3.36.47 \(\int \frac {e^{81 x^2+216 x^2 \log (-e^{-4+x}+x^2-\log (x))+144 x^2 \log ^2(-e^{-4+x}+x^2-\log (x))} (216 x-594 x^3+e^{-4+x} (162 x+216 x^2)+162 x \log (x)+(288 x-1008 x^3+e^{-4+x} (432 x+288 x^2)+432 x \log (x)) \log (-e^{-4+x}+x^2-\log (x))+(288 e^{-4+x} x-288 x^3+288 x \log (x)) \log ^2(-e^{-4+x}+x^2-\log (x)))}{e^{-4+x}-x^2+\log (x)} \, dx\)
Optimal. Leaf size=29 \[ e^{9 x^2 \left (3+4 \log \left (-e^{-4+x}+x^2-\log (x)\right )\right )^2} \]
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Rubi [A] time = 10.34, antiderivative size = 53, normalized size of antiderivative = 1.83,
number of steps used = 1, number of rules used = 1, integrand size = 181, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used
= {6706} \begin {gather*} e^{81 x^2+144 x^2 \log ^2\left (x^2-e^{x-4}-\log (x)\right )} \left (x^2-e^{x-4}-\log (x)\right )^{216 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(E^(81*x^2 + 216*x^2*Log[-E^(-4 + x) + x^2 - Log[x]] + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(216*x -
594*x^3 + E^(-4 + x)*(162*x + 216*x^2) + 162*x*Log[x] + (288*x - 1008*x^3 + E^(-4 + x)*(432*x + 288*x^2) + 43
2*x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]] + (288*E^(-4 + x)*x - 288*x^3 + 288*x*Log[x])*Log[-E^(-4 + x) + x^
2 - Log[x]]^2))/(E^(-4 + x) - x^2 + Log[x]),x]
[Out]
E^(81*x^2 + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(-E^(-4 + x) + x^2 - Log[x])^(216*x^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} &=\exp \left (81 x^2+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right ) \left (-e^{-4+x}+x^2-\log (x)\right )^{216 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 53, normalized size = 1.83 \begin {gather*} e^{81 x^2+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (-e^{-4+x}+x^2-\log (x)\right )^{216 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(E^(81*x^2 + 216*x^2*Log[-E^(-4 + x) + x^2 - Log[x]] + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(2
16*x - 594*x^3 + E^(-4 + x)*(162*x + 216*x^2) + 162*x*Log[x] + (288*x - 1008*x^3 + E^(-4 + x)*(432*x + 288*x^2
) + 432*x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]] + (288*E^(-4 + x)*x - 288*x^3 + 288*x*Log[x])*Log[-E^(-4 + x
) + x^2 - Log[x]]^2))/(E^(-4 + x) - x^2 + Log[x]),x]
[Out]
E^(81*x^2 + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(-E^(-4 + x) + x^2 - Log[x])^(216*x^2)
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fricas [A] time = 0.68, size = 49, normalized size = 1.69 \begin {gather*} e^{\left (144 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \relax (x)\right )^{2} + 216 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \relax (x)\right ) + 81 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2)^2+(432*x*log(x)+(288*x^2+432*x)*exp
(x-4)-1008*x^3+288*x)*log(-log(x)-exp(x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x
^2*log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(log(x)+exp(x-4)-x^2),x, algorithm="f
ricas")
[Out]
e^(144*x^2*log(x^2 - e^(x - 4) - log(x))^2 + 216*x^2*log(x^2 - e^(x - 4) - log(x)) + 81*x^2)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2)^2+(432*x*log(x)+(288*x^2+432*x)*exp
(x-4)-1008*x^3+288*x)*log(-log(x)-exp(x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x
^2*log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(log(x)+exp(x-4)-x^2),x, algorithm="g
iac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}Sign error %%%{ln(`
w`),0%%%}Si
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maple [A] time = 0.05, size = 49, normalized size = 1.69
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\(\left (-\ln \relax (x )-{\mathrm e}^{x -4}+x^{2}\right )^{216 x^{2}} {\mathrm e}^{9 x^{2} \left (16 \ln \left (-\ln \relax (x )-{\mathrm e}^{x -4}+x^{2}\right )^{2}+9\right )}\) |
\(49\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((288*x*ln(x)+288*x*exp(x-4)-288*x^3)*ln(-ln(x)-exp(x-4)+x^2)^2+(432*x*ln(x)+(288*x^2+432*x)*exp(x-4)-1008
*x^3+288*x)*ln(-ln(x)-exp(x-4)+x^2)+162*x*ln(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x^2*ln(-ln(x)-
exp(x-4)+x^2)^2+216*x^2*ln(-ln(x)-exp(x-4)+x^2)+81*x^2)/(ln(x)+exp(x-4)-x^2),x,method=_RETURNVERBOSE)
[Out]
(-ln(x)-exp(x-4)+x^2)^(216*x^2)*exp(9*x^2*(16*ln(-ln(x)-exp(x-4)+x^2)^2+9))
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maxima [B] time = 0.58, size = 55, normalized size = 1.90 \begin {gather*} e^{\left (144 \, x^{2} \log \left (x^{2} e^{4} - e^{4} \log \relax (x) - e^{x}\right )^{2} - 936 \, x^{2} \log \left (x^{2} e^{4} - e^{4} \log \relax (x) - e^{x}\right ) + 1521 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2)^2+(432*x*log(x)+(288*x^2+432*x)*exp
(x-4)-1008*x^3+288*x)*log(-log(x)-exp(x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x
^2*log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(log(x)+exp(x-4)-x^2),x, algorithm="m
axima")
[Out]
e^(144*x^2*log(x^2*e^4 - e^4*log(x) - e^x)^2 - 936*x^2*log(x^2*e^4 - e^4*log(x) - e^x) + 1521*x^2)
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mupad [B] time = 2.44, size = 50, normalized size = 1.72 \begin {gather*} {\mathrm {e}}^{144\,x^2\,{\ln \left (x^2-{\mathrm {e}}^{-4}\,{\mathrm {e}}^x-\ln \relax (x)\right )}^2}\,{\mathrm {e}}^{81\,x^2}\,{\left (x^2-{\mathrm {e}}^{-4}\,{\mathrm {e}}^x-\ln \relax (x)\right )}^{216\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((exp(144*x^2*log(x^2 - log(x) - exp(x - 4))^2 + 216*x^2*log(x^2 - log(x) - exp(x - 4)) + 81*x^2)*(216*x +
exp(x - 4)*(162*x + 216*x^2) + log(x^2 - log(x) - exp(x - 4))*(288*x + exp(x - 4)*(432*x + 288*x^2) + 432*x*lo
g(x) - 1008*x^3) + log(x^2 - log(x) - exp(x - 4))^2*(288*x*exp(x - 4) + 288*x*log(x) - 288*x^3) + 162*x*log(x)
- 594*x^3))/(exp(x - 4) + log(x) - x^2),x)
[Out]
exp(144*x^2*log(x^2 - exp(-4)*exp(x) - log(x))^2)*exp(81*x^2)*(x^2 - exp(-4)*exp(x) - log(x))^(216*x^2)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((288*x*ln(x)+288*x*exp(x-4)-288*x**3)*ln(-ln(x)-exp(x-4)+x**2)**2+(432*x*ln(x)+(288*x**2+432*x)*exp
(x-4)-1008*x**3+288*x)*ln(-ln(x)-exp(x-4)+x**2)+162*x*ln(x)+(216*x**2+162*x)*exp(x-4)-594*x**3+216*x)*exp(144*
x**2*ln(-ln(x)-exp(x-4)+x**2)**2+216*x**2*ln(-ln(x)-exp(x-4)+x**2)+81*x**2)/(ln(x)+exp(x-4)-x**2),x)
[Out]
Timed out
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