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

Optimal. Leaf size=29 \[ \log (3) \left (x+3 \left (3-x+x \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right )\right ) \]

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Rubi [F]  time = 1.52, 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 {e^x (3-3 x) \log (3)+\left (-2 e^x \log (3)-2 x \log (3)\right ) \log \left (\frac {e^x+x}{x}\right )+\left (3 e^x \log (3)+3 x \log (3)\right ) \log \left (\frac {e^x+x}{x}\right ) \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )}{\left (e^x+x\right ) \log \left (\frac {e^x+x}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.24, size = 25, normalized size = 0.86 \begin {gather*} \log (3) \left (-2 x+3 x \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.03, size = 25, normalized size = 0.86 \begin {gather*} 3 \, x \log \relax (3) \log \left (\frac {3}{\log \left (\frac {x + e^{x}}{x}\right )}\right ) - 2 \, x \log \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+3*x*log(3))*log(1/x*(exp(x)+x))*log(3/log(1/x*(exp(x)+x)))+(-2*log(3)*exp(x)-2*x*l
og(3))*log(1/x*(exp(x)+x))+(-3*x+3)*log(3)*exp(x))/(exp(x)+x)/log(1/x*(exp(x)+x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+3*x*log(3))*log(1/x*(exp(x)+x))*log(3/log(1/x*(exp(x)+x)))+(-2*log(3)*exp(x)-2*x*l
og(3))*log(1/x*(exp(x)+x))+(-3*x+3)*log(3)*exp(x))/(exp(x)+x)/log(1/x*(exp(x)+x)),x, algorithm="giac")

[Out]

integrate(-(3*(x - 1)*e^x*log(3) - 3*(x*log(3) + e^x*log(3))*log((x + e^x)/x)*log(3/log((x + e^x)/x)) + 2*(x*l
og(3) + e^x*log(3))*log((x + e^x)/x))/((x + e^x)*log((x + e^x)/x)), x)

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maple [C]  time = 1.03, size = 797, normalized size = 27.48

method	result	size
risch	\(-3 \ln \relax (3) x \ln \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}-2 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{x}+x \right )\right )+\frac {\ln \relax (3) \left (-3 i \pi \,\mathrm {csgn}\left (\frac {i}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right ) \mathrm {csgn}\left (\frac {1}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right )-3 i \pi \mathrm {csgn}\left (\frac {1}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right )^{2}-3 i \pi \,\mathrm {csgn}\left (\frac {i}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right ) \mathrm {csgn}\left (\frac {1}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right )^{2}-3 i \pi \mathrm {csgn}\left (\frac {1}{-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}+x \right )}{x}\right )^{3}+2 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+x \right )}\right )^{3}+3 i \pi +6 \ln \relax (6)-4\right ) x}{2}\)	\(797\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*ln(3)*x*ln(Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))-Pi*csgn(I/x)*csgn(I/x*(exp(x)+x))^2-Pi*csgn
(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I/x*(exp(x)+x))^3-2*I*ln(x)+2*I*ln(exp(x)+x))+1/2*ln(3)*(-3*I*Pi
*csgn(I/(-Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))+Pi*csgn(I/x)*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I*(
exp(x)+x))*csgn(I/x*(exp(x)+x))^2-Pi*csgn(I/x*(exp(x)+x))^3+2*I*ln(x)-2*I*ln(exp(x)+x)))*csgn(1/(-Pi*csgn(I/x)
*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))+Pi*csgn(I/x)*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I*(exp(x)+x))*csgn(I/x*(e
xp(x)+x))^2-Pi*csgn(I/x*(exp(x)+x))^3+2*I*ln(x)-2*I*ln(exp(x)+x)))-3*I*Pi*csgn(1/(-Pi*csgn(I/x)*csgn(I*(exp(x)
+x))*csgn(I/x*(exp(x)+x))+Pi*csgn(I/x)*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2-Pi*
csgn(I/x*(exp(x)+x))^3+2*I*ln(x)-2*I*ln(exp(x)+x)))^2-3*I*Pi*csgn(I/(-Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x
*(exp(x)+x))+Pi*csgn(I/x)*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2-Pi*csgn(I/x*(exp
(x)+x))^3+2*I*ln(x)-2*I*ln(exp(x)+x)))*csgn(1/(-Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))+Pi*csgn(I
/x)*csgn(I/x*(exp(x)+x))^2+Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2-Pi*csgn(I/x*(exp(x)+x))^3+2*I*ln(x)-2*
I*ln(exp(x)+x)))^2-3*I*Pi*csgn(1/(-Pi*csgn(I/x)*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))+Pi*csgn(I/x)*csgn(I/x*
(exp(x)+x))^2+Pi*csgn(I*(exp(x)+x))*csgn(I/x*(exp(x)+x))^2-Pi*csgn(I/x*(exp(x)+x))^3+2*I*ln(x)-2*I*ln(exp(x)+x
)))^3+3*I*Pi+6*ln(6)-4)*x

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maxima [A]  time = 0.49, size = 30, normalized size = 1.03 \begin {gather*} -3 \, x \log \relax (3) \log \left (\log \left (x + e^{x}\right ) - \log \relax (x)\right ) + {\left (3 \, \log \relax (3)^{2} - 2 \, \log \relax (3)\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*log(3)*exp(x)+3*x*log(3))*log(1/x*(exp(x)+x))*log(3/log(1/x*(exp(x)+x)))+(-2*log(3)*exp(x)-2*x*l
og(3))*log(1/x*(exp(x)+x))+(-3*x+3)*log(3)*exp(x))/(exp(x)+x)/log(1/x*(exp(x)+x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.01, size = 25, normalized size = 0.86 \begin {gather*} 3\,x\,\ln \relax (3)\,\ln \left (\frac {3}{\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )}\right )-2\,x\,\ln \relax (3) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*log(3)*log(3/log((x + exp(x))/x)) - 2*x*log(3)

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

[Out]

Timed out

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