Integrand size = 104, antiderivative size = 20 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\log \left (\log \left (4+e^{e^{\log ^2\left (\frac {(3+x)^2}{e^6}\right )}}\right )\right ) \] Output:
ln(ln(4+exp(exp(ln((3+x)^2/exp(3)^2)^2))))
Time = 2.95 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\log \left (\log \left (4+e^{e^{\left (-6+\log \left ((3+x)^2\right )\right )^2}}\right )\right ) \] Input:
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]
Output:
Log[Log[4 + E^E^(-6 + Log[(3 + x)^2])^2]]
Time = 5.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {27, 7235}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {4 \log \left (\frac {x^2+6 x+9}{e^6}\right ) \exp \left (\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )+e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}\right )}{\left ((x+3) e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}}+4 x+12\right ) \log \left (e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}}+4\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {\exp \left (\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )+e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}\right ) \log \left (\frac {x^2+6 x+9}{e^6}\right )}{\left (4 x+e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}} (x+3)+12\right ) \log \left (4+e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}}\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \log \left (\log \left (e^{e^{\log ^2\left (\frac {x^2+6 x+9}{e^6}\right )}}+4\right )\right )\) |
Input:
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]
Output:
Log[Log[4 + E^E^Log[(9 + 6*x + x^2)/E^6]^2]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 1.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15
method | result | size |
parallelrisch | \(\ln \left (\ln \left ({\mathrm e}^{{\mathrm e}^{{\ln \left (\left (x^{2}+6 x +9\right ) {\mathrm e}^{-6}\right )}^{2}}}+4\right )\right )\) | \(23\) |
Input:
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,method=_RETURNVERBOSE)
Output:
ln(ln(exp(exp(ln((x^2+6*x+9)/exp(3)^2)^2))+4))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.45 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\log \left (\log \left ({\left (e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2} + e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )} + 4 \, e^{\left (\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )} e^{\left (-\log \left ({\left (x^{2} + 6 \, x + 9\right )} e^{\left (-6\right )}\right )^{2}\right )}\right )\right ) \] Input:
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="f ricas")
Output:
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)))
Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\log {\left (\log {\left (e^{e^{\log {\left (\frac {x^{2} + 6 x + 9}{e^{6}} \right )}^{2}}} + 4 \right )} \right )} \] Input:
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)
Output:
log(log(exp(exp(log((x**2 + 6*x + 9)*exp(-6))**2)) + 4))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (17) = 34\).
Time = 0.48 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.85 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\log \left (\log \left (e^{\left (\frac {e^{\left (4 \, \log \left (x + 3\right )^{2} + 36\right )}}{x^{24} + 72 \, x^{23} + 2484 \, x^{22} + 54648 \, x^{21} + 860706 \, x^{20} + 10328472 \, x^{19} + 98120484 \, x^{18} + 756929448 \, x^{17} + 4825425231 \, x^{16} + 25735601232 \, x^{15} + 115810205544 \, x^{14} + 442184421168 \, x^{13} + 1437099368796 \, x^{12} + 3979659790512 \, x^{11} + 9380626649064 \, x^{10} + 18761253298128 \, x^{9} + 31659614940591 \, x^{8} + 44695926974952 \, x^{7} + 52145248137444 \, x^{6} + 49400761393368 \, x^{5} + 37050571045026 \, x^{4} + 21171754882872 \, x^{3} + 8661172452084 \, x^{2} + 2259436291848 \, x + 282429536481}\right )} + 4\right )\right ) \] Input:
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="m axima")
Output:
log(log(e^(e^(4*log(x + 3)^2 + 36)/(x^24 + 72*x^23 + 2484*x^22 + 54648*x^2 1 + 860706*x^20 + 10328472*x^19 + 98120484*x^18 + 756929448*x^17 + 4825425 231*x^16 + 25735601232*x^15 + 115810205544*x^14 + 442184421168*x^13 + 1437 099368796*x^12 + 3979659790512*x^11 + 9380626649064*x^10 + 18761253298128* x^9 + 31659614940591*x^8 + 44695926974952*x^7 + 52145248137444*x^6 + 49400 761393368*x^5 + 37050571045026*x^4 + 21171754882872*x^3 + 8661172452084*x^ 2 + 2259436291848*x + 282429536481)) + 4))
Exception generated. \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
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="g iac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Taylor sign with unsigned limit Err or: Bad Argument Value
Time = 0.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\ln \left (\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{{\ln \left ({\mathrm {e}}^{-6}\,x^2+6\,{\mathrm {e}}^{-6}\,x+9\,{\mathrm {e}}^{-6}\right )}^2}}+4\right )\right ) \] Input:
int((4*log(exp(-6)*(6*x + x^2 + 9))*exp(log(exp(-6)*(6*x + x^2 + 9))^2)*ex p(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)
Output:
log(log(exp(exp(log(9*exp(-6) + 6*x*exp(-6) + x^2*exp(-6))^2)) + 4))
Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {4 e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}+\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )} \log \left (\frac {9+6 x+x^2}{e^6}\right )}{\left (12+4 x+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}} (3+x)\right ) \log \left (4+e^{e^{\log ^2\left (\frac {9+6 x+x^2}{e^6}\right )}}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (e^{e^{\mathrm {log}\left (\frac {x^{2}+6 x +9}{e^{6}}\right )^{2}}}+4\right )\right ) \] Input:
int(4*log((x^2+6*x+9)/exp(3)^2)*exp(log((x^2+6*x+9)/exp(3)^2)^2)*exp(exp(l og((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)
Output:
log(log(e**(e**(log((x**2 + 6*x + 9)/e**6)**2)) + 4))