3.53.33 \(\int \frac {-2-6 x+2 x \log (x)+e^{e^{x^2}-x} (-93 x+93 x^2-186 e^{x^2} x^3+(31 x-31 x^2+62 e^{x^2} x^3) \log (x))}{-6 x+2 x \log (x)} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {31}{2} e^{e^{x^2}-x} x-\log (5 (-3+\log (x))) \]

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Rubi [F]  time = 7.37, 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 {-2-6 x+2 x \log (x)+e^{e^{x^2}-x} \left (-93 x+93 x^2-186 e^{x^2} x^3+\left (31 x-31 x^2+62 e^{x^2} x^3\right ) \log (x)\right )}{-6 x+2 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 - 6*x + 2*x*Log[x] + E^(E^x^2 - x)*(-93*x + 93*x^2 - 186*E^x^2*x^3 + (31*x - 31*x^2 + 62*E^x^2*x^3)*Lo
g[x]))/(-6*x + 2*x*Log[x]),x]

[Out]

x - Log[3 - Log[x]] + (31*Defer[Int][E^(E^x^2 - x), x])/2 - (31*Defer[Int][E^(E^x^2 - x)*x, x])/2 + 31*Defer[I
nt][E^(E^x^2 + (-1 + x)*x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2-6 x+2 x \log (x)+e^{e^{x^2}-x} \left (-93 x+93 x^2-186 e^{x^2} x^3+\left (31 x-31 x^2+62 e^{x^2} x^3\right ) \log (x)\right )}{x (-6+2 \log (x))} \, dx\\ &=\int \frac {2+6 x-2 x \log (x)-e^{e^{x^2}-x} \left (-93 x+93 x^2-186 e^{x^2} x^3+\left (31 x-31 x^2+62 e^{x^2} x^3\right ) \log (x)\right )}{2 x (3-\log (x))} \, dx\\ &=\frac {1}{2} \int \frac {2+6 x-2 x \log (x)-e^{e^{x^2}-x} \left (-93 x+93 x^2-186 e^{x^2} x^3+\left (31 x-31 x^2+62 e^{x^2} x^3\right ) \log (x)\right )}{x (3-\log (x))} \, dx\\ &=\frac {1}{2} \int \frac {2+6 x-31 e^{e^{x^2}-x} x \left (1-x+2 e^{x^2} x^2\right ) (-3+\log (x))-2 x \log (x)}{x (3-\log (x))} \, dx\\ &=\frac {1}{2} \int \left (62 e^{e^{x^2}+(-1+x) x} x^2+\frac {e^{-x} \left (-2 e^x-93 e^{e^{x^2}} x-6 e^x x+93 e^{e^{x^2}} x^2+31 e^{e^{x^2}} x \log (x)+2 e^x x \log (x)-31 e^{e^{x^2}} x^2 \log (x)\right )}{x (-3+\log (x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-x} \left (-2 e^x-93 e^{e^{x^2}} x-6 e^x x+93 e^{e^{x^2}} x^2+31 e^{e^{x^2}} x \log (x)+2 e^x x \log (x)-31 e^{e^{x^2}} x^2 \log (x)\right )}{x (-3+\log (x))} \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx\\ &=\frac {1}{2} \int \frac {e^{-x} \left (-93 e^{e^{x^2}} (-1+x) x+2 e^x (1+3 x)-2 e^x x \log (x)+31 e^{e^{x^2}} (-1+x) x \log (x)\right )}{x (3-\log (x))} \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx\\ &=\frac {1}{2} \int \left (-31 e^{e^{x^2}-x} (-1+x)+\frac {2 (-1-3 x+x \log (x))}{x (-3+\log (x))}\right ) \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx\\ &=-\left (\frac {31}{2} \int e^{e^{x^2}-x} (-1+x) \, dx\right )+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx+\int \frac {-1-3 x+x \log (x)}{x (-3+\log (x))} \, dx\\ &=-\left (\frac {31}{2} \int \left (-e^{e^{x^2}-x}+e^{e^{x^2}-x} x\right ) \, dx\right )+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx+\int \left (1-\frac {1}{x (-3+\log (x))}\right ) \, dx\\ &=x+\frac {31}{2} \int e^{e^{x^2}-x} \, dx-\frac {31}{2} \int e^{e^{x^2}-x} x \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx-\int \frac {1}{x (-3+\log (x))} \, dx\\ &=x+\frac {31}{2} \int e^{e^{x^2}-x} \, dx-\frac {31}{2} \int e^{e^{x^2}-x} x \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,-3+\log (x)\right )\\ &=x-\log (3-\log (x))+\frac {31}{2} \int e^{e^{x^2}-x} \, dx-\frac {31}{2} \int e^{e^{x^2}-x} x \, dx+31 \int e^{e^{x^2}+(-1+x) x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{2} \left (2 x+31 e^{e^{x^2}-x} x-2 \log (3-\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 6*x + 2*x*Log[x] + E^(E^x^2 - x)*(-93*x + 93*x^2 - 186*E^x^2*x^3 + (31*x - 31*x^2 + 62*E^x^2*x
^3)*Log[x]))/(-6*x + 2*x*Log[x]),x]

[Out]

(2*x + 31*E^(E^x^2 - x)*x - 2*Log[3 - Log[x]])/2

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fricas [A]  time = 0.55, size = 21, normalized size = 0.78 \begin {gather*} \frac {31}{2} \, x e^{\left (-x + e^{\left (x^{2}\right )}\right )} + x - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((62*x^3*exp(x^2)-31*x^2+31*x)*log(x)-186*x^3*exp(x^2)+93*x^2-93*x)*exp(exp(x^2)-x)+2*x*log(x)-6*x-
2)/(2*x*log(x)-6*x),x, algorithm="fricas")

[Out]

31/2*x*e^(-x + e^(x^2)) + x - log(log(x) - 3)

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giac [A]  time = 0.14, size = 21, normalized size = 0.78 \begin {gather*} \frac {31}{2} \, x e^{\left (-x + e^{\left (x^{2}\right )}\right )} + x - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((62*x^3*exp(x^2)-31*x^2+31*x)*log(x)-186*x^3*exp(x^2)+93*x^2-93*x)*exp(exp(x^2)-x)+2*x*log(x)-6*x-
2)/(2*x*log(x)-6*x),x, algorithm="giac")

[Out]

31/2*x*e^(-x + e^(x^2)) + x - log(log(x) - 3)

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maple [A]  time = 0.06, size = 22, normalized size = 0.81

method	result	size
risch	\(x -\ln \left (\ln \relax (x )-3\right )+\frac {31 x \,{\mathrm e}^{{\mathrm e}^{x^{2}}-x}}{2}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((62*x^3*exp(x^2)-31*x^2+31*x)*ln(x)-186*x^3*exp(x^2)+93*x^2-93*x)*exp(exp(x^2)-x)+2*x*ln(x)-6*x-2)/(2*x*
ln(x)-6*x),x,method=_RETURNVERBOSE)

[Out]

x-ln(ln(x)-3)+31/2*x*exp(exp(x^2)-x)

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maxima [A]  time = 0.37, size = 28, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, {\left (2 \, x e^{x} + 31 \, x e^{\left (e^{\left (x^{2}\right )}\right )}\right )} e^{\left (-x\right )} - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((62*x^3*exp(x^2)-31*x^2+31*x)*log(x)-186*x^3*exp(x^2)+93*x^2-93*x)*exp(exp(x^2)-x)+2*x*log(x)-6*x-
2)/(2*x*log(x)-6*x),x, algorithm="maxima")

[Out]

1/2*(2*x*e^x + 31*x*e^(e^(x^2)))*e^(-x) - log(log(x) - 3)

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mupad [B]  time = 3.42, size = 21, normalized size = 0.78 \begin {gather*} x-\ln \left (\ln \relax (x)-3\right )+\frac {31\,x\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}-x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(exp(x^2) - x)*(93*x - log(x)*(31*x + 62*x^3*exp(x^2) - 31*x^2) + 186*x^3*exp(x^2) - 93*x^2) - 2
*x*log(x) + 2)/(6*x - 2*x*log(x)),x)

[Out]

x - log(log(x) - 3) + (31*x*exp(exp(x^2) - x))/2

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sympy [A]  time = 62.88, size = 20, normalized size = 0.74 \begin {gather*} \frac {31 x e^{- x + e^{x^{2}}}}{2} + x - \log {\left (\log {\relax (x )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((62*x**3*exp(x**2)-31*x**2+31*x)*ln(x)-186*x**3*exp(x**2)+93*x**2-93*x)*exp(exp(x**2)-x)+2*x*ln(x)
-6*x-2)/(2*x*ln(x)-6*x),x)

[Out]

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

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