3.19.76 \(\int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} (2 x-8 e^{x^4} x^4)}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx\)

Optimal. Leaf size=26 \[ \log \left (\frac {1}{50} \left (-1-e^{e^{-4-e^{x^4}+x}}+x\right )^2\right ) \]

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Rubi [F]  time = 2.67, 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 {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x + E^(-4 - E^x^4 + E^(-4 - E^x^4 + x) + x)*(2*x - 8*E^x^4*x^4))/(x + E^E^(-4 - E^x^4 + x)*x - x^2),x]

[Out]

-2*Defer[Int][(1 + E^E^(-4 - E^x^4 + x) - x)^(-1), x] + 2*Defer[Int][E^(-4 - E^x^4 + E^(-4 - E^x^4 + x) + x)/(
1 + E^E^(-4 - E^x^4 + x) - x), x] - 8*Defer[Int][(E^(-4 - E^x^4 + E^(-4 - E^x^4 + x) + x + x^4)*x^3)/(1 + E^E^
(-4 - E^x^4 + x) - x), x]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 22, normalized size = 0.85 \begin {gather*} 2 \log \left (1+e^{e^{-4-e^{x^4}+x}}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + E^(-4 - E^x^4 + E^(-4 - E^x^4 + x) + x)*(2*x - 8*E^x^4*x^4))/(x + E^E^(-4 - E^x^4 + x)*x - x
^2),x]

[Out]

2*Log[1 + E^E^(-4 - E^x^4 + x) - x]

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fricas [A]  time = 0.71, size = 19, normalized size = 0.73 \begin {gather*} 2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^4*exp(x^4)+2*x)*exp(-exp(x^4)+x-4)*exp(exp(-exp(x^4)+x-4))-2*x)/(x*exp(exp(-exp(x^4)+x-4))-x^
2+x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.33, size = 54, normalized size = 2.08 \begin {gather*} -2 \, x + 2 \, e^{\left (x^{4}\right )} + 2 \, \log \left (-x e^{\left (x - e^{\left (x^{4}\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )} + e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^4*exp(x^4)+2*x)*exp(-exp(x^4)+x-4)*exp(exp(-exp(x^4)+x-4))-2*x)/(x*exp(exp(-exp(x^4)+x-4))-x^
2+x),x, algorithm="giac")

[Out]

-2*x + 2*e^(x^4) + 2*log(-x*e^(x - e^(x^4)) + e^(x - e^(x^4) + e^(x - e^(x^4) - 4)) + e^(x - e^(x^4)))

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maple [A]  time = 0.06, size = 20, normalized size = 0.77

method	result	size
risch	\(2 \ln \left ({\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x^{4}}+x -4}}-x +1\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*x^4*exp(x^4)+2*x)*exp(-exp(x^4)+x-4)*exp(exp(-exp(x^4)+x-4))-2*x)/(x*exp(exp(-exp(x^4)+x-4))-x^2+x),x
,method=_RETURNVERBOSE)

[Out]

2*ln(exp(exp(-exp(x^4)+x-4))-x+1)

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maxima [A]  time = 0.56, size = 19, normalized size = 0.73 \begin {gather*} 2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x^4*exp(x^4)+2*x)*exp(-exp(x^4)+x-4)*exp(exp(-exp(x^4)+x-4))-2*x)/(x*exp(exp(-exp(x^4)+x-4))-x^
2+x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.26, size = 21, normalized size = 0.81 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{x^4}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}-x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - exp(exp(x - exp(x^4) - 4))*exp(x - exp(x^4) - 4)*(2*x - 8*x^4*exp(x^4)))/(x + x*exp(exp(x - exp(x^
4) - 4)) - x^2),x)

[Out]

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

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sympy [A]  time = 0.55, size = 17, normalized size = 0.65 \begin {gather*} 2 \log {\left (- x + e^{e^{x - e^{x^{4}} - 4}} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*x**4*exp(x**4)+2*x)*exp(-exp(x**4)+x-4)*exp(exp(-exp(x**4)+x-4))-2*x)/(x*exp(exp(-exp(x**4)+x-4
))-x**2+x),x)

[Out]

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

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