3.32.44 \(\int \frac {-9 x^5+3 x^6+(60 x^2-280 x^3+80 x^4) \log (\frac {e^{4 x}}{-3 x+x^2})+(60 x^2-20 x^3) \log ^2(\frac {e^{4 x}}{-3 x+x^2})+(36-168 x+48 x^2) \log ^3(\frac {e^{4 x}}{-3 x+x^2})+(36-12 x) \log ^4(\frac {e^{4 x}}{-3 x+x^2})}{-9 x^5+3 x^6} \, dx\)

Optimal. Leaf size=29 \[ x+\left (\frac {5}{3}+\frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2}\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 1.19, 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 {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{-9 x^5+3 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9*x^5 + 3*x^6 + (60*x^2 - 280*x^3 + 80*x^4)*Log[E^(4*x)/(-3*x + x^2)] + (60*x^2 - 20*x^3)*Log[E^(4*x)/(-
3*x + x^2)]^2 + (36 - 168*x + 48*x^2)*Log[E^(4*x)/(-3*x + x^2)]^3 + (36 - 12*x)*Log[E^(4*x)/(-3*x + x^2)]^4)/(
-9*x^5 + 3*x^6),x]

[Out]

-5/(3*x^2) + 130/(3*x) + x - (250*Log[3 - x])/27 + (10*Log[-(E^(4*x)/((3 - x)*x))])/(3*x^2) - (260*Log[-(E^(4*
x)/((3 - x)*x))])/(9*x) + (3370*Log[x])/27 - (20*Defer[Int][Log[E^(4*x)/((-3 + x)*x)]/(-3 + x), x])/27 + (20*D
efer[Int][Log[E^(4*x)/((-3 + x)*x)]/x, x])/27 - (20*Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^2/x^3, x])/3 - (4*Def
er[Int][Log[E^(4*x)/((-3 + x)*x)]^3/(-3 + x), x])/81 - 4*Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^3/x^5, x] + (52*
Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^3/x^4, x])/3 + (4*Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^3/x^3, x])/9 + (4*
Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^3/x^2, x])/27 + (4*Defer[Int][Log[E^(4*x)/((-3 + x)*x)]^3/x, x])/81 - 4*D
efer[Int][Log[E^(4*x)/((-3 + x)*x)]^4/x^5, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9 x^5+3 x^6+\left (60 x^2-280 x^3+80 x^4\right ) \log \left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (60 x^2-20 x^3\right ) \log ^2\left (\frac {e^{4 x}}{-3 x+x^2}\right )+\left (36-168 x+48 x^2\right ) \log ^3\left (\frac {e^{4 x}}{-3 x+x^2}\right )+(36-12 x) \log ^4\left (\frac {e^{4 x}}{-3 x+x^2}\right )}{x^5 (-9+3 x)} \, dx\\ &=\int \left (1+\frac {20 \left (3-14 x+4 x^2\right ) \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 (-3+x) x^3}-\frac {20 \log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^3}+\frac {4 \left (3-14 x+4 x^2\right ) \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^5}-\frac {4 \log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5}\right ) \, dx\\ &=x+4 \int \frac {\left (3-14 x+4 x^2\right ) \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx+\frac {20}{3} \int \frac {\left (3-14 x+4 x^2\right ) \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{(-3+x) x^3} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx\\ &=x-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx+4 \int \left (-\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{81 (-3+x)}-\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5}+\frac {13 \log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^4}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 x^3}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{27 x^2}+\frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{81 x}\right ) \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {20}{3} \int \left (-\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 (-3+x)}-\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3}+\frac {13 \log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^2}+\frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{9 x}\right ) \, dx\\ &=x-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx\\ &=x+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-\frac {10}{3} \int \frac {-3+14 x-4 x^2}{(3-x) x^3} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \frac {-3+14 x-4 x^2}{(3-x) x^2} \, dx\\ &=x+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-\frac {10}{3} \int \left (-\frac {1}{9 (-3+x)}-\frac {1}{x^3}+\frac {13}{3 x^2}+\frac {1}{9 x}\right ) \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx+\frac {260}{9} \int \left (-\frac {1}{3 (-3+x)}-\frac {1}{x^2}+\frac {13}{3 x}\right ) \, dx\\ &=-\frac {5}{3 x^2}+\frac {130}{3 x}+x-\frac {250}{27} \log (3-x)+\frac {10 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{3 x^2}-\frac {260 \log \left (-\frac {e^{4 x}}{(3-x) x}\right )}{9 x}+\frac {3370 \log (x)}{27}-\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {4}{81} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx+\frac {4}{27} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^2} \, dx+\frac {4}{9} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx-\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{-3+x} \, dx+\frac {20}{27} \int \frac {\log \left (\frac {e^{4 x}}{(-3+x) x}\right )}{x} \, dx-4 \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-4 \int \frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^5} \, dx-\frac {20}{3} \int \frac {\log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^3} \, dx+\frac {52}{3} \int \frac {\log ^3\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.89, size = 50, normalized size = 1.72 \begin {gather*} -\frac {928}{3}+x+\frac {10 \log ^2\left (\frac {e^{4 x}}{(-3+x) x}\right )}{3 x^2}+\frac {\log ^4\left (\frac {e^{4 x}}{(-3+x) x}\right )}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x^5 + 3*x^6 + (60*x^2 - 280*x^3 + 80*x^4)*Log[E^(4*x)/(-3*x + x^2)] + (60*x^2 - 20*x^3)*Log[E^(4
*x)/(-3*x + x^2)]^2 + (36 - 168*x + 48*x^2)*Log[E^(4*x)/(-3*x + x^2)]^3 + (36 - 12*x)*Log[E^(4*x)/(-3*x + x^2)
]^4)/(-9*x^5 + 3*x^6),x]

[Out]

-928/3 + x + (10*Log[E^(4*x)/((-3 + x)*x)]^2)/(3*x^2) + Log[E^(4*x)/((-3 + x)*x)]^4/x^4

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 52, normalized size = 1.79 \begin {gather*} \frac {3 \, x^{5} + 10 \, x^{2} \log \left (\frac {e^{\left (4 \, x\right )}}{x^{2} - 3 \, x}\right )^{2} + 3 \, \log \left (\frac {e^{\left (4 \, x\right )}}{x^{2} - 3 \, x}\right )^{4}}{3 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+36)*log(exp(x)^4/(x^2-3*x))^4+(48*x^2-168*x+36)*log(exp(x)^4/(x^2-3*x))^3+(-20*x^3+60*x^2)*l
og(exp(x)^4/(x^2-3*x))^2+(80*x^4-280*x^3+60*x^2)*log(exp(x)^4/(x^2-3*x))+3*x^6-9*x^5)/(3*x^6-9*x^5),x, algorit
hm="fricas")

[Out]

1/3*(3*x^5 + 10*x^2*log(e^(4*x)/(x^2 - 3*x))^2 + 3*log(e^(4*x)/(x^2 - 3*x))^4)/x^4

________________________________________________________________________________________

giac [B]  time = 2.67, size = 59, normalized size = 2.03 \begin {gather*} x - \frac {848 \, \log \left (x^{2} - 3 \, x\right )}{3 \, x} + \frac {298 \, \log \left (x^{2} - 3 \, x\right )^{2}}{3 \, x^{2}} - \frac {16 \, \log \left (x^{2} - 3 \, x\right )^{3}}{x^{3}} + \frac {\log \left (x^{2} - 3 \, x\right )^{4}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+36)*log(exp(x)^4/(x^2-3*x))^4+(48*x^2-168*x+36)*log(exp(x)^4/(x^2-3*x))^3+(-20*x^3+60*x^2)*l
og(exp(x)^4/(x^2-3*x))^2+(80*x^4-280*x^3+60*x^2)*log(exp(x)^4/(x^2-3*x))+3*x^6-9*x^5)/(3*x^6-9*x^5),x, algorit
hm="giac")

[Out]

x - 848/3*log(x^2 - 3*x)/x + 298/3*log(x^2 - 3*x)^2/x^2 - 16*log(x^2 - 3*x)^3/x^3 + log(x^2 - 3*x)^4/x^4

________________________________________________________________________________________

maple [C]  time = 55.15, size = 489742, normalized size = 16887.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(489742\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x+36)*ln(exp(x)^4/(x^2-3*x))^4+(48*x^2-168*x+36)*ln(exp(x)^4/(x^2-3*x))^3+(-20*x^3+60*x^2)*ln(exp(x)
^4/(x^2-3*x))^2+(80*x^4-280*x^3+60*x^2)*ln(exp(x)^4/(x^2-3*x))+3*x^6-9*x^5)/(3*x^6-9*x^5),x,method=_RETURNVERB
OSE)

[Out]

result too large to display

________________________________________________________________________________________

maxima [B]  time = 0.51, size = 118, normalized size = 4.07 \begin {gather*} x - \frac {12 \, {\left (4 \, x - \log \relax (x)\right )} \log \left (x - 3\right )^{3} - 3 \, \log \left (x - 3\right )^{4} + 848 \, x^{3} \log \relax (x) - 298 \, x^{2} \log \relax (x)^{2} + 48 \, x \log \relax (x)^{3} - 3 \, \log \relax (x)^{4} - 2 \, {\left (149 \, x^{2} - 72 \, x \log \relax (x) + 9 \, \log \relax (x)^{2}\right )} \log \left (x - 3\right )^{2} + 4 \, {\left (212 \, x^{3} - 149 \, x^{2} \log \relax (x) + 36 \, x \log \relax (x)^{2} - 3 \, \log \relax (x)^{3}\right )} \log \left (x - 3\right )}{3 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+36)*log(exp(x)^4/(x^2-3*x))^4+(48*x^2-168*x+36)*log(exp(x)^4/(x^2-3*x))^3+(-20*x^3+60*x^2)*l
og(exp(x)^4/(x^2-3*x))^2+(80*x^4-280*x^3+60*x^2)*log(exp(x)^4/(x^2-3*x))+3*x^6-9*x^5)/(3*x^6-9*x^5),x, algorit
hm="maxima")

[Out]

x - 1/3*(12*(4*x - log(x))*log(x - 3)^3 - 3*log(x - 3)^4 + 848*x^3*log(x) - 298*x^2*log(x)^2 + 48*x*log(x)^3 -
3*log(x)^4 - 2*(149*x^2 - 72*x*log(x) + 9*log(x)^2)*log(x - 3)^2 + 4*(212*x^3 - 149*x^2*log(x) + 36*x*log(x)^
2 - 3*log(x)^3)*log(x - 3))/x^4

________________________________________________________________________________________

mupad [B]  time = 2.17, size = 53, normalized size = 1.83 \begin {gather*} \frac {x^5+\frac {10\,x^2\,{\ln \left (-\frac {{\mathrm {e}}^{4\,x}}{3\,x-x^2}\right )}^2}{3}+{\ln \left (-\frac {{\mathrm {e}}^{4\,x}}{3\,x-x^2}\right )}^4}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-exp(4*x)/(3*x - x^2))^3*(48*x^2 - 168*x + 36) + log(-exp(4*x)/(3*x - x^2))*(60*x^2 - 280*x^3 + 80*x
^4) + log(-exp(4*x)/(3*x - x^2))^2*(60*x^2 - 20*x^3) - log(-exp(4*x)/(3*x - x^2))^4*(12*x - 36) - 9*x^5 + 3*x^
6)/(9*x^5 - 3*x^6),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 41, normalized size = 1.41 \begin {gather*} x + \frac {10 \log {\left (\frac {e^{4 x}}{x^{2} - 3 x} \right )}^{2}}{3 x^{2}} + \frac {\log {\left (\frac {e^{4 x}}{x^{2} - 3 x} \right )}^{4}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+36)*ln(exp(x)**4/(x**2-3*x))**4+(48*x**2-168*x+36)*ln(exp(x)**4/(x**2-3*x))**3+(-20*x**3+60*
x**2)*ln(exp(x)**4/(x**2-3*x))**2+(80*x**4-280*x**3+60*x**2)*ln(exp(x)**4/(x**2-3*x))+3*x**6-9*x**5)/(3*x**6-9
*x**5),x)

[Out]

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

________________________________________________________________________________________