3.40.33 \(\int \frac {e^{-\frac {2 (-2 x^3+2 x^4)}{-4-6 x^2+6 x^3+e^4 (-x^2+x^3)}} (-16-48 x^2+96 x^3-100 x^4+96 x^5-84 x^6+24 x^7+e^8 (-x^4+2 x^5-x^6)+e^4 (-8 x^2+8 x^3-12 x^4+28 x^5-20 x^6+4 x^7))}{16+48 x^2-48 x^3+36 x^4-72 x^5+36 x^6+e^8 (x^4-2 x^5+x^6)+e^4 (8 x^2-8 x^3+12 x^4-24 x^5+12 x^6)} \, dx\)

Optimal. Leaf size=31 \[ 3-e^{-\frac {4 x}{6+e^4+\frac {4}{x \left (x-x^2\right )}}} x \]

________________________________________________________________________________________

Rubi [F]  time = 12.27, 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 (-\frac {2 \left (-2 x^3+2 x^4\right )}{-4-6 x^2+6 x^3+e^4 \left (-x^2+x^3\right )}\right ) \left (-16-48 x^2+96 x^3-100 x^4+96 x^5-84 x^6+24 x^7+e^8 \left (-x^4+2 x^5-x^6\right )+e^4 \left (-8 x^2+8 x^3-12 x^4+28 x^5-20 x^6+4 x^7\right )\right )}{16+48 x^2-48 x^3+36 x^4-72 x^5+36 x^6+e^8 \left (x^4-2 x^5+x^6\right )+e^4 \left (8 x^2-8 x^3+12 x^4-24 x^5+12 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16 - 48*x^2 + 96*x^3 - 100*x^4 + 96*x^5 - 84*x^6 + 24*x^7 + E^8*(-x^4 + 2*x^5 - x^6) + E^4*(-8*x^2 + 8*x
^3 - 12*x^4 + 28*x^5 - 20*x^6 + 4*x^7))/(E^((2*(-2*x^3 + 2*x^4))/(-4 - 6*x^2 + 6*x^3 + E^4*(-x^2 + x^3)))*(16
+ 48*x^2 - 48*x^3 + 36*x^4 - 72*x^5 + 36*x^6 + E^8*(x^4 - 2*x^5 + x^6) + E^4*(8*x^2 - 8*x^3 + 12*x^4 - 24*x^5
+ 12*x^6))),x]

[Out]

-Defer[Int][E^((-4*(-1 + x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3)), x] + (4*Defer[Int][x/E^((4*(-1 + x)*x^
3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3)), x])/(6 + E^4) - (64*Defer[Int][1/(E^((4*(-1 + x)*x^3)/(-4 - (6 + E^4
)*x^2 + (6 + E^4)*x^3))*(4 + (6 + E^4)*x^2 - (6 + E^4)*x^3)^2), x])/(6 + E^4) - (192*Defer[Int][x/(E^((4*(-1 +
x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3))*(4 + (6 + E^4)*x^2 - (6 + E^4)*x^3)^2), x])/(6 + E^4) - 16*Defe
r[Int][x^2/(E^((4*(-1 + x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3))*(4 + (6 + E^4)*x^2 - (6 + E^4)*x^3)^2),
x] + (16*Defer[Int][1/(E^((4*(-1 + x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3))*(4 + (6 + E^4)*x^2 - (6 + E^4
)*x^3)), x])/(6 + E^4) + (32*Defer[Int][x/(E^((4*(-1 + x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3))*(4 + (6 +
E^4)*x^2 - (6 + E^4)*x^3)), x])/(6 + E^4)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \left (-16-8 \left (6+e^4\right ) x^2+8 \left (12+e^4\right ) x^3-\left (100+12 e^4+e^8\right ) x^4+2 \left (48+14 e^4+e^8\right ) x^5-\left (84+20 e^4+e^8\right ) x^6+4 \left (6+e^4\right ) x^7\right )}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2} \, dx\\ &=\int \left (-\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right )+\frac {4 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x}{6+e^4}+\frac {16 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \left (-4-12 x-\left (6+e^4\right ) x^2\right )}{\left (6+e^4\right ) \left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2}+\frac {16 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) (1+2 x)}{\left (6+e^4\right ) \left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )}\right ) \, dx\\ &=\frac {4 \int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x \, dx}{6+e^4}+\frac {16 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \left (-4-12 x-\left (6+e^4\right ) x^2\right )}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2} \, dx}{6+e^4}+\frac {16 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) (1+2 x)}{4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3} \, dx}{6+e^4}-\int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \, dx\\ &=\frac {4 \int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x \, dx}{6+e^4}+\frac {16 \int \left (-\frac {4 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right )}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2}-\frac {12 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2}+\frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \left (-6-e^4\right ) x^2}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2}\right ) \, dx}{6+e^4}+\frac {16 \int \left (\frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right )}{4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3}+\frac {2 \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x}{4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3}\right ) \, dx}{6+e^4}-\int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \, dx\\ &=-\left (16 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x^2}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2} \, dx\right )+\frac {4 \int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x \, dx}{6+e^4}+\frac {16 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right )}{4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3} \, dx}{6+e^4}+\frac {32 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x}{4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3} \, dx}{6+e^4}-\frac {64 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right )}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2} \, dx}{6+e^4}-\frac {192 \int \frac {\exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) x}{\left (4+\left (6+e^4\right ) x^2-\left (6+e^4\right ) x^3\right )^2} \, dx}{6+e^4}-\int \exp \left (-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}\right ) \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 36, normalized size = 1.16 \begin {gather*} -e^{-\frac {4 (-1+x) x^3}{-4-\left (6+e^4\right ) x^2+\left (6+e^4\right ) x^3}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 48*x^2 + 96*x^3 - 100*x^4 + 96*x^5 - 84*x^6 + 24*x^7 + E^8*(-x^4 + 2*x^5 - x^6) + E^4*(-8*x^2
+ 8*x^3 - 12*x^4 + 28*x^5 - 20*x^6 + 4*x^7))/(E^((2*(-2*x^3 + 2*x^4))/(-4 - 6*x^2 + 6*x^3 + E^4*(-x^2 + x^3))
)*(16 + 48*x^2 - 48*x^3 + 36*x^4 - 72*x^5 + 36*x^6 + E^8*(x^4 - 2*x^5 + x^6) + E^4*(8*x^2 - 8*x^3 + 12*x^4 - 2
4*x^5 + 12*x^6))),x]

[Out]

-(x/E^((4*(-1 + x)*x^3)/(-4 - (6 + E^4)*x^2 + (6 + E^4)*x^3)))

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 41, normalized size = 1.32 \begin {gather*} -x e^{\left (-\frac {4 \, {\left (x^{4} - x^{3}\right )}}{6 \, x^{3} - 6 \, x^{2} + {\left (x^{3} - x^{2}\right )} e^{4} - 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+2*x^5-x^4)*exp(4)^2+(4*x^7-20*x^6+28*x^5-12*x^4+8*x^3-8*x^2)*exp(4)+24*x^7-84*x^6+96*x^5-100*
x^4+96*x^3-48*x^2-16)/((x^6-2*x^5+x^4)*exp(4)^2+(12*x^6-24*x^5+12*x^4-8*x^3+8*x^2)*exp(4)+36*x^6-72*x^5+36*x^4
-48*x^3+48*x^2+16)/exp((2*x^4-2*x^3)/((x^3-x^2)*exp(4)+6*x^3-6*x^2-4))^2,x, algorithm="fricas")

[Out]

-x*e^(-4*(x^4 - x^3)/(6*x^3 - 6*x^2 + (x^3 - x^2)*e^4 - 4))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+2*x^5-x^4)*exp(4)^2+(4*x^7-20*x^6+28*x^5-12*x^4+8*x^3-8*x^2)*exp(4)+24*x^7-84*x^6+96*x^5-100*
x^4+96*x^3-48*x^2-16)/((x^6-2*x^5+x^4)*exp(4)^2+(12*x^6-24*x^5+12*x^4-8*x^3+8*x^2)*exp(4)+36*x^6-72*x^5+36*x^4
-48*x^3+48*x^2+16)/exp((2*x^4-2*x^3)/((x^3-x^2)*exp(4)+6*x^3-6*x^2-4))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:schur row 5 5.87772e-08Evaluation time: 1.66Unable to convert to real exp(8) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.35, size = 40, normalized size = 1.29

method	result	size
risch	\(-x \,{\mathrm e}^{-\frac {4 x^{3} \left (x -1\right )}{x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{4}+6 x^{3}-6 x^{2}-4}}\)	\(40\)
gosper	\(-x \,{\mathrm e}^{-\frac {4 x^{3} \left (x -1\right )}{x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{4}+6 x^{3}-6 x^{2}-4}}\)	\(42\)
norman	\(\frac {\left (\left (-{\mathrm e}^{4}-6\right ) x^{4}+\left ({\mathrm e}^{4}+6\right ) x^{3}+4 x \right ) {\mathrm e}^{-\frac {2 \left (2 x^{4}-2 x^{3}\right )}{\left (x^{3}-x^{2}\right ) {\mathrm e}^{4}+6 x^{3}-6 x^{2}-4}}}{x^{3} {\mathrm e}^{4}-x^{2} {\mathrm e}^{4}+6 x^{3}-6 x^{2}-4}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^6+2*x^5-x^4)*exp(4)^2+(4*x^7-20*x^6+28*x^5-12*x^4+8*x^3-8*x^2)*exp(4)+24*x^7-84*x^6+96*x^5-100*x^4+96
*x^3-48*x^2-16)/((x^6-2*x^5+x^4)*exp(4)^2+(12*x^6-24*x^5+12*x^4-8*x^3+8*x^2)*exp(4)+36*x^6-72*x^5+36*x^4-48*x^
3+48*x^2+16)/exp((2*x^4-2*x^3)/((x^3-x^2)*exp(4)+6*x^3-6*x^2-4))^2,x,method=_RETURNVERBOSE)

[Out]

-x*exp(-4*x^3*(x-1)/(x^3*exp(4)-x^2*exp(4)+6*x^3-6*x^2-4))

________________________________________________________________________________________

maxima [A]  time = 0.64, size = 50, normalized size = 1.61 \begin {gather*} -x e^{\left (-\frac {16 \, x}{x^{3} {\left (e^{8} + 12 \, e^{4} + 36\right )} - x^{2} {\left (e^{8} + 12 \, e^{4} + 36\right )} - 4 \, e^{4} - 24} - \frac {4 \, x}{e^{4} + 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^6+2*x^5-x^4)*exp(4)^2+(4*x^7-20*x^6+28*x^5-12*x^4+8*x^3-8*x^2)*exp(4)+24*x^7-84*x^6+96*x^5-100*
x^4+96*x^3-48*x^2-16)/((x^6-2*x^5+x^4)*exp(4)^2+(12*x^6-24*x^5+12*x^4-8*x^3+8*x^2)*exp(4)+36*x^6-72*x^5+36*x^4
-48*x^3+48*x^2+16)/exp((2*x^4-2*x^3)/((x^3-x^2)*exp(4)+6*x^3-6*x^2-4))^2,x, algorithm="maxima")

[Out]

-x*e^(-16*x/(x^3*(e^8 + 12*e^4 + 36) - x^2*(e^8 + 12*e^4 + 36) - 4*e^4 - 24) - 4*x/(e^4 + 6))

________________________________________________________________________________________

mupad [B]  time = 4.96, size = 44, normalized size = 1.42 \begin {gather*} -x\,{\mathrm {e}}^{-\frac {4\,x^3-4\,x^4}{x^2\,{\mathrm {e}}^4-x^3\,{\mathrm {e}}^4+6\,x^2-6\,x^3+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(2*x^3 - 2*x^4))/(exp(4)*(x^2 - x^3) + 6*x^2 - 6*x^3 + 4))*(exp(8)*(x^4 - 2*x^5 + x^6) + exp(4)*
(8*x^2 - 8*x^3 + 12*x^4 - 28*x^5 + 20*x^6 - 4*x^7) + 48*x^2 - 96*x^3 + 100*x^4 - 96*x^5 + 84*x^6 - 24*x^7 + 16
))/(exp(8)*(x^4 - 2*x^5 + x^6) + exp(4)*(8*x^2 - 8*x^3 + 12*x^4 - 24*x^5 + 12*x^6) + 48*x^2 - 48*x^3 + 36*x^4
- 72*x^5 + 36*x^6 + 16),x)

[Out]

-x*exp(-(4*x^3 - 4*x^4)/(x^2*exp(4) - x^3*exp(4) + 6*x^2 - 6*x^3 + 4))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**6+2*x**5-x**4)*exp(4)**2+(4*x**7-20*x**6+28*x**5-12*x**4+8*x**3-8*x**2)*exp(4)+24*x**7-84*x**6
+96*x**5-100*x**4+96*x**3-48*x**2-16)/((x**6-2*x**5+x**4)*exp(4)**2+(12*x**6-24*x**5+12*x**4-8*x**3+8*x**2)*ex
p(4)+36*x**6-72*x**5+36*x**4-48*x**3+48*x**2+16)/exp((2*x**4-2*x**3)/((x**3-x**2)*exp(4)+6*x**3-6*x**2-4))**2,
x)

[Out]

Timed out

________________________________________________________________________________________