3.61.45 \(\int \frac {12 \log (3)-4 e^x \log (3)-e^x x \log (3) \log (x)+(-54+36 e^x-6 e^{2 x}) \log ^5(x)}{(54 x-36 e^x x+6 e^{2 x} x) \log ^5(x)} \, dx\) [6045]

Optimal. Leaf size=25 \[ 3-\frac {\log (3)}{6 \left (3-e^x\right ) \log ^4(x)}-\log (x) \]

[Out]

-1/6*ln(3)/ln(x)^4/(-exp(x)+3)-ln(x)+3

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Rubi [F]
time = 0.44, 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 {12 \log (3)-4 e^x \log (3)-e^x x \log (3) \log (x)+\left (-54+36 e^x-6 e^{2 x}\right ) \log ^5(x)}{\left (54 x-36 e^x x+6 e^{2 x} x\right ) \log ^5(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12*Log[3] - 4*E^x*Log[3] - E^x*x*Log[3]*Log[x] + (-54 + 36*E^x - 6*E^(2*x))*Log[x]^5)/((54*x - 36*E^x*x +
6*E^(2*x)*x)*Log[x]^5),x]

[Out]

-Log[x] - (Log[9]*Defer[Int][1/((-3 + E^x)*x*Log[x]^5), x])/3 - (Log[3]*Defer[Int][E^x/((-3 + E^x)^2*Log[x]^4)
, x])/6

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}-\frac {\log (9)}{3 \left (-3+e^x\right ) x \log ^5(x)}-\frac {e^x \log (3)}{6 \left (-3+e^x\right )^2 \log ^4(x)}\right ) \, dx\\ &=-\log (x)-\frac {1}{6} \log (3) \int \frac {e^x}{\left (-3+e^x\right )^2 \log ^4(x)} \, dx-\frac {1}{3} \log (9) \int \frac {1}{\left (-3+e^x\right ) x \log ^5(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.33, size = 22, normalized size = 0.88 \begin {gather*} \frac {\log (9)}{12 \left (-3+e^x\right ) \log ^4(x)}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*Log[3] - 4*E^x*Log[3] - E^x*x*Log[3]*Log[x] + (-54 + 36*E^x - 6*E^(2*x))*Log[x]^5)/((54*x - 36*E
^x*x + 6*E^(2*x)*x)*Log[x]^5),x]

[Out]

Log[9]/(12*(-3 + E^x)*Log[x]^4) - Log[x]

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Maple [A]
time = 0.16, size = 20, normalized size = 0.80

method	result	size
risch	\(-\ln \left (x \right )+\frac {\ln \left (3\right )}{6 \left ({\mathrm e}^{x}-3\right ) \ln \left (x \right )^{4}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*exp(x)^2+36*exp(x)-54)*ln(x)^5-x*ln(3)*exp(x)*ln(x)-4*ln(3)*exp(x)+12*ln(3))/(6*x*exp(x)^2-36*exp(x)*
x+54*x)/ln(x)^5,x,method=_RETURNVERBOSE)

[Out]

-ln(x)+1/6*ln(3)/(exp(x)-3)/ln(x)^4

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Maxima [A]
time = 0.55, size = 25, normalized size = 1.00 \begin {gather*} \frac {\log \left (3\right )}{6 \, {\left (e^{x} \log \left (x\right )^{4} - 3 \, \log \left (x\right )^{4}\right )}} - \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(x)^2+36*exp(x)-54)*log(x)^5-x*log(3)*exp(x)*log(x)-4*log(3)*exp(x)+12*log(3))/(6*x*exp(x)^2
-36*exp(x)*x+54*x)/log(x)^5,x, algorithm="maxima")

[Out]

1/6*log(3)/(e^x*log(x)^4 - 3*log(x)^4) - log(x)

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Fricas [A]
time = 0.37, size = 27, normalized size = 1.08 \begin {gather*} -\frac {6 \, {\left (e^{x} - 3\right )} \log \left (x\right )^{5} - \log \left (3\right )}{6 \, {\left (e^{x} - 3\right )} \log \left (x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(x)^2+36*exp(x)-54)*log(x)^5-x*log(3)*exp(x)*log(x)-4*log(3)*exp(x)+12*log(3))/(6*x*exp(x)^2
-36*exp(x)*x+54*x)/log(x)^5,x, algorithm="fricas")

[Out]

-1/6*(6*(e^x - 3)*log(x)^5 - log(3))/((e^x - 3)*log(x)^4)

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Sympy [A]
time = 0.06, size = 22, normalized size = 0.88 \begin {gather*} - \log {\left (x \right )} + \frac {\log {\left (3 \right )}}{6 e^{x} \log {\left (x \right )}^{4} - 18 \log {\left (x \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(x)**2+36*exp(x)-54)*ln(x)**5-x*ln(3)*exp(x)*ln(x)-4*ln(3)*exp(x)+12*ln(3))/(6*x*exp(x)**2-3
6*exp(x)*x+54*x)/ln(x)**5,x)

[Out]

-log(x) + log(3)/(6*exp(x)*log(x)**4 - 18*log(x)**4)

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Giac [A]
time = 0.40, size = 37, normalized size = 1.48 \begin {gather*} -\frac {6 \, e^{x} \log \left (x\right )^{5} - 18 \, \log \left (x\right )^{5} - \log \left (3\right )}{6 \, {\left (e^{x} \log \left (x\right )^{4} - 3 \, \log \left (x\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*exp(x)^2+36*exp(x)-54)*log(x)^5-x*log(3)*exp(x)*log(x)-4*log(3)*exp(x)+12*log(3))/(6*x*exp(x)^2
-36*exp(x)*x+54*x)/log(x)^5,x, algorithm="giac")

[Out]

-1/6*(6*e^x*log(x)^5 - 18*log(x)^5 - log(3))/(e^x*log(x)^4 - 3*log(x)^4)

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Mupad [B]
time = 5.04, size = 800, normalized size = 32.00 \begin {gather*} \frac {\frac {\ln \left (3\right )}{6\,\left ({\mathrm {e}}^x-3\right )}+\frac {x\,{\mathrm {e}}^x\,\ln \left (3\right )\,\ln \left (x\right )}{24\,{\left ({\mathrm {e}}^x-3\right )}^2}}{{\ln \left (x\right )}^4}-\ln \left (x\right )+\frac {\frac {3\,\ln \left (3\right )\,x^4}{16}-\frac {7\,\ln \left (3\right )\,x^3}{24}-\frac {3\,\ln \left (3\right )\,x^2}{16}+\frac {5\,\ln \left (3\right )\,x}{48}}{{\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x+9}+\frac {x^2\,\ln \left (3\right )-\frac {5\,x^4\,\ln \left (3\right )}{4}}{9\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-27\,{\mathrm {e}}^x+27}+\frac {\frac {\ln \left (3\right )\,x^4}{180}-\frac {\ln \left (3\right )\,x^3}{36}+\frac {\ln \left (3\right )\,x^2}{72}+\frac {\ln \left (3\right )\,x}{72}-\frac {\ln \left (3\right )}{720}}{{\mathrm {e}}^x-3}+\frac {\frac {9\,\ln \left (3\right )\,x^4}{4}+\frac {9\,\ln \left (3\right )\,x^3}{4}}{54\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-108\,{\mathrm {e}}^x+81}+\frac {\frac {x\,\left (3\,{\mathrm {e}}^x\,\ln \left (3\right )-{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+3\,x\,{\mathrm {e}}^x\,\ln \left (3\right )+x\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )\right )}{72\,{\left ({\mathrm {e}}^x-3\right )}^3}+\frac {x\,\ln \left (x\right )\,\left ({\mathrm {e}}^{3\,x}\,\ln \left (3\right )-6\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+9\,{\mathrm {e}}^x\,\ln \left (3\right )+27\,x\,{\mathrm {e}}^x\,\ln \left (3\right )-3\,x\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+9\,x^2\,{\mathrm {e}}^x\,\ln \left (3\right )+12\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+x^2\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )\right )}{144\,{\left ({\mathrm {e}}^x-3\right )}^4}}{{\ln \left (x\right )}^2}-\frac {\frac {x\,{\mathrm {e}}^x\,\ln \left (3\right )}{24\,{\left ({\mathrm {e}}^x-3\right )}^2}+\frac {x\,{\mathrm {e}}^x\,\ln \left (3\right )\,\ln \left (x\right )\,\left (3\,x-{\mathrm {e}}^x+x\,{\mathrm {e}}^x+3\right )}{72\,{\left ({\mathrm {e}}^x-3\right )}^3}}{{\ln \left (x\right )}^3}-\frac {\frac {9\,\ln \left (3\right )}{80}+{\mathrm {e}}^{2\,x}\,\left (\frac {33\,\ln \left (3\right )\,x^4}{40}-\frac {15\,\ln \left (3\right )\,x^2}{8}+\frac {3\,\ln \left (3\right )}{40}\right )+\frac {27\,x\,\ln \left (3\right )}{16}+{\mathrm {e}}^x\,\left (\frac {39\,\ln \left (3\right )\,x^4}{40}+\frac {15\,\ln \left (3\right )\,x^3}{4}+\frac {15\,\ln \left (3\right )\,x^2}{8}-\frac {9\,\ln \left (3\right )\,x}{8}-\frac {3\,\ln \left (3\right )}{20}\right )+\frac {45\,x^2\,\ln \left (3\right )}{16}+\frac {9\,x^3\,\ln \left (3\right )}{8}+\frac {9\,x^4\,\ln \left (3\right )}{80}+{\mathrm {e}}^{3\,x}\,\left (\frac {13\,\ln \left (3\right )\,x^4}{120}-\frac {5\,\ln \left (3\right )\,x^3}{12}+\frac {5\,\ln \left (3\right )\,x^2}{24}+\frac {\ln \left (3\right )\,x}{8}-\frac {\ln \left (3\right )}{60}\right )+{\mathrm {e}}^{4\,x}\,\left (\frac {\ln \left (3\right )\,x^4}{720}-\frac {\ln \left (3\right )\,x^3}{72}+\frac {5\,\ln \left (3\right )\,x^2}{144}-\frac {\ln \left (3\right )\,x}{48}+\frac {\ln \left (3\right )}{720}\right )}{270\,{\mathrm {e}}^{2\,x}-90\,{\mathrm {e}}^{3\,x}+15\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{5\,x}-405\,{\mathrm {e}}^x+243}-\frac {\frac {x\,\left ({\mathrm {e}}^{3\,x}\,\ln \left (3\right )-6\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+9\,{\mathrm {e}}^x\,\ln \left (3\right )+27\,x\,{\mathrm {e}}^x\,\ln \left (3\right )-3\,x\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+9\,x^2\,{\mathrm {e}}^x\,\ln \left (3\right )+12\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+x^2\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )\right )}{144\,{\left ({\mathrm {e}}^x-3\right )}^4}+\frac {x\,\ln \left (x\right )\,\left (9\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )-27\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )-{\mathrm {e}}^{4\,x}\,\ln \left (3\right )+27\,{\mathrm {e}}^x\,\ln \left (3\right )+189\,x\,{\mathrm {e}}^x\,\ln \left (3\right )-63\,x\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )-21\,x\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+7\,x\,{\mathrm {e}}^{4\,x}\,\ln \left (3\right )+162\,x^2\,{\mathrm {e}}^x\,\ln \left (3\right )+27\,x^3\,{\mathrm {e}}^x\,\ln \left (3\right )+162\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )-54\,x^2\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+99\,x^3\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )-6\,x^2\,{\mathrm {e}}^{4\,x}\,\ln \left (3\right )+33\,x^3\,{\mathrm {e}}^{3\,x}\,\ln \left (3\right )+x^3\,{\mathrm {e}}^{4\,x}\,\ln \left (3\right )\right )}{144\,{\left ({\mathrm {e}}^x-3\right )}^5}}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*exp(x)*log(3) - 12*log(3) + log(x)^5*(6*exp(2*x) - 36*exp(x) + 54) + x*exp(x)*log(3)*log(x))/(log(x)^5
*(54*x + 6*x*exp(2*x) - 36*x*exp(x))),x)

[Out]

(log(3)/(6*(exp(x) - 3)) + (x*exp(x)*log(3)*log(x))/(24*(exp(x) - 3)^2))/log(x)^4 - log(x) + ((5*x*log(3))/48
- (3*x^2*log(3))/16 - (7*x^3*log(3))/24 + (3*x^4*log(3))/16)/(exp(2*x) - 6*exp(x) + 9) + (x^2*log(3) - (5*x^4*
log(3))/4)/(9*exp(2*x) - exp(3*x) - 27*exp(x) + 27) + ((x*log(3))/72 - log(3)/720 + (x^2*log(3))/72 - (x^3*log
(3))/36 + (x^4*log(3))/180)/(exp(x) - 3) + ((9*x^3*log(3))/4 + (9*x^4*log(3))/4)/(54*exp(2*x) - 12*exp(3*x) +
exp(4*x) - 108*exp(x) + 81) + ((x*(3*exp(x)*log(3) - exp(2*x)*log(3) + 3*x*exp(x)*log(3) + x*exp(2*x)*log(3)))
/(72*(exp(x) - 3)^3) + (x*log(x)*(exp(3*x)*log(3) - 6*exp(2*x)*log(3) + 9*exp(x)*log(3) + 27*x*exp(x)*log(3) -
3*x*exp(3*x)*log(3) + 9*x^2*exp(x)*log(3) + 12*x^2*exp(2*x)*log(3) + x^2*exp(3*x)*log(3)))/(144*(exp(x) - 3)^
4))/log(x)^2 - ((x*exp(x)*log(3))/(24*(exp(x) - 3)^2) + (x*exp(x)*log(3)*log(x)*(3*x - exp(x) + x*exp(x) + 3))
/(72*(exp(x) - 3)^3))/log(x)^3 - ((9*log(3))/80 + exp(2*x)*((3*log(3))/40 - (15*x^2*log(3))/8 + (33*x^4*log(3)
)/40) + (27*x*log(3))/16 + exp(x)*((15*x^2*log(3))/8 - (9*x*log(3))/8 - (3*log(3))/20 + (15*x^3*log(3))/4 + (3
9*x^4*log(3))/40) + (45*x^2*log(3))/16 + (9*x^3*log(3))/8 + (9*x^4*log(3))/80 + exp(3*x)*((x*log(3))/8 - log(3
)/60 + (5*x^2*log(3))/24 - (5*x^3*log(3))/12 + (13*x^4*log(3))/120) + exp(4*x)*(log(3)/720 - (x*log(3))/48 + (
5*x^2*log(3))/144 - (x^3*log(3))/72 + (x^4*log(3))/720))/(270*exp(2*x) - 90*exp(3*x) + 15*exp(4*x) - exp(5*x)
- 405*exp(x) + 243) - ((x*(exp(3*x)*log(3) - 6*exp(2*x)*log(3) + 9*exp(x)*log(3) + 27*x*exp(x)*log(3) - 3*x*ex
p(3*x)*log(3) + 9*x^2*exp(x)*log(3) + 12*x^2*exp(2*x)*log(3) + x^2*exp(3*x)*log(3)))/(144*(exp(x) - 3)^4) + (x
*log(x)*(9*exp(3*x)*log(3) - 27*exp(2*x)*log(3) - exp(4*x)*log(3) + 27*exp(x)*log(3) + 189*x*exp(x)*log(3) - 6
3*x*exp(2*x)*log(3) - 21*x*exp(3*x)*log(3) + 7*x*exp(4*x)*log(3) + 162*x^2*exp(x)*log(3) + 27*x^3*exp(x)*log(3
) + 162*x^2*exp(2*x)*log(3) - 54*x^2*exp(3*x)*log(3) + 99*x^3*exp(2*x)*log(3) - 6*x^2*exp(4*x)*log(3) + 33*x^3
*exp(3*x)*log(3) + x^3*exp(4*x)*log(3)))/(144*(exp(x) - 3)^5))/log(x)

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