3.37.74 \(\int \frac {e^{144 x^2-72 x^3+9 x^4} (-1-1152 x+1152 x^2-360 x^3+36 x^4)}{32-16 x+2 x^2+(16-8 x+x^2) \log (\log (4))} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^{9 (-4+x)^2 x^2}}{(-4+x) (2+\log (\log (4)))} \]

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Rubi [B]  time = 0.25, antiderivative size = 77, normalized size of antiderivative = 3.08, number of steps used = 1, number of rules used = 1, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2288} \begin {gather*} -\frac {e^{9 x^4-72 x^3+144 x^2} \left (-x^4+10 x^3-32 x^2+32 x\right )}{\left (x^3-6 x^2+8 x\right ) \left (2 x^2+\left (x^2-8 x+16\right ) \log (\log (4))-16 x+32\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(144*x^2 - 72*x^3 + 9*x^4)*(-1 - 1152*x + 1152*x^2 - 360*x^3 + 36*x^4))/(32 - 16*x + 2*x^2 + (16 - 8*x
+ x^2)*Log[Log[4]]),x]

[Out]

-((E^(144*x^2 - 72*x^3 + 9*x^4)*(32*x - 32*x^2 + 10*x^3 - x^4))/((8*x - 6*x^2 + x^3)*(32 - 16*x + 2*x^2 + (16
- 8*x + x^2)*Log[Log[4]])))

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^{144 x^2-72 x^3+9 x^4} \left (32 x-32 x^2+10 x^3-x^4\right )}{\left (8 x-6 x^2+x^3\right ) \left (32-16 x+2 x^2+\left (16-8 x+x^2\right ) \log (\log (4))\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 25, normalized size = 1.00 \begin {gather*} \frac {e^{9 (-4+x)^2 x^2}}{(-4+x) (2+\log (\log (4)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(144*x^2 - 72*x^3 + 9*x^4)*(-1 - 1152*x + 1152*x^2 - 360*x^3 + 36*x^4))/(32 - 16*x + 2*x^2 + (16
- 8*x + x^2)*Log[Log[4]]),x]

[Out]

E^(9*(-4 + x)^2*x^2)/((-4 + x)*(2 + Log[Log[4]]))

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fricas [A]  time = 0.76, size = 34, normalized size = 1.36 \begin {gather*} \frac {e^{\left (9 \, x^{4} - 72 \, x^{3} + 144 \, x^{2}\right )}}{{\left (x - 4\right )} \log \left (2 \, \log \relax (2)\right ) + 2 \, x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^4-360*x^3+1152*x^2-1152*x-1)*exp(9*x^4-72*x^3+144*x^2)/((x^2-8*x+16)*log(2*log(2))+2*x^2-16*x+
32),x, algorithm="fricas")

[Out]

e^(9*x^4 - 72*x^3 + 144*x^2)/((x - 4)*log(2*log(2)) + 2*x - 8)

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giac [A]  time = 0.22, size = 43, normalized size = 1.72 \begin {gather*} \frac {e^{\left (9 \, x^{4} - 72 \, x^{3} + 144 \, x^{2}\right )}}{x \log \relax (2) + x \log \left (\log \relax (2)\right ) + 2 \, x - 4 \, \log \relax (2) - 4 \, \log \left (\log \relax (2)\right ) - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^4-360*x^3+1152*x^2-1152*x-1)*exp(9*x^4-72*x^3+144*x^2)/((x^2-8*x+16)*log(2*log(2))+2*x^2-16*x+
32),x, algorithm="giac")

[Out]

e^(9*x^4 - 72*x^3 + 144*x^2)/(x*log(2) + x*log(log(2)) + 2*x - 4*log(2) - 4*log(log(2)) - 8)

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maple [A]  time = 0.15, size = 27, normalized size = 1.08

method	result	size
risch	\(\frac {{\mathrm e}^{9 \left (x -4\right )^{2} x^{2}}}{\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )+2\right ) \left (x -4\right )}\)	\(27\)
norman	\(\frac {{\mathrm e}^{9 x^{4}-72 x^{3}+144 x^{2}}}{\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )+2\right ) \left (x -4\right )}\)	\(33\)
gosper	\(\frac {{\mathrm e}^{9 x^{4}-72 x^{3}+144 x^{2}}}{x \ln \left (2 \ln \relax (2)\right )-4 \ln \left (2 \ln \relax (2)\right )+2 x -8}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*x^4-360*x^3+1152*x^2-1152*x-1)*exp(9*x^4-72*x^3+144*x^2)/((x^2-8*x+16)*ln(2*ln(2))+2*x^2-16*x+32),x,me
thod=_RETURNVERBOSE)

[Out]

1/(ln(2)+ln(ln(2))+2)*exp(9*(x-4)^2*x^2)/(x-4)

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maxima [A]  time = 0.60, size = 40, normalized size = 1.60 \begin {gather*} \frac {e^{\left (9 \, x^{4} - 72 \, x^{3} + 144 \, x^{2}\right )}}{x {\left (\log \relax (2) + \log \left (\log \relax (2)\right ) + 2\right )} - 4 \, \log \relax (2) - 4 \, \log \left (\log \relax (2)\right ) - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^4-360*x^3+1152*x^2-1152*x-1)*exp(9*x^4-72*x^3+144*x^2)/((x^2-8*x+16)*log(2*log(2))+2*x^2-16*x+
32),x, algorithm="maxima")

[Out]

e^(9*x^4 - 72*x^3 + 144*x^2)/(x*(log(2) + log(log(2)) + 2) - 4*log(2) - 4*log(log(2)) - 8)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{9\,x^4-72\,x^3+144\,x^2}\,\left (-36\,x^4+360\,x^3-1152\,x^2+1152\,x+1\right )}{\ln \left (2\,\ln \relax (2)\right )\,\left (x^2-8\,x+16\right )-16\,x+2\,x^2+32} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(144*x^2 - 72*x^3 + 9*x^4)*(1152*x - 1152*x^2 + 360*x^3 - 36*x^4 + 1))/(log(2*log(2))*(x^2 - 8*x + 16
) - 16*x + 2*x^2 + 32),x)

[Out]

int(-(exp(144*x^2 - 72*x^3 + 9*x^4)*(1152*x - 1152*x^2 + 360*x^3 - 36*x^4 + 1))/(log(2*log(2))*(x^2 - 8*x + 16
) - 16*x + 2*x^2 + 32), x)

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sympy [A]  time = 0.22, size = 44, normalized size = 1.76 \begin {gather*} \frac {e^{9 x^{4} - 72 x^{3} + 144 x^{2}}}{x \log {\left (\log {\relax (2 )} \right )} + x \log {\relax (2 )} + 2 x - 8 - 4 \log {\relax (2 )} - 4 \log {\left (\log {\relax (2 )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x**4-360*x**3+1152*x**2-1152*x-1)*exp(9*x**4-72*x**3+144*x**2)/((x**2-8*x+16)*ln(2*ln(2))+2*x**2
-16*x+32),x)

[Out]

exp(9*x**4 - 72*x**3 + 144*x**2)/(x*log(log(2)) + x*log(2) + 2*x - 8 - 4*log(2) - 4*log(log(2)))

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