3.49.47 \(\int \frac {e^{\frac {(64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8) \log ^2(3)}{x^2}} (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8) \log ^2(3)}{x^3} \, dx\)

Optimal. Leaf size=29 \[ e^{\frac {4 (4-x)^2 \left (1+x^2 (2+x)\right )^2 \log ^2(3)}{x^2}} \]

________________________________________________________________________________________

Rubi [A]  time = 4.33, antiderivative size = 44, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {12, 6706} \begin {gather*} \exp \left (\frac {4 \left (x^8-4 x^7-12 x^6+34 x^5+52 x^4+65 x^2-8 x+16\right ) \log ^2(3)}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(((64 - 32*x + 260*x^2 + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^8)*Log[3]^2)/x^2)*(-128 + 32*x + 416
*x^4 + 408*x^5 - 192*x^6 - 80*x^7 + 24*x^8)*Log[3]^2)/x^3,x]

[Out]

E^((4*(16 - 8*x + 65*x^2 + 52*x^4 + 34*x^5 - 12*x^6 - 4*x^7 + x^8)*Log[3]^2)/x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\log ^2(3) \int \frac {\exp \left (\frac {\left (64-32 x+260 x^2+208 x^4+136 x^5-48 x^6-16 x^7+4 x^8\right ) \log ^2(3)}{x^2}\right ) \left (-128+32 x+416 x^4+408 x^5-192 x^6-80 x^7+24 x^8\right )}{x^3} \, dx\\ &=\exp \left (\frac {4 \left (16-8 x+65 x^2+52 x^4+34 x^5-12 x^6-4 x^7+x^8\right ) \log ^2(3)}{x^2}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 4.18, size = 29, normalized size = 1.00 \begin {gather*} e^{\frac {4 \left (-4+x-8 x^2-2 x^3+x^4\right )^2 \log ^2(3)}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(((64 - 32*x + 260*x^2 + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^8)*Log[3]^2)/x^2)*(-128 + 32*x
+ 416*x^4 + 408*x^5 - 192*x^6 - 80*x^7 + 24*x^8)*Log[3]^2)/x^3,x]

[Out]

E^((4*(-4 + x - 8*x^2 - 2*x^3 + x^4)^2*Log[3]^2)/x^2)

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 43, normalized size = 1.48 \begin {gather*} e^{\left (\frac {4 \, {\left (x^{8} - 4 \, x^{7} - 12 \, x^{6} + 34 \, x^{5} + 52 \, x^{4} + 65 \, x^{2} - 8 \, x + 16\right )} \log \relax (3)^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4*x^8-16*x^7-48*x^6+136*x^5+208*x^4+2
60*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(4*(x^8 - 4*x^7 - 12*x^6 + 34*x^5 + 52*x^4 + 65*x^2 - 8*x + 16)*log(3)^2/x^2)

________________________________________________________________________________________

giac [B]  time = 0.15, size = 71, normalized size = 2.45 \begin {gather*} e^{\left (4 \, x^{6} \log \relax (3)^{2} - 16 \, x^{5} \log \relax (3)^{2} - 48 \, x^{4} \log \relax (3)^{2} + 136 \, x^{3} \log \relax (3)^{2} + 208 \, x^{2} \log \relax (3)^{2} + 260 \, \log \relax (3)^{2} - \frac {32 \, \log \relax (3)^{2}}{x} + \frac {64 \, \log \relax (3)^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4*x^8-16*x^7-48*x^6+136*x^5+208*x^4+2
60*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, algorithm="giac")

[Out]

e^(4*x^6*log(3)^2 - 16*x^5*log(3)^2 - 48*x^4*log(3)^2 + 136*x^3*log(3)^2 + 208*x^2*log(3)^2 + 260*log(3)^2 - 3
2*log(3)^2/x + 64*log(3)^2/x^2)

________________________________________________________________________________________

maple [A]  time = 0.17, size = 28, normalized size = 0.97

method	result	size
risch	\({\mathrm e}^{\frac {4 \left (x -4\right )^{2} \left (x^{3}+2 x^{2}+1\right )^{2} \ln \relax (3)^{2}}{x^{2}}}\)	\(28\)
gosper	\({\mathrm e}^{\frac {4 \left (x^{8}-4 x^{7}-12 x^{6}+34 x^{5}+52 x^{4}+65 x^{2}-8 x +16\right ) \ln \relax (3)^{2}}{x^{2}}}\)	\(44\)
norman	\({\mathrm e}^{\frac {\left (4 x^{8}-16 x^{7}-48 x^{6}+136 x^{5}+208 x^{4}+260 x^{2}-32 x +64\right ) \ln \relax (3)^{2}}{x^{2}}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*ln(3)^2*exp((4*x^8-16*x^7-48*x^6+136*x^5+208*x^4+260*x^2-
32*x+64)*ln(3)^2/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(4*(x-4)^2*(x^3+2*x^2+1)^2*ln(3)^2/x^2)

________________________________________________________________________________________

maxima [B]  time = 0.68, size = 71, normalized size = 2.45 \begin {gather*} e^{\left (4 \, x^{6} \log \relax (3)^{2} - 16 \, x^{5} \log \relax (3)^{2} - 48 \, x^{4} \log \relax (3)^{2} + 136 \, x^{3} \log \relax (3)^{2} + 208 \, x^{2} \log \relax (3)^{2} + 260 \, \log \relax (3)^{2} - \frac {32 \, \log \relax (3)^{2}}{x} + \frac {64 \, \log \relax (3)^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^8-80*x^7-192*x^6+408*x^5+416*x^4+32*x-128)*log(3)^2*exp((4*x^8-16*x^7-48*x^6+136*x^5+208*x^4+2
60*x^2-32*x+64)*log(3)^2/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(4*x^6*log(3)^2 - 16*x^5*log(3)^2 - 48*x^4*log(3)^2 + 136*x^3*log(3)^2 + 208*x^2*log(3)^2 + 260*log(3)^2 - 3
2*log(3)^2/x + 64*log(3)^2/x^2)

________________________________________________________________________________________

mupad [B]  time = 3.62, size = 78, normalized size = 2.69 \begin {gather*} {\mathrm {e}}^{4\,x^6\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-16\,x^5\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{-\frac {32\,{\ln \relax (3)}^2}{x}}\,{\mathrm {e}}^{-48\,x^4\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{\frac {64\,{\ln \relax (3)}^2}{x^2}}\,{\mathrm {e}}^{136\,x^3\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{208\,x^2\,{\ln \relax (3)}^2}\,{\mathrm {e}}^{260\,{\ln \relax (3)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(3)^2*(260*x^2 - 32*x + 208*x^4 + 136*x^5 - 48*x^6 - 16*x^7 + 4*x^8 + 64))/x^2)*log(3)^2*(32*x +
416*x^4 + 408*x^5 - 192*x^6 - 80*x^7 + 24*x^8 - 128))/x^3,x)

[Out]

exp(4*x^6*log(3)^2)*exp(-16*x^5*log(3)^2)*exp(-(32*log(3)^2)/x)*exp(-48*x^4*log(3)^2)*exp((64*log(3)^2)/x^2)*e
xp(136*x^3*log(3)^2)*exp(208*x^2*log(3)^2)*exp(260*log(3)^2)

________________________________________________________________________________________

sympy [A]  time = 0.36, size = 44, normalized size = 1.52 \begin {gather*} e^{\frac {\left (4 x^{8} - 16 x^{7} - 48 x^{6} + 136 x^{5} + 208 x^{4} + 260 x^{2} - 32 x + 64\right ) \log {\relax (3 )}^{2}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x**8-80*x**7-192*x**6+408*x**5+416*x**4+32*x-128)*ln(3)**2*exp((4*x**8-16*x**7-48*x**6+136*x**5+
208*x**4+260*x**2-32*x+64)*ln(3)**2/x**2)/x**3,x)

[Out]

exp((4*x**8 - 16*x**7 - 48*x**6 + 136*x**5 + 208*x**4 + 260*x**2 - 32*x + 64)*log(3)**2/x**2)

________________________________________________________________________________________