3.34.66 \(\int \frac {e^{200+40 x+2 x^2-2 x^3} (-2+(40 x+4 x^2-6 x^3) \log (x) \log (\log (x)))}{x \log (x) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=21 \[ \frac {e^{-2 x^3+2 (10+x)^2}}{\log ^2(\log (x))} \]

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Rubi [B]  time = 0.42, antiderivative size = 52, normalized size of antiderivative = 2.48, number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2288} \begin {gather*} \frac {e^{-2 x^3+2 x^2+40 x+200} \left (-3 x^3+2 x^2+20 x\right )}{x \left (-3 x^2+2 x+20\right ) \log ^2(\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(200 + 40*x + 2*x^2 - 2*x^3)*(-2 + (40*x + 4*x^2 - 6*x^3)*Log[x]*Log[Log[x]]))/(x*Log[x]*Log[Log[x]]^3)
,x]

[Out]

(E^(200 + 40*x + 2*x^2 - 2*x^3)*(20*x + 2*x^2 - 3*x^3))/(x*(20 + 2*x - 3*x^2)*Log[Log[x]]^2)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{200+40 x+2 x^2-2 x^3} \left (20 x+2 x^2-3 x^3\right )}{x \left (20+2 x-3 x^2\right ) \log ^2(\log (x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 23, normalized size = 1.10 \begin {gather*} \frac {e^{200+40 x+2 x^2-2 x^3}}{\log ^2(\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(200 + 40*x + 2*x^2 - 2*x^3)*(-2 + (40*x + 4*x^2 - 6*x^3)*Log[x]*Log[Log[x]]))/(x*Log[x]*Log[Log[
x]]^3),x]

[Out]

E^(200 + 40*x + 2*x^2 - 2*x^3)/Log[Log[x]]^2

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fricas [A]  time = 0.57, size = 22, normalized size = 1.05 \begin {gather*} e^{\left (-2 \, x^{3} + 2 \, x^{2} + 40 \, x - 2 \, \log \left (\log \left (\log \relax (x)\right )\right ) + 200\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3+4*x^2+40*x)*log(x)*log(log(x))-2)*exp(-log(log(log(x)))-x^3+x^2+20*x+100)^2/x/log(x)/log(lo
g(x)),x, algorithm="fricas")

[Out]

e^(-2*x^3 + 2*x^2 + 40*x - 2*log(log(log(x))) + 200)

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giac [A]  time = 0.21, size = 22, normalized size = 1.05 \begin {gather*} e^{\left (-2 \, x^{3} + 2 \, x^{2} + 40 \, x - 2 \, \log \left (\log \left (\log \relax (x)\right )\right ) + 200\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3+4*x^2+40*x)*log(x)*log(log(x))-2)*exp(-log(log(log(x)))-x^3+x^2+20*x+100)^2/x/log(x)/log(lo
g(x)),x, algorithm="giac")

[Out]

e^(-2*x^3 + 2*x^2 + 40*x - 2*log(log(log(x))) + 200)

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maple [A]  time = 0.03, size = 23, normalized size = 1.10

method	result	size
risch	\(\frac {{\mathrm e}^{-2 x^{3}+2 x^{2}+40 x +200}}{\ln \left (\ln \relax (x )\right )^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^3+4*x^2+40*x)*ln(x)*ln(ln(x))-2)*exp(-ln(ln(ln(x)))-x^3+x^2+20*x+100)^2/x/ln(x)/ln(ln(x)),x,method=
_RETURNVERBOSE)

[Out]

1/ln(ln(x))^2*exp(-2*x^3+2*x^2+40*x+200)

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maxima [A]  time = 0.65, size = 22, normalized size = 1.05 \begin {gather*} \frac {e^{\left (-2 \, x^{3} + 2 \, x^{2} + 40 \, x + 200\right )}}{\log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3+4*x^2+40*x)*log(x)*log(log(x))-2)*exp(-log(log(log(x)))-x^3+x^2+20*x+100)^2/x/log(x)/log(lo
g(x)),x, algorithm="maxima")

[Out]

e^(-2*x^3 + 2*x^2 + 40*x + 200)/log(log(x))^2

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mupad [B]  time = 2.08, size = 24, normalized size = 1.14 \begin {gather*} \frac {{\mathrm {e}}^{40\,x}\,{\mathrm {e}}^{200}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{-2\,x^3}}{{\ln \left (\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(40*x - 2*log(log(log(x))) + 2*x^2 - 2*x^3 + 200)*(log(log(x))*log(x)*(40*x + 4*x^2 - 6*x^3) - 2))/(x*
log(log(x))*log(x)),x)

[Out]

(exp(40*x)*exp(200)*exp(2*x^2)*exp(-2*x^3))/log(log(x))^2

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sympy [A]  time = 0.38, size = 22, normalized size = 1.05 \begin {gather*} \frac {e^{- 2 x^{3} + 2 x^{2} + 40 x + 200}}{\log {\left (\log {\relax (x )} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**3+4*x**2+40*x)*ln(x)*ln(ln(x))-2)*exp(-ln(ln(ln(x)))-x**3+x**2+20*x+100)**2/x/ln(x)/ln(ln(x)
),x)

[Out]

exp(-2*x**3 + 2*x**2 + 40*x + 200)/log(log(x))**2

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