3.50.68 \(\int \frac {e^{\frac {256}{\log ^2(5-2 x-x^2+\log (x))}} (-512+1024 x+1024 x^2)}{(5 x-2 x^2-x^3+x \log (x)) \log ^3(5-2 x-x^2+\log (x))} \, dx\)

Optimal. Leaf size=19 \[ e^{\frac {256}{\log ^2\left (5-2 x-x^2+\log (x)\right )}} \]

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Rubi [A]  time = 0.32, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6706} \begin {gather*} e^{\frac {256}{\log ^2\left (-x^2-2 x+\log (x)+5\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(256/Log[5 - 2*x - x^2 + Log[x]]^2)*(-512 + 1024*x + 1024*x^2))/((5*x - 2*x^2 - x^3 + x*Log[x])*Log[5 -
2*x - x^2 + Log[x]]^3),x]

[Out]

E^(256/Log[5 - 2*x - x^2 + Log[x]]^2)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{\frac {256}{\log ^2\left (5-2 x-x^2+\log (x)\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 19, normalized size = 1.00 \begin {gather*} e^{\frac {256}{\log ^2\left (5-2 x-x^2+\log (x)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(256/Log[5 - 2*x - x^2 + Log[x]]^2)*(-512 + 1024*x + 1024*x^2))/((5*x - 2*x^2 - x^3 + x*Log[x])*L
og[5 - 2*x - x^2 + Log[x]]^3),x]

[Out]

E^(256/Log[5 - 2*x - x^2 + Log[x]]^2)

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fricas [A]  time = 0.75, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (\frac {256}{\log \left (-x^{2} - 2 \, x + \log \relax (x) + 5\right )^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^2+1024*x-512)*exp(256/log(log(x)-x^2-2*x+5)^2)/(x*log(x)-x^3-2*x^2+5*x)/log(log(x)-x^2-2*x+5
)^3,x, algorithm="fricas")

[Out]

e^(256/log(-x^2 - 2*x + log(x) + 5)^2)

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giac [A]  time = 0.27, size = 18, normalized size = 0.95 \begin {gather*} e^{\left (\frac {256}{\log \left (-x^{2} - 2 \, x + \log \relax (x) + 5\right )^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^2+1024*x-512)*exp(256/log(log(x)-x^2-2*x+5)^2)/(x*log(x)-x^3-2*x^2+5*x)/log(log(x)-x^2-2*x+5
)^3,x, algorithm="giac")

[Out]

e^(256/log(-x^2 - 2*x + log(x) + 5)^2)

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maple [A]  time = 0.02, size = 19, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {256}{\ln \left (\ln \relax (x )-x^{2}-2 x +5\right )^{2}}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1024*x^2+1024*x-512)*exp(256/ln(ln(x)-x^2-2*x+5)^2)/(x*ln(x)-x^3-2*x^2+5*x)/ln(ln(x)-x^2-2*x+5)^3,x,metho
d=_RETURNVERBOSE)

[Out]

exp(256/ln(ln(x)-x^2-2*x+5)^2)

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maxima [B]  time = 0.45, size = 101, normalized size = 5.32 \begin {gather*} \frac {2 \, x^{2} e^{\left (\frac {256}{\log \left (-x^{2} - 2 \, x + \log \relax (x) + 5\right )^{2}}\right )}}{2 \, x^{2} + 2 \, x - 1} + \frac {2 \, x e^{\left (\frac {256}{\log \left (-x^{2} - 2 \, x + \log \relax (x) + 5\right )^{2}}\right )}}{2 \, x^{2} + 2 \, x - 1} - \frac {e^{\left (\frac {256}{\log \left (-x^{2} - 2 \, x + \log \relax (x) + 5\right )^{2}}\right )}}{2 \, x^{2} + 2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x^2+1024*x-512)*exp(256/log(log(x)-x^2-2*x+5)^2)/(x*log(x)-x^3-2*x^2+5*x)/log(log(x)-x^2-2*x+5
)^3,x, algorithm="maxima")

[Out]

2*x^2*e^(256/log(-x^2 - 2*x + log(x) + 5)^2)/(2*x^2 + 2*x - 1) + 2*x*e^(256/log(-x^2 - 2*x + log(x) + 5)^2)/(2
*x^2 + 2*x - 1) - e^(256/log(-x^2 - 2*x + log(x) + 5)^2)/(2*x^2 + 2*x - 1)

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mupad [B]  time = 3.50, size = 18, normalized size = 0.95 \begin {gather*} {\mathrm {e}}^{\frac {256}{{\ln \left (\ln \relax (x)-2\,x-x^2+5\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(256/log(log(x) - 2*x - x^2 + 5)^2)*(1024*x + 1024*x^2 - 512))/(log(log(x) - 2*x - x^2 + 5)^3*(5*x + x
*log(x) - 2*x^2 - x^3)),x)

[Out]

exp(256/log(log(x) - 2*x - x^2 + 5)^2)

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sympy [A]  time = 2.22, size = 17, normalized size = 0.89 \begin {gather*} e^{\frac {256}{\log {\left (- x^{2} - 2 x + \log {\relax (x )} + 5 \right )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*x**2+1024*x-512)*exp(256/ln(ln(x)-x**2-2*x+5)**2)/(x*ln(x)-x**3-2*x**2+5*x)/ln(ln(x)-x**2-2*x+
5)**3,x)

[Out]

exp(256/log(-x**2 - 2*x + log(x) + 5)**2)

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