3.16.80 \(\int \frac {e^{\frac {4+3 x^2-x^3}{12 x^2+2 e^{2 x} x^2}} (-48-6 x^3+e^{2 x} (-8-8 x-7 x^3+2 x^4))}{72 x^3+24 e^{2 x} x^3+2 e^{4 x} x^3} \, dx\)

Optimal. Leaf size=33 \[ e^{\frac {e^{-x} \left (3+\frac {4}{x^2}-x\right )}{12 e^{-x}+2 e^x}} \]

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Rubi [F]  time = 9.07, 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 {e^{\frac {4+3 x^2-x^3}{12 x^2+2 e^{2 x} x^2}} \left (-48-6 x^3+e^{2 x} \left (-8-8 x-7 x^3+2 x^4\right )\right )}{72 x^3+24 e^{2 x} x^3+2 e^{4 x} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 + 3*x^2 - x^3)/(12*x^2 + 2*E^(2*x)*x^2))*(-48 - 6*x^3 + E^(2*x)*(-8 - 8*x - 7*x^3 + 2*x^4)))/(72*x^
3 + 24*E^(2*x)*x^3 + 2*E^(4*x)*x^3),x]

[Out]

18*Defer[Int][E^((4 + 3*x^2 - x^3)/(2*(6 + E^(2*x))*x^2))/(6 + E^(2*x))^2, x] - (7*Defer[Int][E^((4 + 3*x^2 -
x^3)/(2*(6 + E^(2*x))*x^2))/(6 + E^(2*x)), x])/2 - 4*Defer[Int][E^((4 + 3*x^2 - x^3)/(2*(6 + E^(2*x))*x^2))/((
6 + E^(2*x))*x^3), x] + 24*Defer[Int][E^((4 + 3*x^2 - x^3)/(2*(6 + E^(2*x))*x^2))/((6 + E^(2*x))^2*x^2), x] -
4*Defer[Int][E^((4 + 3*x^2 - x^3)/(2*(6 + E^(2*x))*x^2))/((6 + E^(2*x))*x^2), x] - 6*Defer[Int][(E^((4 + 3*x^2
- x^3)/(2*(6 + E^(2*x))*x^2))*x)/(6 + E^(2*x))^2, x] + Defer[Int][(E^((4 + 3*x^2 - x^3)/(2*(6 + E^(2*x))*x^2)
)*x)/(6 + E^(2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-48-6 x^3+e^{2 x} \left (-8-8 x-7 x^3+2 x^4\right )\right )}{2 \left (6+e^{2 x}\right )^2 x^3} \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-48-6 x^3+e^{2 x} \left (-8-8 x-7 x^3+2 x^4\right )\right )}{\left (6+e^{2 x}\right )^2 x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {12 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-4-3 x^2+x^3\right )}{\left (6+e^{2 x}\right )^2 x^2}+\frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-8-8 x-7 x^3+2 x^4\right )}{\left (6+e^{2 x}\right ) x^3}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-8-8 x-7 x^3+2 x^4\right )}{\left (6+e^{2 x}\right ) x^3} \, dx-6 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \left (-4-3 x^2+x^3\right )}{\left (6+e^{2 x}\right )^2 x^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {7 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{6+e^{2 x}}-\frac {8 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right ) x^3}-\frac {8 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right ) x^2}+\frac {2 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} x}{6+e^{2 x}}\right ) \, dx-6 \int \left (-\frac {3 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right )^2}-\frac {4 e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right )^2 x^2}+\frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} x}{\left (6+e^{2 x}\right )^2}\right ) \, dx\\ &=-\left (\frac {7}{2} \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{6+e^{2 x}} \, dx\right )-4 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right ) x^3} \, dx-4 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right ) x^2} \, dx-6 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} x}{\left (6+e^{2 x}\right )^2} \, dx+18 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right )^2} \, dx+24 \int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}}}{\left (6+e^{2 x}\right )^2 x^2} \, dx+\int \frac {e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} x}{6+e^{2 x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.12, size = 30, normalized size = 0.91 \begin {gather*} e^{\frac {4+3 x^2-x^3}{2 \left (6+e^{2 x}\right ) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 + 3*x^2 - x^3)/(12*x^2 + 2*E^(2*x)*x^2))*(-48 - 6*x^3 + E^(2*x)*(-8 - 8*x - 7*x^3 + 2*x^4)))/
(72*x^3 + 24*E^(2*x)*x^3 + 2*E^(4*x)*x^3),x]

[Out]

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

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fricas [A]  time = 0.59, size = 29, normalized size = 0.88 \begin {gather*} e^{\left (-\frac {x^{3} - 3 \, x^{2} - 4}{2 \, {\left (x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-7*x^3-8*x-8)*exp(x)^2-6*x^3-48)*exp((-x^3+3*x^2+4)/(2*exp(x)^2*x^2+12*x^2))/(2*x^3*exp(x)^4+
24*exp(x)^2*x^3+72*x^3),x, algorithm="fricas")

[Out]

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

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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(((2*x^4-7*x^3-8*x-8)*exp(x)^2-6*x^3-48)*exp((-x^3+3*x^2+4)/(2*exp(x)^2*x^2+12*x^2))/(2*x^3*exp(x)^4+
24*exp(x)^2*x^3+72*x^3),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%%%{-1152,[1,23]%%%}+%%%{18432,[1,22]%%%}+%%%{-117792,[1,21]
%%%}+%%%{39

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maple [A]  time = 0.09, size = 25, normalized size = 0.76

method	result	size
risch	\({\mathrm e}^{-\frac {x^{3}-3 x^{2}-4}{2 x^{2} \left ({\mathrm e}^{2 x}+6\right )}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4-7*x^3-8*x-8)*exp(x)^2-6*x^3-48)*exp((-x^3+3*x^2+4)/(2*exp(x)^2*x^2+12*x^2))/(2*x^3*exp(x)^4+24*exp
(x)^2*x^3+72*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.62, size = 41, normalized size = 1.24 \begin {gather*} e^{\left (-\frac {x}{2 \, {\left (e^{\left (2 \, x\right )} + 6\right )}} + \frac {2}{x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2}} + \frac {3}{2 \, {\left (e^{\left (2 \, x\right )} + 6\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4-7*x^3-8*x-8)*exp(x)^2-6*x^3-48)*exp((-x^3+3*x^2+4)/(2*exp(x)^2*x^2+12*x^2))/(2*x^3*exp(x)^4+
24*exp(x)^2*x^3+72*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.18, size = 47, normalized size = 1.42 \begin {gather*} {\mathrm {e}}^{\frac {3}{2\,{\mathrm {e}}^{2\,x}+12}}\,{\mathrm {e}}^{\frac {2}{x^2\,{\mathrm {e}}^{2\,x}+6\,x^2}}\,{\mathrm {e}}^{-\frac {x}{2\,{\mathrm {e}}^{2\,x}+12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((3*x^2 - x^3 + 4)/(2*x^2*exp(2*x) + 12*x^2))*(exp(2*x)*(8*x + 7*x^3 - 2*x^4 + 8) + 6*x^3 + 48))/(24*
x^3*exp(2*x) + 2*x^3*exp(4*x) + 72*x^3),x)

[Out]

exp(3/(2*exp(2*x) + 12))*exp(2/(x^2*exp(2*x) + 6*x^2))*exp(-x/(2*exp(2*x) + 12))

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sympy [A]  time = 0.40, size = 26, normalized size = 0.79 \begin {gather*} e^{\frac {- x^{3} + 3 x^{2} + 4}{2 x^{2} e^{2 x} + 12 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4-7*x**3-8*x-8)*exp(x)**2-6*x**3-48)*exp((-x**3+3*x**2+4)/(2*exp(x)**2*x**2+12*x**2))/(2*x**3
*exp(x)**4+24*exp(x)**2*x**3+72*x**3),x)

[Out]

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

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