3.31.76 \(\int \frac {2 x \log (3)+6 e^x x \log (3)+(-6 e^x \log (3)-2 x \log (3)) \log (3 e^x+x)+(3 e^x \log (3)+x \log (3)) \log (3 e^x+x) \log (-\frac {x}{\log (3 e^x+x)})}{(3 e^x+x) \log (3 e^x+x) \log ^3(-\frac {x}{\log (3 e^x+x)})} \, dx\)

Optimal. Leaf size=20 \[ \frac {x \log (3)}{\log ^2\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \]

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Rubi [F]  time = 2.72, 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 {2 x \log (3)+6 e^x x \log (3)+\left (-6 e^x \log (3)-2 x \log (3)\right ) \log \left (3 e^x+x\right )+\left (3 e^x \log (3)+x \log (3)\right ) \log \left (3 e^x+x\right ) \log \left (-\frac {x}{\log \left (3 e^x+x\right )}\right )}{\left (3 e^x+x\right ) \log \left (3 e^x+x\right ) \log ^3\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x*Log[3] + 6*E^x*x*Log[3] + (-6*E^x*Log[3] - 2*x*Log[3])*Log[3*E^x + x] + (3*E^x*Log[3] + x*Log[3])*Log
[3*E^x + x]*Log[-(x/Log[3*E^x + x])])/((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^3),x]

[Out]

-2*Log[3]*Defer[Int][Log[-(x/Log[3*E^x + x])]^(-3), x] + 2*Log[3]*Defer[Int][x/(Log[3*E^x + x]*Log[-(x/Log[3*E
^x + x])]^3), x] + 2*Log[3]*Defer[Int][x/((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^3), x] - 2*Log[3
]*Defer[Int][x^2/((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^3), x] + Log[3]*Defer[Int][Log[-(x/Log[3
*E^x + x])]^(-2), x]

Rubi steps

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

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Mathematica [A]  time = 0.43, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \log (3)}{\log ^2\left (-\frac {x}{\log \left (3 e^x+x\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x*Log[3] + 6*E^x*x*Log[3] + (-6*E^x*Log[3] - 2*x*Log[3])*Log[3*E^x + x] + (3*E^x*Log[3] + x*Log[3
])*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])])/((3*E^x + x)*Log[3*E^x + x]*Log[-(x/Log[3*E^x + x])]^3),x]

[Out]

(x*Log[3])/Log[-(x/Log[3*E^x + x])]^2

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fricas [A]  time = 0.77, size = 19, normalized size = 0.95 \begin {gather*} \frac {x \log \relax (3)}{\log \left (-\frac {x}{\log \left (x + 3 \, e^{x}\right )}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+x))+(-6*log(3)*exp(x)-2*x*log(3))*lo
g(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*log(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorith
m="fricas")

[Out]

x*log(3)/log(-x/log(x + 3*e^x))^2

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giac [A]  time = 1.31, size = 37, normalized size = 1.85 \begin {gather*} \frac {x \log \relax (3)}{\log \left (-x\right )^{2} - 2 \, \log \left (-x\right ) \log \left (\log \left (x + 3 \, e^{x}\right )\right ) + \log \left (\log \left (x + 3 \, e^{x}\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+x))+(-6*log(3)*exp(x)-2*x*log(3))*lo
g(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*log(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorith
m="giac")

[Out]

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

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maple [C]  time = 0.56, size = 156, normalized size = 7.80

method	result	size
risch	\(-\frac {4 \ln \relax (3) x}{\left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )^{2}+2 \pi \mathrm {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i x}{\ln \left (3 \,{\mathrm e}^{x}+x \right )}\right )^{3}-2 \pi +2 i \ln \relax (x )-2 i \ln \left (\ln \left (3 \,{\mathrm e}^{x}+x \right )\right )\right )^{2}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*ln(3)*x/(Pi*csgn(I*x)*csgn(I/ln(3*exp(x)+x))*csgn(I*x/ln(3*exp(x)+x))-Pi*csgn(I*x)*csgn(I*x/ln(3*exp(x)+x))
^2+2*Pi*csgn(I*x/ln(3*exp(x)+x))^2-Pi*csgn(I/ln(3*exp(x)+x))*csgn(I*x/ln(3*exp(x)+x))^2-Pi*csgn(I*x/ln(3*exp(x
)+x))^3-2*Pi+2*I*ln(x)-2*I*ln(ln(3*exp(x)+x)))^2

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maxima [A]  time = 0.71, size = 37, normalized size = 1.85 \begin {gather*} \frac {x \log \relax (3)}{\log \left (-x\right )^{2} - 2 \, \log \left (-x\right ) \log \left (\log \left (x + 3 \, e^{x}\right )\right ) + \log \left (\log \left (x + 3 \, e^{x}\right )\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+x*log(3))*log(3*exp(x)+x)*log(-x/log(3*exp(x)+x))+(-6*log(3)*exp(x)-2*x*log(3))*lo
g(3*exp(x)+x)+6*x*log(3)*exp(x)+2*x*log(3))/(3*exp(x)+x)/log(3*exp(x)+x)/log(-x/log(3*exp(x)+x))^3,x, algorith
m="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {2\,x\,\ln \relax (3)-\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (2\,x\,\ln \relax (3)+6\,{\mathrm {e}}^x\,\ln \relax (3)\right )+6\,x\,{\mathrm {e}}^x\,\ln \relax (3)+\ln \left (-\frac {x}{\ln \left (x+3\,{\mathrm {e}}^x\right )}\right )\,\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (x\,\ln \relax (3)+3\,{\mathrm {e}}^x\,\ln \relax (3)\right )}{{\ln \left (-\frac {x}{\ln \left (x+3\,{\mathrm {e}}^x\right )}\right )}^3\,\ln \left (x+3\,{\mathrm {e}}^x\right )\,\left (x+3\,{\mathrm {e}}^x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*log(3) - log(x + 3*exp(x))*(2*x*log(3) + 6*exp(x)*log(3)) + 6*x*exp(x)*log(3) + log(-x/log(x + 3*exp(
x)))*log(x + 3*exp(x))*(x*log(3) + 3*exp(x)*log(3)))/(log(-x/log(x + 3*exp(x)))^3*log(x + 3*exp(x))*(x + 3*exp
(x))),x)

[Out]

int((2*x*log(3) - log(x + 3*exp(x))*(2*x*log(3) + 6*exp(x)*log(3)) + 6*x*exp(x)*log(3) + log(-x/log(x + 3*exp(
x)))*log(x + 3*exp(x))*(x*log(3) + 3*exp(x)*log(3)))/(log(-x/log(x + 3*exp(x)))^3*log(x + 3*exp(x))*(x + 3*exp
(x))), x)

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sympy [A]  time = 0.92, size = 19, normalized size = 0.95 \begin {gather*} \frac {x \log {\relax (3 )}}{\log {\left (- \frac {x}{\log {\left (x + 3 e^{x} \right )}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(3)/log(-x/log(x + 3*exp(x)))**2

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