3.58.95 \(\int \frac {4 e^{e^{\log ^2(\frac {9+6 x+x^2}{e^6})}+\log ^2(\frac {9+6 x+x^2}{e^6})} \log (\frac {9+6 x+x^2}{e^6})}{(12+4 x+e^{e^{\log ^2(\frac {9+6 x+x^2}{e^6})}} (3+x)) \log (4+e^{e^{\log ^2(\frac {9+6 x+x^2}{e^6})}})} \, dx\)

Optimal. Leaf size=20 \[ \log \left (\log \left (4+e^{e^{\log ^2\left (\frac {(3+x)^2}{e^6}\right )}}\right )\right ) \]

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Rubi [A]  time = 7.05, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 3, integrand size = 104, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {12, 6741, 6684} \begin {gather*} \log \left (\log \left (e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}}+4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^(E^Log[(9 + 6*x + x^2)/E^6]^2 + Log[(9 + 6*x + x^2)/E^6]^2)*Log[(9 + 6*x + x^2)/E^6])/((12 + 4*x + E^
E^Log[(9 + 6*x + x^2)/E^6]^2*(3 + x))*Log[4 + E^E^Log[(9 + 6*x + x^2)/E^6]^2]),x]

[Out]

Log[Log[4 + E^E^Log[(9 + 6*x + x^2)/E^6]^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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {\exp \left (e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )\right ) \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx\\ &=\log \left (\log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.12, size = 18, normalized size = 0.90 \begin {gather*} \log \left (\log \left (4+e^{e^{\left (-6+\log \left ((3+x)^2\right )\right )^2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(E^Log[(9 + 6*x + x^2)/E^6]^2 + Log[(9 + 6*x + x^2)/E^6]^2)*Log[(9 + 6*x + x^2)/E^6])/((12 + 4*
x + E^E^Log[(9 + 6*x + x^2)/E^6]^2*(3 + x))*Log[4 + E^E^Log[(9 + 6*x + x^2)/E^6]^2]),x]

[Out]

Log[Log[4 + E^E^(-6 + Log[(3 + x)^2])^2]]

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fricas [B]  time = 0.66, size = 69, normalized size = 3.45 \begin {gather*} \log \left (\log \left ({\left (e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2} + e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )} + 4 \, e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )} e^{\left (-\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log((x^2+6*x+9)/exp(3)^2)*exp(log((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))/(
(3+x)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4*x+12)/log(exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4),x, algorithm=
"fricas")

[Out]

log(log((e^(log((x^2 + 6*x + 9)*e^(-6))^2 + e^(log((x^2 + 6*x + 9)*e^(-6))^2)) + 4*e^(log((x^2 + 6*x + 9)*e^(-
6))^2))*e^(-log((x^2 + 6*x + 9)*e^(-6))^2)))

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giac [B]  time = 3.88, size = 150, normalized size = 7.50 \begin {gather*} \frac {1}{2} \, \log \left (18446744073709551616 \, \pi ^{2} + \log \left (e^{\left (\frac {e^{\left (\log \left (x^{2} + 6 \, x + 9\right )^{2} + 36\right )}}{x^{24} + 72 \, x^{23} + 2484 \, x^{22} + 54648 \, x^{21} + 860706 \, x^{20} + 10328472 \, x^{19} + 98120484 \, x^{18} + 756929448 \, x^{17} + 4825425231 \, x^{16} + 25735601232 \, x^{15} + 115810205544 \, x^{14} + 442184421168 \, x^{13} + 1437099368796 \, x^{12} + 3979659790512 \, x^{11} + 9380626649064 \, x^{10} + 18761253298128 \, x^{9} + 31659614940591 \, x^{8} + 44695926974952 \, x^{7} + 52145248137444 \, x^{6} + 49400761393368 \, x^{5} + 37050571045026 \, x^{4} + 21171754882872 \, x^{3} + 8661172452084 \, x^{2} + 2259436291848 \, x + 282429536481}\right )} + 4\right )^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log((x^2+6*x+9)/exp(3)^2)*exp(log((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))/(
(3+x)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4*x+12)/log(exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4),x, algorithm=
"giac")

[Out]

1/2*log(18446744073709551616*pi^2 + log(e^(e^(log(x^2 + 6*x + 9)^2 + 36)/(x^24 + 72*x^23 + 2484*x^22 + 54648*x
^21 + 860706*x^20 + 10328472*x^19 + 98120484*x^18 + 756929448*x^17 + 4825425231*x^16 + 25735601232*x^15 + 1158
10205544*x^14 + 442184421168*x^13 + 1437099368796*x^12 + 3979659790512*x^11 + 9380626649064*x^10 + 18761253298
128*x^9 + 31659614940591*x^8 + 44695926974952*x^7 + 52145248137444*x^6 + 49400761393368*x^5 + 37050571045026*x
^4 + 21171754882872*x^3 + 8661172452084*x^2 + 2259436291848*x + 282429536481)) + 4)^2)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {4 \ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right ) {\mathrm e}^{\ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right )^{2}} {\mathrm e}^{{\mathrm e}^{\ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right )^{2}}}}{\left (\left (3+x \right ) {\mathrm e}^{{\mathrm e}^{\ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right )^{2}}}+4 x +12\right ) \ln \left ({\mathrm e}^{{\mathrm e}^{\ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right )^{2}}}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*ln((x^2+6*x+9)/exp(3)^2)*exp(ln((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(ln((x^2+6*x+9)/exp(3)^2)^2))/((3+x)*exp
(exp(ln((x^2+6*x+9)/exp(3)^2)^2))+4*x+12)/ln(exp(exp(ln((x^2+6*x+9)/exp(3)^2)^2))+4),x)

[Out]

int(4*ln((x^2+6*x+9)/exp(3)^2)*exp(ln((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(ln((x^2+6*x+9)/exp(3)^2)^2))/((3+x)*exp
(exp(ln((x^2+6*x+9)/exp(3)^2)^2))+4*x+12)/ln(exp(exp(ln((x^2+6*x+9)/exp(3)^2)^2))+4),x)

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maxima [B]  time = 0.82, size = 137, normalized size = 6.85 \begin {gather*} \log \left (\log \left (e^{\left (\frac {e^{\left (4 \, \log \left (x + 3\right )^{2} + 36\right )}}{x^{24} + 72 \, x^{23} + 2484 \, x^{22} + 54648 \, x^{21} + 860706 \, x^{20} + 10328472 \, x^{19} + 98120484 \, x^{18} + 756929448 \, x^{17} + 4825425231 \, x^{16} + 25735601232 \, x^{15} + 115810205544 \, x^{14} + 442184421168 \, x^{13} + 1437099368796 \, x^{12} + 3979659790512 \, x^{11} + 9380626649064 \, x^{10} + 18761253298128 \, x^{9} + 31659614940591 \, x^{8} + 44695926974952 \, x^{7} + 52145248137444 \, x^{6} + 49400761393368 \, x^{5} + 37050571045026 \, x^{4} + 21171754882872 \, x^{3} + 8661172452084 \, x^{2} + 2259436291848 \, x + 282429536481}\right )} + 4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*log((x^2+6*x+9)/exp(3)^2)*exp(log((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))/(
(3+x)*exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4*x+12)/log(exp(exp(log((x^2+6*x+9)/exp(3)^2)^2))+4),x, algorithm=
"maxima")

[Out]

log(log(e^(e^(4*log(x + 3)^2 + 36)/(x^24 + 72*x^23 + 2484*x^22 + 54648*x^21 + 860706*x^20 + 10328472*x^19 + 98
120484*x^18 + 756929448*x^17 + 4825425231*x^16 + 25735601232*x^15 + 115810205544*x^14 + 442184421168*x^13 + 14
37099368796*x^12 + 3979659790512*x^11 + 9380626649064*x^10 + 18761253298128*x^9 + 31659614940591*x^8 + 4469592
6974952*x^7 + 52145248137444*x^6 + 49400761393368*x^5 + 37050571045026*x^4 + 21171754882872*x^3 + 866117245208
4*x^2 + 2259436291848*x + 282429536481)) + 4))

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mupad [B]  time = 5.51, size = 25, normalized size = 1.25 \begin {gather*} \ln \left (\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{{\ln \left ({\mathrm {e}}^{-6}\,x^2+6\,{\mathrm {e}}^{-6}\,x+9\,{\mathrm {e}}^{-6}\right )}^2}}+4\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(exp(-6)*(6*x + x^2 + 9))*exp(log(exp(-6)*(6*x + x^2 + 9))^2)*exp(exp(log(exp(-6)*(6*x + x^2 + 9))^2
)))/(log(exp(exp(log(exp(-6)*(6*x + x^2 + 9))^2)) + 4)*(4*x + exp(exp(log(exp(-6)*(6*x + x^2 + 9))^2))*(x + 3)
+ 12)),x)

[Out]

log(log(exp(exp(log(9*exp(-6) + 6*x*exp(-6) + x^2*exp(-6))^2)) + 4))

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sympy [A]  time = 1.33, size = 22, normalized size = 1.10 \begin {gather*} \log {\left (\log {\left (e^{e^{\log {\left (\frac {x^{2} + 6 x + 9}{e^{6}} \right )}^{2}}} + 4 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*ln((x**2+6*x+9)/exp(3)**2)*exp(ln((x**2+6*x+9)/exp(3)**2)**2)*exp(exp(ln((x**2+6*x+9)/exp(3)**2)**
2))/((3+x)*exp(exp(ln((x**2+6*x+9)/exp(3)**2)**2))+4*x+12)/ln(exp(exp(ln((x**2+6*x+9)/exp(3)**2)**2))+4),x)

[Out]

log(log(exp(exp(log((x**2 + 6*x + 9)*exp(-6))**2)) + 4))

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