3.103.32 \(\int \frac {e^{x^6} x^2+e^{x^6} (1-2 x) \log (\log (3))+e^{e^{e^x-x^6}} (-e^{x^6}+e^{e^x} (e^x x-6 x^6+(-e^x+6 x^5) \log (\log (3))))}{e^{x^6} x^2-2 e^{x^6} x \log (\log (3))+e^{x^6} \log ^2(\log (3))} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^{e^{e^x-x^6}}-x+x^2}{x-\log (\log (3))} \]

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Rubi [F]  time = 3.57, 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^{x^6} x^2+e^{x^6} (1-2 x) \log (\log (3))+e^{e^{e^x-x^6}} \left (-e^{x^6}+e^{e^x} \left (e^x x-6 x^6+\left (-e^x+6 x^5\right ) \log (\log (3))\right )\right )}{e^{x^6} x^2-2 e^{x^6} x \log (\log (3))+e^{x^6} \log ^2(\log (3))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x^6*x^2 + E^x^6*(1 - 2*x)*Log[Log[3]] + E^E^(E^x - x^6)*(-E^x^6 + E^E^x*(E^x*x - 6*x^6 + (-E^x + 6*x^5)
*Log[Log[3]])))/(E^x^6*x^2 - 2*E^x^6*x*Log[Log[3]] + E^x^6*Log[Log[3]]^2),x]

[Out]

x - ((1 - Log[Log[3]])*Log[Log[3]])/(x - Log[Log[3]]) - 6*Log[Log[3]]^4*Defer[Int][E^(E^x + E^(E^x - x^6) - x^
6), x] - 6*Log[Log[3]]^3*Defer[Int][E^(E^x + E^(E^x - x^6) - x^6)*x, x] - 6*Log[Log[3]]^2*Defer[Int][E^(E^x +
E^(E^x - x^6) - x^6)*x^2, x] - 6*Log[Log[3]]*Defer[Int][E^(E^x + E^(E^x - x^6) - x^6)*x^3, x] - 6*Defer[Int][E
^(E^x + E^(E^x - x^6) - x^6)*x^4, x] - Defer[Int][E^E^(E^x - x^6)/(x - Log[Log[3]])^2, x] - 6*Log[Log[3]]^5*De
fer[Int][E^(E^x + E^(E^x - x^6) - x^6)/(x - Log[Log[3]]), x] + Defer[Int][E^(E^x + E^(E^x - x^6) + x - x^6)/(x
- Log[Log[3]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x^6} \left (e^{x^6} x^2+e^{x^6} (1-2 x) \log (\log (3))+e^{e^{e^x-x^6}} \left (-e^{x^6}+e^{e^x} \left (e^x x-6 x^6+\left (-e^x+6 x^5\right ) \log (\log (3))\right )\right )\right )}{(x-\log (\log (3)))^2} \, dx\\ &=\int \left (\frac {e^{e^x+e^{e^x-x^6}-x^6} \left (e^x-6 x^5\right )}{x-\log (\log (3))}+\frac {-e^{e^{e^x-x^6}}+x^2+\log (\log (3))-2 x \log (\log (3))}{(x-\log (\log (3)))^2}\right ) \, dx\\ &=\int \frac {e^{e^x+e^{e^x-x^6}-x^6} \left (e^x-6 x^5\right )}{x-\log (\log (3))} \, dx+\int \frac {-e^{e^{e^x-x^6}}+x^2+\log (\log (3))-2 x \log (\log (3))}{(x-\log (\log (3)))^2} \, dx\\ &=\int \left (\frac {e^{e^x+e^{e^x-x^6}+x-x^6}}{x-\log (\log (3))}-\frac {6 e^{e^x+e^{e^x-x^6}-x^6} x^5}{x-\log (\log (3))}\right ) \, dx+\int \left (-\frac {e^{e^{e^x-x^6}}}{(x-\log (\log (3)))^2}+\frac {x^2+\log (\log (3))-2 x \log (\log (3))}{(x-\log (\log (3)))^2}\right ) \, dx\\ &=-\left (6 \int \frac {e^{e^x+e^{e^x-x^6}-x^6} x^5}{x-\log (\log (3))} \, dx\right )-\int \frac {e^{e^{e^x-x^6}}}{(x-\log (\log (3)))^2} \, dx+\int \frac {e^{e^x+e^{e^x-x^6}+x-x^6}}{x-\log (\log (3))} \, dx+\int \frac {x^2+\log (\log (3))-2 x \log (\log (3))}{(x-\log (\log (3)))^2} \, dx\\ &=-\left (6 \int \left (e^{e^x+e^{e^x-x^6}-x^6} x^4+e^{e^x+e^{e^x-x^6}-x^6} x^3 \log (\log (3))+e^{e^x+e^{e^x-x^6}-x^6} x^2 \log ^2(\log (3))+e^{e^x+e^{e^x-x^6}-x^6} x \log ^3(\log (3))+e^{e^x+e^{e^x-x^6}-x^6} \log ^4(\log (3))+\frac {e^{e^x+e^{e^x-x^6}-x^6} \log ^5(\log (3))}{x-\log (\log (3))}\right ) \, dx\right )-\int \frac {e^{e^{e^x-x^6}}}{(x-\log (\log (3)))^2} \, dx+\int \frac {e^{e^x+e^{e^x-x^6}+x-x^6}}{x-\log (\log (3))} \, dx+\int \left (1-\frac {(-1+\log (\log (3))) \log (\log (3))}{(x-\log (\log (3)))^2}\right ) \, dx\\ &=x-\frac {(1-\log (\log (3))) \log (\log (3))}{x-\log (\log (3))}-6 \int e^{e^x+e^{e^x-x^6}-x^6} x^4 \, dx-(6 \log (\log (3))) \int e^{e^x+e^{e^x-x^6}-x^6} x^3 \, dx-\left (6 \log ^2(\log (3))\right ) \int e^{e^x+e^{e^x-x^6}-x^6} x^2 \, dx-\left (6 \log ^3(\log (3))\right ) \int e^{e^x+e^{e^x-x^6}-x^6} x \, dx-\left (6 \log ^4(\log (3))\right ) \int e^{e^x+e^{e^x-x^6}-x^6} \, dx-\left (6 \log ^5(\log (3))\right ) \int \frac {e^{e^x+e^{e^x-x^6}-x^6}}{x-\log (\log (3))} \, dx-\int \frac {e^{e^{e^x-x^6}}}{(x-\log (\log (3)))^2} \, dx+\int \frac {e^{e^x+e^{e^x-x^6}+x-x^6}}{x-\log (\log (3))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 42, normalized size = 1.40 \begin {gather*} \frac {e^{e^{e^x-x^6}}+x^2-x \log (\log (3))+(-1+\log (\log (3))) \log (\log (3))}{x-\log (\log (3))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x^6*x^2 + E^x^6*(1 - 2*x)*Log[Log[3]] + E^E^(E^x - x^6)*(-E^x^6 + E^E^x*(E^x*x - 6*x^6 + (-E^x +
6*x^5)*Log[Log[3]])))/(E^x^6*x^2 - 2*E^x^6*x*Log[Log[3]] + E^x^6*Log[Log[3]]^2),x]

[Out]

(E^E^(E^x - x^6) + x^2 - x*Log[Log[3]] + (-1 + Log[Log[3]])*Log[Log[3]])/(x - Log[Log[3]])

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fricas [A]  time = 0.61, size = 37, normalized size = 1.23 \begin {gather*} \frac {x^{2} - {\left (x + 1\right )} \log \left (\log \relax (3)\right ) + \log \left (\log \relax (3)\right )^{2} + e^{\left (e^{\left (-x^{6} + e^{x}\right )}\right )}}{x - \log \left (\log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-exp(x)+6*x^5)*log(log(3))+exp(x)*x-6*x^6)*exp(exp(x))-exp(x^6))*exp(exp(exp(x))/exp(x^6))+(1-2*
x)*exp(x^6)*log(log(3))+x^2*exp(x^6))/(exp(x^6)*log(log(3))^2-2*x*exp(x^6)*log(log(3))+x^2*exp(x^6)),x, algori
thm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} e^{\left (x^{6}\right )} - {\left (2 \, x - 1\right )} e^{\left (x^{6}\right )} \log \left (\log \relax (3)\right ) - {\left ({\left (6 \, x^{6} - x e^{x} - {\left (6 \, x^{5} - e^{x}\right )} \log \left (\log \relax (3)\right )\right )} e^{\left (e^{x}\right )} + e^{\left (x^{6}\right )}\right )} e^{\left (e^{\left (-x^{6} + e^{x}\right )}\right )}}{x^{2} e^{\left (x^{6}\right )} - 2 \, x e^{\left (x^{6}\right )} \log \left (\log \relax (3)\right ) + e^{\left (x^{6}\right )} \log \left (\log \relax (3)\right )^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-exp(x)+6*x^5)*log(log(3))+exp(x)*x-6*x^6)*exp(exp(x))-exp(x^6))*exp(exp(exp(x))/exp(x^6))+(1-2*
x)*exp(x^6)*log(log(3))+x^2*exp(x^6))/(exp(x^6)*log(log(3))^2-2*x*exp(x^6)*log(log(3))+x^2*exp(x^6)),x, algori
thm="giac")

[Out]

integrate((x^2*e^(x^6) - (2*x - 1)*e^(x^6)*log(log(3)) - ((6*x^6 - x*e^x - (6*x^5 - e^x)*log(log(3)))*e^(e^x)
+ e^(x^6))*e^(e^(-x^6 + e^x)))/(x^2*e^(x^6) - 2*x*e^(x^6)*log(log(3)) + e^(x^6)*log(log(3))^2), x)

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maple [A]  time = 0.09, size = 53, normalized size = 1.77

method	result	size
risch	\(x -\frac {\ln \left (\ln \relax (3)\right )^{2}}{\ln \left (\ln \relax (3)\right )-x}+\frac {\ln \left (\ln \relax (3)\right )}{\ln \left (\ln \relax (3)\right )-x}-\frac {{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}-x^{6}}}}{\ln \left (\ln \relax (3)\right )-x}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.51, size = 104, normalized size = 3.47 \begin {gather*} 2 \, {\left (\frac {\log \left (\log \relax (3)\right )}{x - \log \left (\log \relax (3)\right )} - \log \left (x - \log \left (\log \relax (3)\right )\right )\right )} \log \left (\log \relax (3)\right ) + 2 \, \log \left (x - \log \left (\log \relax (3)\right )\right ) \log \left (\log \relax (3)\right ) + \frac {x^{2} - x \log \left (\log \relax (3)\right ) - \log \left (\log \relax (3)\right )^{2}}{x - \log \left (\log \relax (3)\right )} + \frac {e^{\left (e^{\left (-x^{6} + e^{x}\right )}\right )}}{x - \log \left (\log \relax (3)\right )} - \frac {\log \left (\log \relax (3)\right )}{x - \log \left (\log \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-exp(x)+6*x^5)*log(log(3))+exp(x)*x-6*x^6)*exp(exp(x))-exp(x^6))*exp(exp(exp(x))/exp(x^6))+(1-2*
x)*exp(x^6)*log(log(3))+x^2*exp(x^6))/(exp(x^6)*log(log(3))^2-2*x*exp(x^6)*log(log(3))+x^2*exp(x^6)),x, algori
thm="maxima")

[Out]

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

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mupad [B]  time = 6.33, size = 41, normalized size = 1.37 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-x^6}}-\ln \left (\ln \relax (3)\right )+{\ln \left (\ln \relax (3)\right )}^2-x\,\ln \left (\ln \relax (3)\right )+x^2}{x-\ln \left (\ln \relax (3)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(exp(x))*exp(-x^6))*(exp(x^6) + exp(exp(x))*(log(log(3))*(exp(x) - 6*x^5) - x*exp(x) + 6*x^6)) -
x^2*exp(x^6) + exp(x^6)*log(log(3))*(2*x - 1))/(x^2*exp(x^6) + exp(x^6)*log(log(3))^2 - 2*x*exp(x^6)*log(log(3
))),x)

[Out]

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

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sympy [A]  time = 0.81, size = 37, normalized size = 1.23 \begin {gather*} x + \frac {e^{e^{- x^{6}} e^{e^{x}}}}{x - \log {\left (\log {\relax (3 )} \right )}} + \frac {- \log {\left (\log {\relax (3 )} \right )} + \log {\left (\log {\relax (3 )} \right )}^{2}}{x - \log {\left (\log {\relax (3 )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-exp(x)+6*x**5)*ln(ln(3))+exp(x)*x-6*x**6)*exp(exp(x))-exp(x**6))*exp(exp(exp(x))/exp(x**6))+(1-
2*x)*exp(x**6)*ln(ln(3))+x**2*exp(x**6))/(exp(x**6)*ln(ln(3))**2-2*x*exp(x**6)*ln(ln(3))+x**2*exp(x**6)),x)

[Out]

x + exp(exp(-x**6)*exp(exp(x)))/(x - log(log(3))) + (-log(log(3)) + log(log(3))**2)/(x - log(log(3)))

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