3.78.64 \(\int e^{-40 x^4+20 x^5+(-8 x^3+4 x^4) \log (2)-4 \log ^2(2)} (-160 x^3+100 x^4+(-24 x^2+16 x^3) \log (2)) \, dx\)

Optimal. Leaf size=26 \[ -3+e^{4 \left (-\log ^2(2)+(-2+x) x^3 (5 x+\log (2))\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 1, number of rules used = 1, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6706} \begin {gather*} 2^{4 x^4-8 x^3} e^{20 x^5-40 x^4-4 \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-40*x^4 + 20*x^5 + (-8*x^3 + 4*x^4)*Log[2] - 4*Log[2]^2)*(-160*x^3 + 100*x^4 + (-24*x^2 + 16*x^3)*Log[2
]),x]

[Out]

2^(-8*x^3 + 4*x^4)*E^(-40*x^4 + 20*x^5 - 4*Log[2]^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} &=2^{-8 x^3+4 x^4} e^{-40 x^4+20 x^5-4 \log ^2(2)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.58, size = 30, normalized size = 1.15 \begin {gather*} 2^{4 (-2+x) x^3} e^{-40 x^4+20 x^5-4 \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-40*x^4 + 20*x^5 + (-8*x^3 + 4*x^4)*Log[2] - 4*Log[2]^2)*(-160*x^3 + 100*x^4 + (-24*x^2 + 16*x^3)
*Log[2]),x]

[Out]

2^(4*(-2 + x)*x^3)*E^(-40*x^4 + 20*x^5 - 4*Log[2]^2)

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 31, normalized size = 1.19 \begin {gather*} e^{\left (20 \, x^{5} - 40 \, x^{4} + 4 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-24*x^2)*log(2)+100*x^4-160*x^3)*exp(-4*log(2)^2+(4*x^4-8*x^3)*log(2)+20*x^5-40*x^4),x, algo
rithm="fricas")

[Out]

e^(20*x^5 - 40*x^4 + 4*(x^4 - 2*x^3)*log(2) - 4*log(2)^2)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 32, normalized size = 1.23 \begin {gather*} e^{\left (20 \, x^{5} + 4 \, x^{4} \log \relax (2) - 40 \, x^{4} - 8 \, x^{3} \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-24*x^2)*log(2)+100*x^4-160*x^3)*exp(-4*log(2)^2+(4*x^4-8*x^3)*log(2)+20*x^5-40*x^4),x, algo
rithm="giac")

[Out]

e^(20*x^5 + 4*x^4*log(2) - 40*x^4 - 8*x^3*log(2) - 4*log(2)^2)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 29, normalized size = 1.12

method	result	size
risch	\(16^{\left (x -2\right ) x^{3}} {\mathrm e}^{-4 \ln \relax (2)^{2}+20 x^{5}-40 x^{4}}\)	\(29\)
gosper	\({\mathrm e}^{4 x^{4} \ln \relax (2)+20 x^{5}-8 x^{3} \ln \relax (2)-40 x^{4}-4 \ln \relax (2)^{2}}\)	\(33\)
derivativedivides	\({\mathrm e}^{-4 \ln \relax (2)^{2}+\left (4 x^{4}-8 x^{3}\right ) \ln \relax (2)+20 x^{5}-40 x^{4}}\)	\(33\)
default	\({\mathrm e}^{-4 \ln \relax (2)^{2}+\left (4 x^{4}-8 x^{3}\right ) \ln \relax (2)+20 x^{5}-40 x^{4}}\)	\(33\)
norman	\({\mathrm e}^{-4 \ln \relax (2)^{2}+\left (4 x^{4}-8 x^{3}\right ) \ln \relax (2)+20 x^{5}-40 x^{4}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^3-24*x^2)*ln(2)+100*x^4-160*x^3)*exp(-4*ln(2)^2+(4*x^4-8*x^3)*ln(2)+20*x^5-40*x^4),x,method=_RETURN
VERBOSE)

[Out]

16^((x-2)*x^3)*exp(-4*ln(2)^2+20*x^5-40*x^4)

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 32, normalized size = 1.23 \begin {gather*} e^{\left (20 \, x^{5} + 4 \, x^{4} \log \relax (2) - 40 \, x^{4} - 8 \, x^{3} \log \relax (2) - 4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-24*x^2)*log(2)+100*x^4-160*x^3)*exp(-4*log(2)^2+(4*x^4-8*x^3)*log(2)+20*x^5-40*x^4),x, algo
rithm="maxima")

[Out]

e^(20*x^5 + 4*x^4*log(2) - 40*x^4 - 8*x^3*log(2) - 4*log(2)^2)

________________________________________________________________________________________

mupad [B]  time = 7.35, size = 36, normalized size = 1.38 \begin {gather*} \frac {2^{4\,x^4}\,{\mathrm {e}}^{-4\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{20\,x^5}\,{\mathrm {e}}^{-40\,x^4}}{2^{8\,x^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(20*x^5 - 4*log(2)^2 - 40*x^4 - log(2)*(8*x^3 - 4*x^4))*(log(2)*(24*x^2 - 16*x^3) + 160*x^3 - 100*x^4)
,x)

[Out]

(2^(4*x^4)*exp(-4*log(2)^2)*exp(20*x^5)*exp(-40*x^4))/2^(8*x^3)

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 31, normalized size = 1.19 \begin {gather*} e^{20 x^{5} - 40 x^{4} + \left (4 x^{4} - 8 x^{3}\right ) \log {\relax (2 )} - 4 \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**3-24*x**2)*ln(2)+100*x**4-160*x**3)*exp(-4*ln(2)**2+(4*x**4-8*x**3)*ln(2)+20*x**5-40*x**4),x
)

[Out]

exp(20*x**5 - 40*x**4 + (4*x**4 - 8*x**3)*log(2) - 4*log(2)**2)

________________________________________________________________________________________