42*x)*exp(((x^
4+x^2)*exp(4/exp(3))^2+(8*x^4+8*x^2)*exp(4/exp(3))+16*x^4+16*x^2-5)/(x^2+1))/(x^4+2*x^2+1),x, algorithm="maxim
a")

[Out]

-e^(x^2*e^(8*e^(-3)) + 8*x^2*e^(4*e^(-3)) + 16*x^2 - 5/(x^2 + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (28) = 56\).

Time = 0.35 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.32 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=-e^{\left (\frac {x^{4} e^{\left (8 \, e^{\left (-3\right )}\right )}}{x^{2} + 1} + \frac {8 \, x^{4} e^{\left (4 \, e^{\left (-3\right )}\right )}}{x^{2} + 1} + \frac {16 \, x^{4}}{x^{2} + 1} + \frac {x^{2} e^{\left (8 \, e^{\left (-3\right )}\right )}}{x^{2} + 1} + \frac {8 \, x^{2} e^{\left (4 \, e^{\left (-3\right )}\right )}}{x^{2} + 1} + \frac {16 \, x^{2}}{x^{2} + 1} - \frac {5}{x^{2} + 1}\right )} \]

[In]

integrate(((-2*x^5-4*x^3-2*x)*exp(4/exp(3))^2+(-16*x^5-32*x^3-16*x)*exp(4/exp(3))-32*x^5-64*x^3-42*x)*exp(((x^
4+x^2)*exp(4/exp(3))^2+(8*x^4+8*x^2)*exp(4/exp(3))+16*x^4+16*x^2-5)/(x^2+1))/(x^4+2*x^2+1),x, algorithm="giac"
)

[Out]

-e^(x^4*e^(8*e^(-3))/(x^2 + 1) + 8*x^4*e^(4*e^(-3))/(x^2 + 1) + 16*x^4/(x^2 + 1) + x^2*e^(8*e^(-3))/(x^2 + 1)
+ 8*x^2*e^(4*e^(-3))/(x^2 + 1) + 16*x^2/(x^2 + 1) - 5/(x^2 + 1))

Mupad [B] (verification not implemented)

Time = 10.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.48 \[ \int \frac {e^{\frac {-5+16 x^2+16 x^4+e^{\frac {8}{e^3}} \left (x^2+x^4\right )+e^{\frac {4}{e^3}} \left (8 x^2+8 x^4\right )}{1+x^2}} \left (-42 x-64 x^3-32 x^5+e^{\frac {4}{e^3}} \left (-16 x-32 x^3-16 x^5\right )+e^{\frac {8}{e^3}} \left (-2 x-4 x^3-2 x^5\right )\right )}{1+2 x^2+x^4} \, dx=-{\mathrm {e}}^{\frac {16\,x^2}{x^2+1}}\,{\mathrm {e}}^{\frac {16\,x^4}{x^2+1}}\,{\mathrm {e}}^{-\frac {5}{x^2+1}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-3}}}{x^2+1}}\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-3}}}{x^2+1}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-3}}}{x^2+1}}\,{\mathrm {e}}^{\frac {8\,x^4\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-3}}}{x^2+1}} \]

[In]

int(-(exp((exp(4*exp(-3))*(8*x^2 + 8*x^4) + exp(8*exp(-3))*(x^2 + x^4) + 16*x^2 + 16*x^4 - 5)/(x^2 + 1))*(42*x
+ exp(8*exp(-3))*(2*x + 4*x^3 + 2*x^5) + exp(4*exp(-3))*(16*x + 32*x^3 + 16*x^5) + 64*x^3 + 32*x^5))/(2*x^2 +
x^4 + 1),x)

[Out]

-exp((16*x^2)/(x^2 + 1))*exp((16*x^4)/(x^2 + 1))*exp(-5/(x^2 + 1))*exp((x^2*exp(8*exp(-3)))/(x^2 + 1))*exp((x^
4*exp(8*exp(-3)))/(x^2 + 1))*exp((8*x^2*exp(4*exp(-3)))/(x^2 + 1))*exp((8*x^4*exp(4*exp(-3)))/(x^2 + 1))