3.7.28 \(\int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} (-8+2 x+4 x^2-2 x^3))}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} (16 x^4-8 x^5+x^6)+e^{\frac {4}{\log ^2(3)}} (-32 x^5+16 x^6-2 x^7)} \, dx\)

Optimal. Leaf size=31 \[ \frac {2 e^{-\frac {1}{x^2}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right ) (-4+x) x} \]

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Rubi [F]  time = 5.60, 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 {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-1 + x^2*Log[2])/x^2)*(8*x - 2*x^2 - 8*x^3 + 3*x^4 + E^(4/Log[3]^2)*(-8 + 2*x + 4*x^2 - 2*x^3)))/(16*
x^6 - 8*x^7 + x^8 + E^(8/Log[3]^2)*(16*x^4 - 8*x^5 + x^6) + E^(4/Log[3]^2)*(-32*x^5 + 16*x^6 - 2*x^7)),x]

[Out]

-1/8*(E^(-x^(-2) - 8/Log[3]^2)*(4 + E^(4/Log[3]^2))) - E^(-x^(-2) - 4/Log[3]^2)/(2*x) + (Sqrt[Pi]*Erf[x^(-1)])
/(4*E^(4/Log[3]^2)) + ((16 + 4*E^(4/Log[3]^2) - 7*E^(8/Log[3]^2))*Sqrt[Pi]*Erf[x^(-1)])/(32*E^(12/Log[3]^2)) +
((64 + 16*E^(4/Log[3]^2) + 4*E^(8/Log[3]^2) + E^(12/Log[3]^2))*ExpIntegralEi[-x^(-2)])/(128*E^(16/Log[3]^2))
- (2*Defer[Int][E^(-x^(-2) - 4/Log[3]^2)/(E^(4/Log[3]^2) - x)^2, x])/(4 - E^(4/Log[3]^2)) - (4*Defer[Int][E^(-
x^(-2) - 16/Log[3]^2)/(E^(4/Log[3]^2) - x), x])/(4 - E^(4/Log[3]^2)) + Defer[Int][1/(E^x^(-2)*(-4 + x)^2), x]/
(2*(4 - E^(4/Log[3]^2))) - Defer[Int][1/(E^x^(-2)*(-4 + x)), x]/(64*(4 - E^(4/Log[3]^2)))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-\frac {1}{x^2}} \left (-8 e^{\frac {4}{\log ^2(3)}}+2 \left (4+e^{\frac {4}{\log ^2(3)}}\right ) x-2 \left (1-2 e^{\frac {4}{\log ^2(3)}}\right ) x^2-2 \left (4+e^{\frac {4}{\log ^2(3)}}\right ) x^3+3 x^4\right )}{(4-x)^2 \left (e^{\frac {4}{\log ^2(3)}}-x\right )^2 x^4} \, dx\\ &=2 \int \frac {e^{-\frac {1}{x^2}} \left (-8 e^{\frac {4}{\log ^2(3)}}+2 \left (4+e^{\frac {4}{\log ^2(3)}}\right ) x-2 \left (1-2 e^{\frac {4}{\log ^2(3)}}\right ) x^2-2 \left (4+e^{\frac {4}{\log ^2(3)}}\right ) x^3+3 x^4\right )}{(4-x)^2 \left (e^{\frac {4}{\log ^2(3)}}-x\right )^2 x^4} \, dx\\ &=2 \int \left (\frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{\left (-4+e^{\frac {4}{\log ^2(3)}}\right ) \left (e^{\frac {4}{\log ^2(3)}}-x\right )^2}+\frac {2 e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{\left (-4+e^{\frac {4}{\log ^2(3)}}\right ) \left (e^{\frac {4}{\log ^2(3)}}-x\right )}-\frac {e^{-\frac {1}{x^2}}}{4 \left (-4+e^{\frac {4}{\log ^2(3)}}\right ) (-4+x)^2}+\frac {e^{-\frac {1}{x^2}}}{128 \left (-4+e^{\frac {4}{\log ^2(3)}}\right ) (-4+x)}-\frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{2 x^4}-\frac {e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}} \left (4+e^{\frac {4}{\log ^2(3)}}\right )}{8 x^3}+\frac {e^{-\frac {1}{x^2}-\frac {12}{\log ^2(3)}} \left (-16-4 e^{\frac {4}{\log ^2(3)}}+7 e^{\frac {8}{\log ^2(3)}}\right )}{32 x^2}-\frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}} \left (64+16 e^{\frac {4}{\log ^2(3)}}+4 e^{\frac {8}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right )}{128 x}\right ) \, dx\\ &=\frac {1}{4} \left (-4-e^{\frac {4}{\log ^2(3)}}\right ) \int \frac {e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}}}{x^3} \, dx-\frac {\int \frac {e^{-\frac {1}{x^2}}}{-4+x} \, dx}{64 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}+\frac {\int \frac {e^{-\frac {1}{x^2}}}{(-4+x)^2} \, dx}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}-\frac {2 \int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right )^2} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}-\frac {4 \int \frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{e^{\frac {4}{\log ^2(3)}}-x} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}+\frac {1}{16} \left (-16-4 e^{\frac {4}{\log ^2(3)}}+7 e^{\frac {8}{\log ^2(3)}}\right ) \int \frac {e^{-\frac {1}{x^2}-\frac {12}{\log ^2(3)}}}{x^2} \, dx+\frac {1}{64} \left (-64-16 e^{\frac {4}{\log ^2(3)}}-4 e^{\frac {8}{\log ^2(3)}}-e^{\frac {12}{\log ^2(3)}}\right ) \int \frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{x} \, dx-\int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{x^4} \, dx\\ &=-\frac {1}{8} e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}} \left (4+e^{\frac {4}{\log ^2(3)}}\right )-\frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{2 x}+\frac {1}{128} e^{-\frac {16}{\log ^2(3)}} \left (64+16 e^{\frac {4}{\log ^2(3)}}+4 e^{\frac {8}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right ) \text {Ei}\left (-\frac {1}{x^2}\right )-\frac {1}{2} \int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{x^2} \, dx-\frac {\int \frac {e^{-\frac {1}{x^2}}}{-4+x} \, dx}{64 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}+\frac {\int \frac {e^{-\frac {1}{x^2}}}{(-4+x)^2} \, dx}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}-\frac {2 \int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right )^2} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}-\frac {4 \int \frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{e^{\frac {4}{\log ^2(3)}}-x} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}+\frac {1}{16} \left (16+4 e^{\frac {4}{\log ^2(3)}}-7 e^{\frac {8}{\log ^2(3)}}\right ) \operatorname {Subst}\left (\int e^{-x^2-\frac {12}{\log ^2(3)}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{8} e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}} \left (4+e^{\frac {4}{\log ^2(3)}}\right )-\frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{2 x}+\frac {1}{32} e^{-\frac {12}{\log ^2(3)}} \left (16+4 e^{\frac {4}{\log ^2(3)}}-7 e^{\frac {8}{\log ^2(3)}}\right ) \sqrt {\pi } \text {erf}\left (\frac {1}{x}\right )+\frac {1}{128} e^{-\frac {16}{\log ^2(3)}} \left (64+16 e^{\frac {4}{\log ^2(3)}}+4 e^{\frac {8}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right ) \text {Ei}\left (-\frac {1}{x^2}\right )+\frac {1}{2} \operatorname {Subst}\left (\int e^{-x^2-\frac {4}{\log ^2(3)}} \, dx,x,\frac {1}{x}\right )-\frac {\int \frac {e^{-\frac {1}{x^2}}}{-4+x} \, dx}{64 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}+\frac {\int \frac {e^{-\frac {1}{x^2}}}{(-4+x)^2} \, dx}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}-\frac {2 \int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right )^2} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}-\frac {4 \int \frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{e^{\frac {4}{\log ^2(3)}}-x} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}\\ &=-\frac {1}{8} e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}} \left (4+e^{\frac {4}{\log ^2(3)}}\right )-\frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{2 x}+\frac {1}{4} e^{-\frac {4}{\log ^2(3)}} \sqrt {\pi } \text {erf}\left (\frac {1}{x}\right )+\frac {1}{32} e^{-\frac {12}{\log ^2(3)}} \left (16+4 e^{\frac {4}{\log ^2(3)}}-7 e^{\frac {8}{\log ^2(3)}}\right ) \sqrt {\pi } \text {erf}\left (\frac {1}{x}\right )+\frac {1}{128} e^{-\frac {16}{\log ^2(3)}} \left (64+16 e^{\frac {4}{\log ^2(3)}}+4 e^{\frac {8}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right ) \text {Ei}\left (-\frac {1}{x^2}\right )-\frac {\int \frac {e^{-\frac {1}{x^2}}}{-4+x} \, dx}{64 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}+\frac {\int \frac {e^{-\frac {1}{x^2}}}{(-4+x)^2} \, dx}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}-\frac {2 \int \frac {e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right )^2} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}-\frac {4 \int \frac {e^{-\frac {1}{x^2}-\frac {16}{\log ^2(3)}}}{e^{\frac {4}{\log ^2(3)}}-x} \, dx}{4-e^{\frac {4}{\log ^2(3)}}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.30, size = 31, normalized size = 1.00 \begin {gather*} \frac {2 e^{-\frac {1}{x^2}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right ) (-4+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-1 + x^2*Log[2])/x^2)*(8*x - 2*x^2 - 8*x^3 + 3*x^4 + E^(4/Log[3]^2)*(-8 + 2*x + 4*x^2 - 2*x^3))
)/(16*x^6 - 8*x^7 + x^8 + E^(8/Log[3]^2)*(16*x^4 - 8*x^5 + x^6) + E^(4/Log[3]^2)*(-32*x^5 + 16*x^6 - 2*x^7)),x
]

[Out]

2/(E^x^(-2)*(E^(4/Log[3]^2) - x)*(-4 + x)*x)

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fricas [A]  time = 0.60, size = 42, normalized size = 1.35 \begin {gather*} -\frac {e^{\left (\frac {x^{2} \log \relax (2) - 1}{x^{2}}\right )}}{x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x\right )} e^{\left (\frac {4}{\log \relax (3)^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*
x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6-32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*
x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6-32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.96, size = 36, normalized size = 1.16

method	result	size
norman	\(\frac {{\mathrm e}^{\frac {x^{2} \ln \relax (2)-1}{x^{2}}}}{x \left (x -4\right ) \left ({\mathrm e}^{\frac {4}{\ln \relax (3)^{2}}}-x \right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+4*x^2+2*x-8)*exp(4/ln(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp((x^2*ln(2)-1)/x^2)/((x^6-8*x^5+16*x^4)*exp
(4/ln(3)^2)^2+(-2*x^7+16*x^6-32*x^5)*exp(4/ln(3)^2)+x^8-8*x^7+16*x^6),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.74, size = 38, normalized size = 1.23 \begin {gather*} -\frac {2 \, e^{\left (-\frac {1}{x^{2}}\right )}}{x^{3} - x^{2} {\left (e^{\left (\frac {4}{\log \relax (3)^{2}}\right )} + 4\right )} + 4 \, x e^{\left (\frac {4}{\log \relax (3)^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*
x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6-32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.99, size = 41, normalized size = 1.32 \begin {gather*} -\frac {2\,{\mathrm {e}}^{-\frac {1}{x^2}}}{4\,x\,{\mathrm {e}}^{\frac {4}{{\ln \relax (3)}^2}}-x^2\,{\mathrm {e}}^{\frac {4}{{\ln \relax (3)}^2}}-4\,x^2+x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^2*log(2) - 1)/x^2)*(8*x + exp(4/log(3)^2)*(2*x + 4*x^2 - 2*x^3 - 8) - 2*x^2 - 8*x^3 + 3*x^4))/(exp
(8/log(3)^2)*(16*x^4 - 8*x^5 + x^6) + 16*x^6 - 8*x^7 + x^8 - exp(4/log(3)^2)*(32*x^5 - 16*x^6 + 2*x^7)),x)

[Out]

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

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sympy [B]  time = 0.34, size = 46, normalized size = 1.48 \begin {gather*} - \frac {e^{\frac {x^{2} \log {\relax (2 )} - 1}{x^{2}}}}{x^{3} - x^{2} e^{\frac {4}{\log {\relax (3 )}^{2}}} - 4 x^{2} + 4 x e^{\frac {4}{\log {\relax (3 )}^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+4*x**2+2*x-8)*exp(4/ln(3)**2)+3*x**4-8*x**3-2*x**2+8*x)*exp((x**2*ln(2)-1)/x**2)/((x**6-8*
x**5+16*x**4)*exp(4/ln(3)**2)**2+(-2*x**7+16*x**6-32*x**5)*exp(4/ln(3)**2)+x**8-8*x**7+16*x**6),x)

[Out]

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

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