∫ −10+e4+4x(2+8x)−2 log(3)+(5−e4+4x+log(3)) log(− 9___________ −5+e4+4x−log(3)) _________________________________________________________________________________________________10x−2e4+4x x+2x log(3)+(−5x+e4+4xx−x log(3)) log(− 9__________

Optimal. Leaf size=32 \[ \log \left (\frac {4}{x \left (2-\log \left (\frac {9}{5-e^{2 (2+2 x)}+\log (3)}\right )\right )^2}\right ) \]

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Rubi [F] time = 2.88, 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 {-10+e^{4+4 x} (2+8 x)-2 \log (3)+\left (5-e^{4+4 x}+\log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )}{10 x-2 e^{4+4 x} x+2 x \log (3)+\left (-5 x+e^{4+4 x} x-x \log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )} \, dx \end {gather*}

Int[(-10 + E^(4 + 4*x)*(2 + 8*x) - 2*Log[3] + (5 - E^(4 + 4*x) + Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[3])])/

(10*x - 2*E^(4 + 4*x)*x + 2*x*Log[3] + (-5*x + E^(4 + 4*x)*x - x*Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[3])]),

-Log[x] - 2*Log[2 - Log[9/(5 - E^(4 + 4*x) + Log[3])]] + 2*Defer[Subst][Defer[Int][1/(x*(2 - Log[9/(5 - x + Lo

g[3])])), x], x, E^(4 + 4*x)] + 2*Defer[Subst][Defer[Int][1/(x*(-2 + Log[9/(5 - x + Log[3])])), x], x, E^(4 +

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10+e^{4+4 x} (2+8 x)-2 \log (3)+\left (5-e^{4+4 x}+\log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )}{-2 e^{4+4 x} x+x (10+2 \log (3))+\left (-5 x+e^{4+4 x} x-x \log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )} \, dx\\ &=\int \frac {-e^{4+4 x} (2+8 x)+10 \left (1+\frac {\log (3)}{5}\right )-\left (5-e^{4+4 x}+\log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )}{x \left (e^{4+4 x}-5 \left (1+\frac {\log (3)}{5}\right )\right ) \left (2-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )} \, dx\\ &=\int \left (\frac {8 (-5-\log (3))}{\left (e^{4+4 x}-5 \left (1+\frac {\log (3)}{5}\right )\right ) \left (2-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )}+\frac {2+8 x-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )}{x \left (-2+\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )}\right ) \, dx\\ &=-\left ((8 (5+\log (3))) \int \frac {1}{\left (e^{4+4 x}-5 \left (1+\frac {\log (3)}{5}\right )\right ) \left (2-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )} \, dx\right )+\int \frac {2+8 x-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )}{x \left (-2+\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )} \, dx\\ &=-\left ((2 (5+\log (3))) \operatorname {Subst}\left (\int \frac {1}{x (-5+x-\log (3)) \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\right )+\int \left (-\frac {1}{x}+\frac {8}{-2+\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )}\right ) \, dx\\ &=-\log (x)+8 \int \frac {1}{-2+\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )} \, dx-(2 (5+\log (3))) \operatorname {Subst}\left (\int \left (-\frac {1}{x (5+\log (3)) \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )}+\frac {1}{(-5+x-\log (3)) (5+\log (3)) \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )}\right ) \, dx,x,e^{4+4 x}\right )\\ &=-\log (x)+2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )-2 \operatorname {Subst}\left (\int \frac {1}{(-5+x-\log (3)) \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (-2+\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\\ &=-\log (x)+2 \operatorname {Subst}\left (\int \frac {5+\log (3)}{x (-5-\log (3)) \left (2-\log \left (\frac {9}{x}\right )\right )} \, dx,x,5-e^{4+4 x}+\log (3)\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (-2+\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\\ &=-\log (x)-2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{x}\right )\right )} \, dx,x,5-e^{4+4 x}+\log (3)\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (-2+\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\\ &=-\log (x)-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,2-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (-2+\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\\ &=-\log (x)-2 \log \left (2-\log \left (\frac {9}{5-e^{4+4 x}+\log (3)}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (2-\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )+2 \operatorname {Subst}\left (\int \frac {1}{x \left (-2+\log \left (\frac {9}{5-x+\log (3)}\right )\right )} \, dx,x,e^{4+4 x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-10+e^{4+4 x} (2+8 x)-2 \log (3)+\left (5-e^{4+4 x}+\log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )}{10 x-2 e^{4+4 x} x+2 x \log (3)+\left (-5 x+e^{4+4 x} x-x \log (3)\right ) \log \left (-\frac {9}{-5+e^{4+4 x}-\log (3)}\right )} \, dx \end {gather*}

Integrate[(-10 + E^(4 + 4*x)*(2 + 8*x) - 2*Log[3] + (5 - E^(4 + 4*x) + Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[

3])])/(10*x - 2*E^(4 + 4*x)*x + 2*x*Log[3] + (-5*x + E^(4 + 4*x)*x - x*Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[

Integrate[(-10 + E^(4 + 4*x)*(2 + 8*x) - 2*Log[3] + (5 - E^(4 + 4*x) + Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[

3])])/(10*x - 2*E^(4 + 4*x)*x + 2*x*Log[3] + (-5*x + E^(4 + 4*x)*x - x*Log[3])*Log[-9/(-5 + E^(4 + 4*x) - Log[

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fricas [A] time = 0.71, size = 27, normalized size = 0.84 \begin {gather*} -\log \relax (x) - 2 \, \log \left (\log \left (-\frac {9}{e^{\left (4 \, x + 4\right )} - \log \relax (3) - 5}\right ) - 2\right ) \end {gather*}

integrate(((-exp(4*x+4)+5+log(3))*log(-9/(exp(4*x+4)-log(3)-5))+(8*x+2)*exp(4*x+4)-2*log(3)-10)/((x*exp(4*x+4)

-x*log(3)-5*x)*log(-9/(exp(4*x+4)-log(3)-5))-2*x*exp(4*x+4)+2*x*log(3)+10*x),x, algorithm="fricas")

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giac [B] time = 0.54, size = 94, normalized size = 2.94 \begin {gather*} -\log \left (\frac {1}{2} \, \pi ^{2} \mathrm {sgn}\left (e^{\left (4 \, x + 4\right )} - \log \relax (3) - 5\right ) + \frac {1}{2} \, \pi ^{2} + 4 \, \log \relax (3)^{2} - 4 \, \log \relax (3) \log \left ({\left | e^{\left (4 \, x + 4\right )} - \log \relax (3) - 5 \right |}\right ) + \log \left ({\left | e^{\left (4 \, x + 4\right )} - \log \relax (3) - 5 \right |}\right )^{2} - 8 \, \log \relax (3) + 4 \, \log \left ({\left | e^{\left (4 \, x + 4\right )} - \log \relax (3) - 5 \right |}\right ) + 4\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

integrate(((-exp(4*x+4)+5+log(3))*log(-9/(exp(4*x+4)-log(3)-5))+(8*x+2)*exp(4*x+4)-2*log(3)-10)/((x*exp(4*x+4)

-x*log(3)-5*x)*log(-9/(exp(4*x+4)-log(3)-5))-2*x*exp(4*x+4)+2*x*log(3)+10*x),x, algorithm="giac")

-log(1/2*pi^2*sgn(e^(4*x + 4) - log(3) - 5) + 1/2*pi^2 + 4*log(3)^2 - 4*log(3)*log(abs(e^(4*x + 4) - log(3) -

5)) + log(abs(e^(4*x + 4) - log(3) - 5))^2 - 8*log(3) + 4*log(abs(e^(4*x + 4) - log(3) - 5)) + 4) - log(abs(x)

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method	result	size

norman	\(-\ln \relax (x )-2 \ln \left (\ln \left (-\frac {9}{{\mathrm e}^{4 x +4}-\ln \relax (3)-5}\right )-2\right )\)	\(28\)
default	\(-\ln \relax (x )-2 \ln \left (-\ln \left (-\frac {9}{{\mathrm e}^{4 x +4}-\ln \relax (3)-5}\right )+2\right )\)	\(30\)
risch	\(-\ln \relax (x )-2 \ln \left (\ln \left (-{\mathrm e}^{4 x +4}+5+\ln \relax (3)\right )+\frac {i \left (4 i \ln \relax (3)-4 i\right )}{2}\right )\)	\(34\)

int(((-exp(4*x+4)+5+ln(3))*ln(-9/(exp(4*x+4)-ln(3)-5))+(8*x+2)*exp(4*x+4)-2*ln(3)-10)/((x*exp(4*x+4)-x*ln(3)-5

*x)*ln(-9/(exp(4*x+4)-ln(3)-5))-2*x*exp(4*x+4)+2*x*ln(3)+10*x),x,method=_RETURNVERBOSE)

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maxima [A] time = 0.56, size = 27, normalized size = 0.84 \begin {gather*} -\log \relax (x) - 2 \, \log \left (-2 \, \log \relax (3) + \log \left (-e^{\left (4 \, x + 4\right )} + \log \relax (3) + 5\right ) + 2\right ) \end {gather*}

integrate(((-exp(4*x+4)+5+log(3))*log(-9/(exp(4*x+4)-log(3)-5))+(8*x+2)*exp(4*x+4)-2*log(3)-10)/((x*exp(4*x+4)

-x*log(3)-5*x)*log(-9/(exp(4*x+4)-log(3)-5))-2*x*exp(4*x+4)+2*x*log(3)+10*x),x, algorithm="maxima")

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2\,\ln \relax (3)-\ln \left (\frac {9}{\ln \relax (3)-{\mathrm {e}}^{4\,x+4}+5}\right )\,\left (\ln \relax (3)-{\mathrm {e}}^{4\,x+4}+5\right )-{\mathrm {e}}^{4\,x+4}\,\left (8\,x+2\right )+10}{10\,x-\ln \left (\frac {9}{\ln \relax (3)-{\mathrm {e}}^{4\,x+4}+5}\right )\,\left (5\,x+x\,\ln \relax (3)-x\,{\mathrm {e}}^{4\,x+4}\right )+2\,x\,\ln \relax (3)-2\,x\,{\mathrm {e}}^{4\,x+4}} \,d x \end {gather*}

int(-(2*log(3) - log(9/(log(3) - exp(4*x + 4) + 5))*(log(3) - exp(4*x + 4) + 5) - exp(4*x + 4)*(8*x + 2) + 10)

/(10*x - log(9/(log(3) - exp(4*x + 4) + 5))*(5*x + x*log(3) - x*exp(4*x + 4)) + 2*x*log(3) - 2*x*exp(4*x + 4))

int(-(2*log(3) - log(9/(log(3) - exp(4*x + 4) + 5))*(log(3) - exp(4*x + 4) + 5) - exp(4*x + 4)*(8*x + 2) + 10)

/(10*x - log(9/(log(3) - exp(4*x + 4) + 5))*(5*x + x*log(3) - x*exp(4*x + 4)) + 2*x*log(3) - 2*x*exp(4*x + 4))

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sympy [A] time = 0.31, size = 26, normalized size = 0.81 \begin {gather*} - \log {\relax (x )} - 2 \log {\left (\log {\left (- \frac {9}{e^{4 x + 4} - 5 - \log {\relax (3 )}} \right )} - 2 \right )} \end {gather*}

integrate(((-exp(4*x+4)+5+ln(3))*ln(-9/(exp(4*x+4)-ln(3)-5))+(8*x+2)*exp(4*x+4)-2*ln(3)-10)/((x*exp(4*x+4)-x*l

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3.26.42 \(\int \frac {-10+e^{4+4 x} (2+8 x)-2 \log (3)+(5-e^{4+4 x}+\log (3)) \log (-\frac {9}{-5+e^{4+4 x}-\log (3)})}{10 x-2 e^{4+4 x} x+2 x \log (3)+(-5 x+e^{4+4 x} x-x \log (3)) \log (-\frac {9}{-5+e^{4+4 x}-\log (3)})} \, dx\)