3.85.85 \(\int \frac {e^{\frac {-192 x+58 x^2+2 x^3+e^{-6+x^2} (-24 x+8 x^2)}{8+e^{-6+x^2}}} (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} (-384+244 x+6 x^2+12 x^3-4 x^4))}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx\)

Optimal. Leaf size=25 \[ 3+e^{2 (-3+x) x \left (4+\frac {x}{8+e^{-6+x^2}}\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 15.39, 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 {-192 x+58 x^2+2 x^3+e^{-6+x^2} \left (-24 x+8 x^2\right )}{8+e^{-6+x^2}}\right ) \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{64+16 e^{-6+x^2}+e^{-12+2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-192*x + 58*x^2 + 2*x^3 + E^(-6 + x^2)*(-24*x + 8*x^2))/(8 + E^(-6 + x^2)))*(-1536 + 928*x + 48*x^2 +
E^(-12 + 2*x^2)*(-24 + 16*x) + E^(-6 + x^2)*(-384 + 244*x + 6*x^2 + 12*x^3 - 4*x^4)))/(64 + 16*E^(-6 + x^2) +
E^(-12 + 2*x^2)),x]

[Out]

-24*Defer[Int][E^((2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2)))), x] + 16*Defer[Int][E^((
2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x, x] - 12*Defer[Int][(E^(6 + (2*(-3 + x)*x
*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x)/(8*E^6 + E^x^2), x] + 6*Defer[Int][(E^(6 + (2*(-3 +
x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x^2)/(8*E^6 + E^x^2), x] - 96*Defer[Int][(E^(12 + (
2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x^3)/(8*E^6 + E^x^2)^2, x] + 12*Defer[Int][
(E^(6 + (2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x^3)/(8*E^6 + E^x^2), x] + 32*Defe
r[Int][(E^(12 + (2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x^4)/(8*E^6 + E^x^2)^2, x]
- 4*Defer[Int][(E^(6 + (2*(-3 + x)*x*(32*E^6 + 4*E^x^2 + E^6*x))/(E^6*(8 + E^(-6 + x^2))))*x^4)/(8*E^6 + E^x^
2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) \left (-1536+928 x+48 x^2+e^{-12+2 x^2} (-24+16 x)+e^{-6+x^2} \left (-384+244 x+6 x^2+12 x^3-4 x^4\right )\right )}{\left (8 e^6+e^{x^2}\right )^2} \, dx\\ &=\int \left (\frac {32 \exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) (-3+x) x^3}{\left (8 e^6+e^{x^2}\right )^2}+8 \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) (-3+2 x)-\frac {2 \exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x \left (6-3 x-6 x^2+2 x^3\right )}{8 e^6+e^{x^2}}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x \left (6-3 x-6 x^2+2 x^3\right )}{8 e^6+e^{x^2}} \, dx\right )+8 \int \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) (-3+2 x) \, dx+32 \int \frac {\exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) (-3+x) x^3}{\left (8 e^6+e^{x^2}\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {6 \exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x}{8 e^6+e^{x^2}}-\frac {3 \exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^2}{8 e^6+e^{x^2}}-\frac {6 \exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^3}{8 e^6+e^{x^2}}+\frac {2 \exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^4}{8 e^6+e^{x^2}}\right ) \, dx\right )+8 \int \left (-3 \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right )+2 \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x\right ) \, dx+32 \int \left (-\frac {3 \exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^3}{\left (8 e^6+e^{x^2}\right )^2}+\frac {\exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^4}{\left (8 e^6+e^{x^2}\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^4}{8 e^6+e^{x^2}} \, dx\right )+6 \int \frac {\exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^2}{8 e^6+e^{x^2}} \, dx-12 \int \frac {\exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x}{8 e^6+e^{x^2}} \, dx+12 \int \frac {\exp \left (6+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^3}{8 e^6+e^{x^2}} \, dx+16 \int \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x \, dx-24 \int \exp \left (\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) \, dx+32 \int \frac {\exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^4}{\left (8 e^6+e^{x^2}\right )^2} \, dx-96 \int \frac {\exp \left (12+\frac {2 (-3+x) x \left (32 e^6+4 e^{x^2}+e^6 x\right )}{e^6 \left (8+e^{-6+x^2}\right )}\right ) x^3}{\left (8 e^6+e^{x^2}\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 36, normalized size = 1.44 \begin {gather*} e^{\frac {2 (-3+x) x \left (4 e^{x^2}+e^6 (32+x)\right )}{8 e^6+e^{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-192*x + 58*x^2 + 2*x^3 + E^(-6 + x^2)*(-24*x + 8*x^2))/(8 + E^(-6 + x^2)))*(-1536 + 928*x + 48
*x^2 + E^(-12 + 2*x^2)*(-24 + 16*x) + E^(-6 + x^2)*(-384 + 244*x + 6*x^2 + 12*x^3 - 4*x^4)))/(64 + 16*E^(-6 +
x^2) + E^(-12 + 2*x^2)),x]

[Out]

E^((2*(-3 + x)*x*(4*E^x^2 + E^6*(32 + x)))/(8*E^6 + E^x^2))

________________________________________________________________________________________

fricas [A]  time = 1.00, size = 40, normalized size = 1.60 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{3} + 29 \, x^{2} + 4 \, {\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} - 6\right )} - 96 \, x\right )}}{e^{\left (x^{2} - 6\right )} + 8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48*x^2+928*x-1536)*exp(((8*x^2-24
*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="fricas")

[Out]

e^(2*(x^3 + 29*x^2 + 4*(x^2 - 3*x)*e^(x^2 - 6) - 96*x)/(e^(x^2 - 6) + 8))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (24 \, x^{2} + 4 \, {\left (2 \, x - 3\right )} e^{\left (2 \, x^{2} - 12\right )} - {\left (2 \, x^{4} - 6 \, x^{3} - 3 \, x^{2} - 122 \, x + 192\right )} e^{\left (x^{2} - 6\right )} + 464 \, x - 768\right )} e^{\left (\frac {2 \, {\left (x^{3} + 29 \, x^{2} + 4 \, {\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} - 6\right )} - 96 \, x\right )}}{e^{\left (x^{2} - 6\right )} + 8}\right )}}{e^{\left (2 \, x^{2} - 12\right )} + 16 \, e^{\left (x^{2} - 6\right )} + 64}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48*x^2+928*x-1536)*exp(((8*x^2-24
*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="giac")

[Out]

integrate(2*(24*x^2 + 4*(2*x - 3)*e^(2*x^2 - 12) - (2*x^4 - 6*x^3 - 3*x^2 - 122*x + 192)*e^(x^2 - 6) + 464*x -
768)*e^(2*(x^3 + 29*x^2 + 4*(x^2 - 3*x)*e^(x^2 - 6) - 96*x)/(e^(x^2 - 6) + 8))/(e^(2*x^2 - 12) + 16*e^(x^2 -
6) + 64), x)

________________________________________________________________________________________

maple [A]  time = 0.26, size = 29, normalized size = 1.16

method	result	size
risch	\({\mathrm e}^{\frac {2 x \left (x -3\right ) \left (x +4 \,{\mathrm e}^{x^{2}-6}+32\right )}{{\mathrm e}^{x^{2}-6}+8}}\)	\(29\)
norman	\(\frac {{\mathrm e}^{x^{2}-6} {\mathrm e}^{\frac {\left (8 x^{2}-24 x \right ) {\mathrm e}^{x^{2}-6}+2 x^{3}+58 x^{2}-192 x}{{\mathrm e}^{x^{2}-6}+8}}+8 \,{\mathrm e}^{\frac {\left (8 x^{2}-24 x \right ) {\mathrm e}^{x^{2}-6}+2 x^{3}+58 x^{2}-192 x}{{\mathrm e}^{x^{2}-6}+8}}}{{\mathrm e}^{x^{2}-6}+8}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48*x^2+928*x-1536)*exp(((8*x^2-24*x)*ex
p(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x,method=_RETURNVERBOSE)

[Out]

exp(2*x*(x-3)*(x+4*exp(x^2-6)+32)/(exp(x^2-6)+8))

________________________________________________________________________________________

maxima [B]  time = 0.71, size = 92, normalized size = 3.68 \begin {gather*} e^{\left (\frac {2 \, x^{3} e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} + \frac {58 \, x^{2} e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} + \frac {8 \, x^{2} e^{\left (x^{2}\right )}}{8 \, e^{6} + e^{\left (x^{2}\right )}} - \frac {192 \, x e^{6}}{8 \, e^{6} + e^{\left (x^{2}\right )}} - \frac {24 \, x e^{\left (x^{2}\right )}}{8 \, e^{6} + e^{\left (x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x-24)*exp(x^2-6)^2+(-4*x^4+12*x^3+6*x^2+244*x-384)*exp(x^2-6)+48*x^2+928*x-1536)*exp(((8*x^2-24
*x)*exp(x^2-6)+2*x^3+58*x^2-192*x)/(exp(x^2-6)+8))/(exp(x^2-6)^2+16*exp(x^2-6)+64),x, algorithm="maxima")

[Out]

e^(2*x^3*e^6/(8*e^6 + e^(x^2)) + 58*x^2*e^6/(8*e^6 + e^(x^2)) + 8*x^2*e^(x^2)/(8*e^6 + e^(x^2)) - 192*x*e^6/(8
*e^6 + e^(x^2)) - 24*x*e^(x^2)/(8*e^6 + e^(x^2)))

________________________________________________________________________________________

mupad [B]  time = 5.45, size = 96, normalized size = 3.84 \begin {gather*} {\mathrm {e}}^{-\frac {24\,x\,{\mathrm {e}}^{x^2}}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{-\frac {192\,x\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{x^2}}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}}\,{\mathrm {e}}^{\frac {58\,x^2\,{\mathrm {e}}^6}{{\mathrm {e}}^{x^2}+8\,{\mathrm {e}}^6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(192*x + exp(x^2 - 6)*(24*x - 8*x^2) - 58*x^2 - 2*x^3)/(exp(x^2 - 6) + 8))*(928*x + exp(2*x^2 - 12)*
(16*x - 24) + exp(x^2 - 6)*(244*x + 6*x^2 + 12*x^3 - 4*x^4 - 384) + 48*x^2 - 1536))/(16*exp(x^2 - 6) + exp(2*x
^2 - 12) + 64),x)

[Out]

exp(-(24*x*exp(x^2))/(exp(x^2) + 8*exp(6)))*exp(-(192*x*exp(6))/(exp(x^2) + 8*exp(6)))*exp((8*x^2*exp(x^2))/(e
xp(x^2) + 8*exp(6)))*exp((2*x^3*exp(6))/(exp(x^2) + 8*exp(6)))*exp((58*x^2*exp(6))/(exp(x^2) + 8*exp(6)))

________________________________________________________________________________________

sympy [A]  time = 0.69, size = 37, normalized size = 1.48 \begin {gather*} e^{\frac {2 x^{3} + 58 x^{2} - 192 x + \left (8 x^{2} - 24 x\right ) e^{x^{2} - 6}}{e^{x^{2} - 6} + 8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x-24)*exp(x**2-6)**2+(-4*x**4+12*x**3+6*x**2+244*x-384)*exp(x**2-6)+48*x**2+928*x-1536)*exp(((8
*x**2-24*x)*exp(x**2-6)+2*x**3+58*x**2-192*x)/(exp(x**2-6)+8))/(exp(x**2-6)**2+16*exp(x**2-6)+64),x)

[Out]

exp((2*x**3 + 58*x**2 - 192*x + (8*x**2 - 24*x)*exp(x**2 - 6))/(exp(x**2 - 6) + 8))

________________________________________________________________________________________