3.84.5 \(\int \frac {e^{e^{\frac {9 x}{6+e^3+\log (\frac {5}{3})+3 \log (x)}}+\frac {9 x}{6+e^3+\log (\frac {5}{3})+3 \log (x)}} (27+9 e^3+9 \log (\frac {5}{3})+27 \log (x))}{36+12 e^3+e^6-(-12-2 e^3) \log (\frac {5}{3})+\log ^2(\frac {5}{3})+(36+6 e^3+6 \log (\frac {5}{3})) \log (x)+9 \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ e^{e^{\frac {3 x}{2+\frac {1}{3} \left (e^3+\log \left (\frac {5}{3}\right )\right )+\log (x)}}} \]

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Rubi [F]  time = 3.98, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right ) \left (27+9 e^3+9 \log \left (\frac {5}{3}\right )+27 \log (x)\right )}{36+12 e^3+e^6-\left (-12-2 e^3\right ) \log \left (\frac {5}{3}\right )+\log ^2\left (\frac {5}{3}\right )+\left (36+6 e^3+6 \log \left (\frac {5}{3}\right )\right ) \log (x)+9 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((9*x)/(6 + E^3 + Log[5/3] + 3*Log[x])) + (9*x)/(6 + E^3 + Log[5/3] + 3*Log[x]))*(27 + 9*E^3 + 9*Log
[5/3] + 27*Log[x]))/(36 + 12*E^3 + E^6 - (-12 - 2*E^3)*Log[5/3] + Log[5/3]^2 + (36 + 6*E^3 + 6*Log[5/3])*Log[x
] + 9*Log[x]^2),x]

[Out]

-27*Defer[Int][E^(E^((9*x)/(6 + E^3 + Log[5/3] + 3*Log[x])) + (9*x)/(6 + E^3 + Log[5/3] + 3*Log[x]))/(6*(1 + (
E^3 + Log[5/3])/6) + 3*Log[x])^2, x] + 9*Defer[Int][E^(E^((9*x)/(6 + E^3 + Log[5/3] + 3*Log[x])) + (9*x)/(6 +
E^3 + Log[5/3] + 3*Log[x]))/(6*(1 + (E^3 + Log[5/3])/6) + 3*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 \exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right ) \left (3 \left (1+\frac {1}{3} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )}{\left (6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )^2} \, dx\\ &=9 \int \frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right ) \left (3 \left (1+\frac {1}{3} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )}{\left (6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )^2} \, dx\\ &=9 \int \left (\frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right ) \left (-18 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+9 \left (1+\frac {1}{3} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )\right )}{3 \left (6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )^2}+\frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right )}{6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)}\right ) \, dx\\ &=9 \int \frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right )}{6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)} \, dx-27 \int \frac {\exp \left (e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}+\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}\right )}{\left (6 \left (1+\frac {1}{6} \left (e^3+\log \left (\frac {5}{3}\right )\right )\right )+3 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.42, size = 22, normalized size = 0.88 \begin {gather*} e^{e^{\frac {9 x}{6+e^3+\log \left (\frac {5}{3}\right )+3 \log (x)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((9*x)/(6 + E^3 + Log[5/3] + 3*Log[x])) + (9*x)/(6 + E^3 + Log[5/3] + 3*Log[x]))*(27 + 9*E^3 +
9*Log[5/3] + 27*Log[x]))/(36 + 12*E^3 + E^6 - (-12 - 2*E^3)*Log[5/3] + Log[5/3]^2 + (36 + 6*E^3 + 6*Log[5/3])
*Log[x] + 9*Log[x]^2),x]

[Out]

E^E^((9*x)/(6 + E^3 + Log[5/3] + 3*Log[x]))

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fricas [B]  time = 0.79, size = 69, normalized size = 2.76 \begin {gather*} e^{\left (\frac {{\left (e^{3} - \log \left (\frac {3}{5}\right ) + 3 \, \log \relax (x) + 6\right )} e^{\left (\frac {9 \, x}{e^{3} - \log \left (\frac {3}{5}\right ) + 3 \, \log \relax (x) + 6}\right )} + 9 \, x}{e^{3} - \log \left (\frac {3}{5}\right ) + 3 \, \log \relax (x) + 6} - \frac {9 \, x}{e^{3} - \log \left (\frac {3}{5}\right ) + 3 \, \log \relax (x) + 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*log(x)-9*log(3/5)+9*exp(3)+27)*exp(9*x/(3*log(x)-log(3/5)+exp(3)+6))*exp(exp(9*x/(3*log(x)-log(3
/5)+exp(3)+6)))/(9*log(x)^2+(-6*log(3/5)+6*exp(3)+36)*log(x)+log(3/5)^2+(-2*exp(3)-12)*log(3/5)+exp(3)^2+12*ex
p(3)+36),x, algorithm="fricas")

[Out]

e^(((e^3 - log(3/5) + 3*log(x) + 6)*e^(9*x/(e^3 - log(3/5) + 3*log(x) + 6)) + 9*x)/(e^3 - log(3/5) + 3*log(x)
+ 6) - 9*x/(e^3 - log(3/5) + 3*log(x) + 6))

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giac [B]  time = 7.83, size = 157, normalized size = 6.28 \begin {gather*} e^{\left (\frac {e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )} \log \relax (5) - e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )} \log \relax (3) + 3 \, e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )} \log \relax (x) + 9 \, x + 6 \, e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )} + e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6} + 3\right )}}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6} - \frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*log(x)-9*log(3/5)+9*exp(3)+27)*exp(9*x/(3*log(x)-log(3/5)+exp(3)+6))*exp(exp(9*x/(3*log(x)-log(3
/5)+exp(3)+6)))/(9*log(x)^2+(-6*log(3/5)+6*exp(3)+36)*log(x)+log(3/5)^2+(-2*exp(3)-12)*log(3/5)+exp(3)^2+12*ex
p(3)+36),x, algorithm="giac")

[Out]

e^((e^(9*x/(e^3 + log(5) - log(3) + 3*log(x) + 6))*log(5) - e^(9*x/(e^3 + log(5) - log(3) + 3*log(x) + 6))*log
(3) + 3*e^(9*x/(e^3 + log(5) - log(3) + 3*log(x) + 6))*log(x) + 9*x + 6*e^(9*x/(e^3 + log(5) - log(3) + 3*log(
x) + 6)) + e^(9*x/(e^3 + log(5) - log(3) + 3*log(x) + 6) + 3))/(e^3 + log(5) - log(3) + 3*log(x) + 6) - 9*x/(e
^3 + log(5) - log(3) + 3*log(x) + 6))

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maple [A]  time = 0.78, size = 22, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {9 x}{3 \ln \relax (x )-\ln \relax (3)+\ln \relax (5)+{\mathrm e}^{3}+6}}}\)	\(22\)
norman	\(\frac {\left ({\mathrm e}^{3}-\ln \relax (3)+\ln \relax (5)+6\right ) {\mathrm e}^{{\mathrm e}^{\frac {9 x}{3 \ln \relax (x )-\ln \left (\frac {3}{5}\right )+{\mathrm e}^{3}+6}}}+3 \ln \relax (x ) {\mathrm e}^{{\mathrm e}^{\frac {9 x}{3 \ln \relax (x )-\ln \left (\frac {3}{5}\right )+{\mathrm e}^{3}+6}}}}{3 \ln \relax (x )-\ln \left (\frac {3}{5}\right )+{\mathrm e}^{3}+6}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*ln(x)-9*ln(3/5)+9*exp(3)+27)*exp(9*x/(3*ln(x)-ln(3/5)+exp(3)+6))*exp(exp(9*x/(3*ln(x)-ln(3/5)+exp(3)+6
)))/(9*ln(x)^2+(-6*ln(3/5)+6*exp(3)+36)*ln(x)+ln(3/5)^2+(-2*exp(3)-12)*ln(3/5)+exp(3)^2+12*exp(3)+36),x,method
=_RETURNVERBOSE)

[Out]

exp(exp(9*x/(3*ln(x)-ln(3)+ln(5)+exp(3)+6)))

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maxima [A]  time = 0.68, size = 21, normalized size = 0.84 \begin {gather*} e^{\left (e^{\left (\frac {9 \, x}{e^{3} + \log \relax (5) - \log \relax (3) + 3 \, \log \relax (x) + 6}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*log(x)-9*log(3/5)+9*exp(3)+27)*exp(9*x/(3*log(x)-log(3/5)+exp(3)+6))*exp(exp(9*x/(3*log(x)-log(3
/5)+exp(3)+6)))/(9*log(x)^2+(-6*log(3/5)+6*exp(3)+36)*log(x)+log(3/5)^2+(-2*exp(3)-12)*log(3/5)+exp(3)^2+12*ex
p(3)+36),x, algorithm="maxima")

[Out]

e^(e^(9*x/(e^3 + log(5) - log(3) + 3*log(x) + 6)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {9\,x}{{\mathrm {e}}^3-\ln \left (\frac {3}{5}\right )+3\,\ln \relax (x)+6}}\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {9\,x}{{\mathrm {e}}^3-\ln \left (\frac {3}{5}\right )+3\,\ln \relax (x)+6}}}\,\left (9\,{\mathrm {e}}^3-9\,\ln \left (\frac {3}{5}\right )+27\,\ln \relax (x)+27\right )}{9\,{\ln \relax (x)}^2+\left (6\,{\mathrm {e}}^3-6\,\ln \left (\frac {3}{5}\right )+36\right )\,\ln \relax (x)+12\,{\mathrm {e}}^3+{\mathrm {e}}^6+{\ln \left (\frac {3}{5}\right )}^2-\ln \left (\frac {3}{5}\right )\,\left (2\,{\mathrm {e}}^3+12\right )+36} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((9*x)/(exp(3) - log(3/5) + 3*log(x) + 6))*exp(exp((9*x)/(exp(3) - log(3/5) + 3*log(x) + 6)))*(9*exp(3
) - 9*log(3/5) + 27*log(x) + 27))/(12*exp(3) + exp(6) + log(x)*(6*exp(3) - 6*log(3/5) + 36) + 9*log(x)^2 + log
(3/5)^2 - log(3/5)*(2*exp(3) + 12) + 36),x)

[Out]

int((exp((9*x)/(exp(3) - log(3/5) + 3*log(x) + 6))*exp(exp((9*x)/(exp(3) - log(3/5) + 3*log(x) + 6)))*(9*exp(3
) - 9*log(3/5) + 27*log(x) + 27))/(12*exp(3) + exp(6) + log(x)*(6*exp(3) - 6*log(3/5) + 36) + 9*log(x)^2 + log
(3/5)^2 - log(3/5)*(2*exp(3) + 12) + 36), x)

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sympy [A]  time = 5.43, size = 20, normalized size = 0.80 \begin {gather*} e^{e^{\frac {9 x}{3 \log {\relax (x )} - \log {\left (\frac {3}{5} \right )} + 6 + e^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*ln(x)-9*ln(3/5)+9*exp(3)+27)*exp(9*x/(3*ln(x)-ln(3/5)+exp(3)+6))*exp(exp(9*x/(3*ln(x)-ln(3/5)+ex
p(3)+6)))/(9*ln(x)**2+(-6*ln(3/5)+6*exp(3)+36)*ln(x)+ln(3/5)**2+(-2*exp(3)-12)*ln(3/5)+exp(3)**2+12*exp(3)+36)
,x)

[Out]

exp(exp(9*x/(3*log(x) - log(3/5) + 6 + exp(3))))

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