3.5.36 \(\int \frac {13 e^3 x+52 x^2+e^x (24 x+12 e^3 x+24 x^2)+(e^x (-12 e^3-24 x)-13 e^3 x-26 x^2) \log (\frac {1}{3} (13 e^3 x+26 x^2+e^x (12 e^3+24 x))) \log (\log (\frac {1}{3} (13 e^3 x+26 x^2+e^x (12 e^3+24 x))))}{(13 e^3 x^3+26 x^4+e^x (12 e^3 x^2+24 x^3)) \log (\frac {1}{3} (13 e^3 x+26 x^2+e^x (12 e^3+24 x)))} \, dx\) [436]

Optimal. Leaf size=27 \[ \frac {\log \left (\log \left (\left (e^3+2 x\right ) \left (\frac {x}{3}+4 \left (e^x+x\right )\right )\right )\right )}{x} \]

[Out]

ln(ln((13/3*x+4*exp(x))*(2*x+exp(3))))/x

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Rubi [F]
time = 1.34, 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 {13 e^3 x+52 x^2+e^x \left (24 x+12 e^3 x+24 x^2\right )+\left (e^x \left (-12 e^3-24 x\right )-13 e^3 x-26 x^2\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right ) \log \left (\log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )\right )}{\left (13 e^3 x^3+26 x^4+e^x \left (12 e^3 x^2+24 x^3\right )\right ) \log \left (\frac {1}{3} \left (13 e^3 x+26 x^2+e^x \left (12 e^3+24 x\right )\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(13*E^3*x + 52*x^2 + E^x*(24*x + 12*E^3*x + 24*x^2) + (E^x*(-12*E^3 - 24*x) - 13*E^3*x - 26*x^2)*Log[(13*E
^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]*Log[Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]])/((13*E^3*x^3 +
26*x^4 + E^x*(12*E^3*x^2 + 24*x^3))*Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]),x]

[Out]

((2 + E^3)*Defer[Int][1/(x*Log[((E^3 + 2*x)*(12*E^x + 13*x))/3]), x])/E^3 - (4*Defer[Int][1/((E^3 + 2*x)*Log[(
(E^3 + 2*x)*(12*E^x + 13*x))/3]), x])/E^3 - 13*Defer[Int][1/((12*E^x + 13*x)*Log[((E^3 + 2*x)*(12*E^x + 13*x))
/3]), x] + 13*Defer[Int][1/(x*(12*E^x + 13*x)*Log[((E^3 + 2*x)*(12*E^x + 13*x))/3]), x] - Defer[Int][Log[Log[(
(E^3 + 2*x)*(12*E^x + 13*x))/3]]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {x \left (13 e^3+12 e^{3+x}+52 x+24 e^x (1+x)\right )}{\left (e^3+2 x\right ) \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}-\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2} \, dx\\ &=\int \left (-\frac {13 (-1+x)}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}+\frac {2 \left (1+\frac {e^3}{2}\right ) x+2 x^2-e^3 \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )-2 x \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2 \left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}\right ) \, dx\\ &=-\left (13 \int \frac {-1+x}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\right )+\int \frac {2 \left (1+\frac {e^3}{2}\right ) x+2 x^2-e^3 \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )-2 x \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2 \left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\\ &=-\left (13 \int \left (\frac {1}{\left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}-\frac {1}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}\right ) \, dx\right )+\int \frac {\frac {x \left (2+e^3+2 x\right )}{\left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}-\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2} \, dx\\ &=-\left (13 \int \frac {1}{\left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\right )+13 \int \frac {1}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx+\int \left (\frac {2+e^3+2 x}{x \left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}-\frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2}\right ) \, dx\\ &=-\left (13 \int \frac {1}{\left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\right )+13 \int \frac {1}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx+\int \frac {2+e^3+2 x}{x \left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx-\int \frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2} \, dx\\ &=-\left (13 \int \frac {1}{\left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\right )+13 \int \frac {1}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx+\int \left (\frac {2+e^3}{e^3 x \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}-\frac {4}{e^3 \left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )}\right ) \, dx-\int \frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2} \, dx\\ &=-\left (13 \int \frac {1}{\left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx\right )+13 \int \frac {1}{x \left (12 e^x+13 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx-\frac {4 \int \frac {1}{\left (e^3+2 x\right ) \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx}{e^3}+\frac {\left (2+e^3\right ) \int \frac {1}{x \log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )} \, dx}{e^3}-\int \frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, size = 26, normalized size = 0.96 \begin {gather*} \frac {\log \left (\log \left (\frac {1}{3} \left (e^3+2 x\right ) \left (12 e^x+13 x\right )\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(13*E^3*x + 52*x^2 + E^x*(24*x + 12*E^3*x + 24*x^2) + (E^x*(-12*E^3 - 24*x) - 13*E^3*x - 26*x^2)*Log
[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]*Log[Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]])/((13*E^3*
x^3 + 26*x^4 + E^x*(12*E^3*x^2 + 24*x^3))*Log[(13*E^3*x + 26*x^2 + E^x*(12*E^3 + 24*x))/3]),x]

[Out]

Log[Log[((E^3 + 2*x)*(12*E^x + 13*x))/3]]/x

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.27, size = 103, normalized size = 3.81

method	result	size
risch	\(\frac {\ln \left (-\ln \left (3\right )+\ln \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right )+\ln \left (2 x +{\mathrm e}^{3}\right )-\frac {i \pi \,\mathrm {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right ) \left (-\mathrm {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right )+\mathrm {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right )\right )\right ) \left (-\mathrm {csgn}\left (i \left (x +\frac {12 \,{\mathrm e}^{x}}{13}\right ) \left (2 x +{\mathrm e}^{3}\right )\right )+\mathrm {csgn}\left (i \left (2 x +{\mathrm e}^{3}\right )\right )\right )}{2}\right )}{x}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2)*ln(l
n(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+52*x^2)/((
12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2),x,me
thod=_RETURNVERBOSE)

[Out]

1/x*ln(-ln(3)+ln(x+12/13*exp(x))+ln(2*x+exp(3))-1/2*I*Pi*csgn(I*(x+12/13*exp(x))*(2*x+exp(3)))*(-csgn(I*(x+12/
13*exp(x))*(2*x+exp(3)))+csgn(I*(x+12/13*exp(x))))*(-csgn(I*(x+12/13*exp(x))*(2*x+exp(3)))+csgn(I*(2*x+exp(3))
)))

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Maxima [A]
time = 0.57, size = 26, normalized size = 0.96 \begin {gather*} \frac {\log \left (-\log \left (3\right ) + \log \left (13 \, x + 12 \, e^{x}\right ) + \log \left (2 \, x + e^{3}\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^
2)*log(log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+5
2*x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3
*x^2),x, algorithm="maxima")

[Out]

log(-log(3) + log(13*x + 12*e^x) + log(2*x + e^3))/x

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Fricas [A]
time = 0.33, size = 27, normalized size = 1.00 \begin {gather*} \frac {\log \left (\log \left (\frac {26}{3} \, x^{2} + \frac {13}{3} \, x e^{3} + 4 \, {\left (2 \, x + e^{3}\right )} e^{x}\right )\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^
2)*log(log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+5
2*x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3
*x^2),x, algorithm="fricas")

[Out]

log(log(26/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)*e^x))/x

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Sympy [A]
time = 2.46, size = 31, normalized size = 1.15 \begin {gather*} \frac {\log {\left (\log {\left (\frac {26 x^{2}}{3} + \frac {13 x e^{3}}{3} + \left (8 x + 4 e^{3}\right ) e^{x} \right )} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x**2)*ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x*
*2)*ln(ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x**2))+(12*x*exp(3)+24*x**2+24*x)*exp(x)+13*x*exp(3)+
52*x**2)/((12*x**2*exp(3)+24*x**3)*exp(x)+13*x**3*exp(3)+26*x**4)/ln(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)
+26/3*x**2),x)

[Out]

log(log(26*x**2/3 + 13*x*exp(3)/3 + (8*x + 4*exp(3))*exp(x)))/x

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-12*exp(3)-24*x)*exp(x)-13*x*exp(3)-26*x^2)*log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^
2)*log(log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3*x^2))+(12*x*exp(3)+24*x^2+24*x)*exp(x)+13*x*exp(3)+5
2*x^2)/((12*x^2*exp(3)+24*x^3)*exp(x)+13*x^3*exp(3)+26*x^4)/log(1/3*(12*exp(3)+24*x)*exp(x)+13/3*x*exp(3)+26/3
*x^2),x, algorithm="giac")

[Out]

integrate(-((26*x^2 + 13*x*e^3 + 12*(2*x + e^3)*e^x)*log(26/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)*e^x)*log(log(26
/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)*e^x)) - 52*x^2 - 13*x*e^3 - 12*(2*x^2 + x*e^3 + 2*x)*e^x)/((26*x^4 + 13*x^
3*e^3 + 12*(2*x^3 + x^2*e^3)*e^x)*log(26/3*x^2 + 13/3*x*e^3 + 4*(2*x + e^3)*e^x)), x)

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Mupad [B]
time = 1.32, size = 47, normalized size = 1.74 \begin {gather*} \frac {\ln \left (\ln \left (\frac {{\mathrm {e}}^x\,\left (24\,x+12\,{\mathrm {e}}^3\right )}{3}+\frac {13\,x\,{\mathrm {e}}^3}{3}+\frac {26\,x^2}{3}\right )\right )\,\left (2\,x^2+{\mathrm {e}}^3\,x\right )}{x^2\,\left (2\,x+{\mathrm {e}}^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x*exp(3) + 52*x^2 + exp(x)*(24*x + 12*x*exp(3) + 24*x^2) - log((exp(x)*(24*x + 12*exp(3)))/3 + (13*x*e
xp(3))/3 + (26*x^2)/3)*log(log((exp(x)*(24*x + 12*exp(3)))/3 + (13*x*exp(3))/3 + (26*x^2)/3))*(exp(x)*(24*x +
12*exp(3)) + 13*x*exp(3) + 26*x^2))/(log((exp(x)*(24*x + 12*exp(3)))/3 + (13*x*exp(3))/3 + (26*x^2)/3)*(exp(x)
*(12*x^2*exp(3) + 24*x^3) + 13*x^3*exp(3) + 26*x^4)),x)

[Out]

(log(log((exp(x)*(24*x + 12*exp(3)))/3 + (13*x*exp(3))/3 + (26*x^2)/3))*(x*exp(3) + 2*x^2))/(x^2*(2*x + exp(3)
))

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