3.50.16 \(\int \frac {1+3 x \log (4)+e^{3+e^{3-x \log (\frac {1}{27} (1+3 x \log (4)))}-x \log (\frac {1}{27} (1+3 x \log (4)))} (-3 x \log (4)+(-1-3 x \log (4)) \log (\frac {1}{27} (1+3 x \log (4))))}{1+3 x \log (4)} \, dx\)

Optimal. Leaf size=32 \[ e^{e^{x \left (\frac {3}{x}-\log \left (\frac {1}{9} x \left (\frac {1}{3 x}+\log (4)\right )\right )\right )}}+x \]

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Rubi [F]  time = 4.12, 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 {1+3 x \log (4)+\exp \left (3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right ) \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x*Log[(1 + 3*x*Log[4])/27])*(-3*x*Log[4] + (
-1 - 3*x*Log[4])*Log[(1 + 3*x*Log[4])/27]))/(1 + 3*x*Log[4]),x]

[Out]

x - Log[64]*Defer[Int][27^x*E^(3 + (27^x*E^3)/(1 + 3*x*Log[4])^x)*x*(1 + x*Log[64])^(-1 - x), x] - Log[1/27 +
(2*x*Log[2])/9]*Defer[Int][(27^x*E^(3 + (27^x*E^3)/(1 + 3*x*Log[4])^x))/(1 + x*Log[64])^x, x] + 6*Log[2]*Defer
[Int][Defer[Int][(27^x*E^(3 + (27^x*E^3)/(1 + 3*x*Log[4])^x))/(1 + x*Log[64])^x, x]/(1 + x*Log[64]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-1-x} \left (x \log (64)+\log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )+x \log (64) \log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )\right )\right ) \, dx\\ &=x-\int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-1-x} \left (x \log (64)+\log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )+x \log (64) \log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )\right ) \, dx\\ &=x-\int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-1-x} \left (x \log (64)+(1+x \log (64)) \log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )\right ) \, dx\\ &=x-\int \left (27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} x \log (64) (1+x \log (64))^{-1-x}+27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right )\right ) \, dx\\ &=x-\log (64) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} x (1+x \log (64))^{-1-x} \, dx-\int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right ) \, dx\\ &=x-\log (64) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} x (1+x \log (64))^{-1-x} \, dx-\log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right ) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \, dx+\int \frac {6 \log (2) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \, dx}{1+x \log (64)} \, dx\\ &=x+(6 \log (2)) \int \frac {\int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \, dx}{1+x \log (64)} \, dx-\log (64) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} x (1+x \log (64))^{-1-x} \, dx-\log \left (\frac {1}{27}+\frac {2}{9} x \log (2)\right ) \int 27^x e^{3+27^x e^3 (1+3 x \log (4))^{-x}} (1+x \log (64))^{-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 6.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x \log (4)+e^{3+e^{3-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )}-x \log \left (\frac {1}{27} (1+3 x \log (4))\right )} \left (-3 x \log (4)+(-1-3 x \log (4)) \log \left (\frac {1}{27} (1+3 x \log (4))\right )\right )}{1+3 x \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x*Log[(1 + 3*x*Log[4])/27])*(-3*x*Log[
4] + (-1 - 3*x*Log[4])*Log[(1 + 3*x*Log[4])/27]))/(1 + 3*x*Log[4]),x]

[Out]

Integrate[(1 + 3*x*Log[4] + E^(3 + E^(3 - x*Log[(1 + 3*x*Log[4])/27]) - x*Log[(1 + 3*x*Log[4])/27])*(-3*x*Log[
4] + (-1 - 3*x*Log[4])*Log[(1 + 3*x*Log[4])/27]))/(1 + 3*x*Log[4]), x]

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fricas [B]  time = 0.97, size = 59, normalized size = 1.84 \begin {gather*} {\left (x e^{\left (-x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) + 3\right )} + e^{\left (-x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) + e^{\left (-x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) + 3\right )} + 3\right )}\right )} e^{\left (x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log(2/9*x*log(2)+1/27)+3)*exp(exp(-x*log
(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/(6*x*log(2)+1),x, algorithm="fricas")

[Out]

(x*e^(-x*log(2/9*x*log(2) + 1/27) + 3) + e^(-x*log(2/9*x*log(2) + 1/27) + e^(-x*log(2/9*x*log(2) + 1/27) + 3)
+ 3))*e^(x*log(2/9*x*log(2) + 1/27) - 3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (6 \, x \log \relax (2) + {\left (6 \, x \log \relax (2) + 1\right )} \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right )\right )} e^{\left (-x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) + e^{\left (-x \log \left (\frac {2}{9} \, x \log \relax (2) + \frac {1}{27}\right ) + 3\right )} + 3\right )} - 6 \, x \log \relax (2) - 1}{6 \, x \log \relax (2) + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log(2/9*x*log(2)+1/27)+3)*exp(exp(-x*log
(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/(6*x*log(2)+1),x, algorithm="giac")

[Out]

integrate(-((6*x*log(2) + (6*x*log(2) + 1)*log(2/9*x*log(2) + 1/27))*e^(-x*log(2/9*x*log(2) + 1/27) + e^(-x*lo
g(2/9*x*log(2) + 1/27) + 3) + 3) - 6*x*log(2) - 1)/(6*x*log(2) + 1), x)

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maple [A]  time = 0.32, size = 18, normalized size = 0.56

method	result	size
default	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \relax (2)}{9}+\frac {1}{27}\right )+3}}\)	\(18\)
norman	\(x +{\mathrm e}^{{\mathrm e}^{-x \ln \left (\frac {2 x \ln \relax (2)}{9}+\frac {1}{27}\right )+3}}\)	\(18\)
risch	\(x +{\mathrm e}^{\left (\frac {2 x \ln \relax (2)}{9}+\frac {1}{27}\right )^{-x} {\mathrm e}^{3}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x*ln(2)-1)*ln(2/9*x*ln(2)+1/27)-6*x*ln(2))*exp(-x*ln(2/9*x*ln(2)+1/27)+3)*exp(exp(-x*ln(2/9*x*ln(2)+
1/27)+3))+6*x*ln(2)+1)/(6*x*ln(2)+1),x,method=_RETURNVERBOSE)

[Out]

x+exp(exp(-x*ln(2/9*x*ln(2)+1/27)+3))

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maxima [B]  time = 0.62, size = 61, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, {\left (\frac {6 \, x}{\log \relax (2)} - \frac {\log \left (6 \, x \log \relax (2) + 1\right )}{\log \relax (2)^{2}}\right )} \log \relax (2) + \frac {\log \left (6 \, x \log \relax (2) + 1\right )}{6 \, \log \relax (2)} + e^{\left (e^{\left (3 \, x \log \relax (3) - x \log \left (6 \, x \log \relax (2) + 1\right ) + 3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*log(2)-1)*log(2/9*x*log(2)+1/27)-6*x*log(2))*exp(-x*log(2/9*x*log(2)+1/27)+3)*exp(exp(-x*log
(2/9*x*log(2)+1/27)+3))+6*x*log(2)+1)/(6*x*log(2)+1),x, algorithm="maxima")

[Out]

1/6*(6*x/log(2) - log(6*x*log(2) + 1)/log(2)^2)*log(2) + 1/6*log(6*x*log(2) + 1)/log(2) + e^(e^(3*x*log(3) - x
*log(6*x*log(2) + 1) + 3))

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mupad [B]  time = 4.22, size = 17, normalized size = 0.53 \begin {gather*} x+{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{{\left (\frac {2\,x\,\ln \relax (2)}{9}+\frac {1}{27}\right )}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*log(2) - exp(exp(3 - x*log((2*x*log(2))/9 + 1/27)))*exp(3 - x*log((2*x*log(2))/9 + 1/27))*(log((2*x*l
og(2))/9 + 1/27)*(6*x*log(2) + 1) + 6*x*log(2)) + 1)/(6*x*log(2) + 1),x)

[Out]

x + exp(exp(3)/((2*x*log(2))/9 + 1/27)^x)

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sympy [A]  time = 1.18, size = 20, normalized size = 0.62 \begin {gather*} x + e^{e^{- x \log {\left (\frac {2 x \log {\relax (2 )}}{9} + \frac {1}{27} \right )} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x*ln(2)-1)*ln(2/9*x*ln(2)+1/27)-6*x*ln(2))*exp(-x*ln(2/9*x*ln(2)+1/27)+3)*exp(exp(-x*ln(2/9*x*
ln(2)+1/27)+3))+6*x*ln(2)+1)/(6*x*ln(2)+1),x)

[Out]

x + exp(exp(-x*log(2*x*log(2)/9 + 1/27) + 3))

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