3.90.18 \(\int \frac {-6+8 x-2 e^4 x+12 x^2-2 x \log (x)}{9 e^{\frac {1}{3} (5 x-e^4 x+3 x^2-x \log (x))}-6 x} \, dx\)

Optimal. Leaf size=27 \[ \log \left (6-4 e^{-\frac {1}{3} x \left (5-e^4+3 x-\log (x)\right )} x\right ) \]

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Rubi [F]  time = 3.27, 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 {-6+8 x-2 e^4 x+12 x^2-2 x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 8*x - 2*E^4*x + 12*x^2 - 2*x*Log[x])/(9*E^((5*x - E^4*x + 3*x^2 - x*Log[x])/3) - 6*x),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6+\left (8-2 e^4\right ) x+12 x^2-2 x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=\int \frac {2 \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=2 \int \frac {-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)}{9 e^{\frac {1}{3} \left (5 x-e^4 x+3 x^2-x \log (x)\right )}-6 x} \, dx\\ &=2 \int \left (\frac {e^{\frac {5 x}{3}+x^2} \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{2 x \left (3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )}+\frac {3-4 \left (1-\frac {e^4}{4}\right ) x-6 x^2+x \log (x)}{6 x}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3-4 \left (1-\frac {e^4}{4}\right ) x-6 x^2+x \log (x)}{x} \, dx+\int \frac {e^{\frac {5 x}{3}+x^2} \left (-3+4 \left (1-\frac {e^4}{4}\right ) x+6 x^2-x \log (x)\right )}{x \left (3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx\\ &=\frac {1}{3} \int \left (\frac {3-\left (4-e^4\right ) x-6 x^2}{x}+\log (x)\right ) \, dx+\int \left (-\frac {e^{\frac {5 x}{3}+x^2} \left (-4+e^4\right )}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}+\frac {3 e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )}-\frac {6 e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}-\frac {e^{\frac {5 x}{3}+x^2} \log (x)}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}}\right ) \, dx\\ &=\frac {1}{3} \int \frac {3-\left (4-e^4\right ) x-6 x^2}{x} \, dx+\frac {1}{3} \int \log (x) \, dx+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\int \frac {e^{\frac {5 x}{3}+x^2} \log (x)}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx\\ &=-\frac {x}{3}+\frac {1}{3} x \log (x)+\frac {1}{3} \int \left (-4+e^4+\frac {3}{x}-6 x\right ) \, dx+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\log (x) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\int \frac {\int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx}{x} \, dx\\ &=-\frac {x}{3}-\frac {1}{3} \left (4-e^4\right ) x-x^2+\log (x)+\frac {1}{3} x \log (x)+3 \int \frac {e^{\frac {5 x}{3}+x^2}}{x \left (-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}\right )} \, dx-6 \int \frac {e^{\frac {5 x}{3}+x^2} x}{-3 e^{\frac {5 x}{3}+x^2}+2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\left (4-e^4\right ) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx-\log (x) \int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx+\int \frac {\int \frac {e^{\frac {5 x}{3}+x^2}}{3 e^{\frac {5 x}{3}+x^2}-2 e^{\frac {e^4 x}{3}} x^{1+\frac {x}{3}}} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.31, size = 57, normalized size = 2.11 \begin {gather*} -\frac {1}{3} x \left (6-e^4+3 x-\log (x)\right )+\frac {1}{3} \left (x+3 \log \left (-2 x+3 e^{-\frac {1}{3} \left (-5+e^4-3 x\right ) x} x^{-x/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 8*x - 2*E^4*x + 12*x^2 - 2*x*Log[x])/(9*E^((5*x - E^4*x + 3*x^2 - x*Log[x])/3) - 6*x),x]

[Out]

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

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fricas [A]  time = 0.45, size = 44, normalized size = 1.63 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x) - \frac {5}{3} \, x + \log \left (-2 \, x + 3 \, e^{\left (x^{2} - \frac {1}{3} \, x e^{4} - \frac {1}{3} \, x \log \relax (x) + \frac {5}{3} \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-2*x*exp(4)+12*x^2+8*x-6)/(9*exp(-1/3*x*log(x)-1/3*x*exp(4)+x^2+5/3*x)-6*x),x, algorithm
="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 44, normalized size = 1.63 \begin {gather*} -x^{2} + \frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x) - \frac {5}{3} \, x + \log \left (2 \, x - 3 \, e^{\left (x^{2} - \frac {1}{3} \, x e^{4} - \frac {1}{3} \, x \log \relax (x) + \frac {5}{3} \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-2*x*exp(4)+12*x^2+8*x-6)/(9*exp(-1/3*x*log(x)-1/3*x*exp(4)+x^2+5/3*x)-6*x),x, algorithm
="giac")

[Out]

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

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maple [A]  time = 0.08, size = 42, normalized size = 1.56

method	result	size
risch	\(\frac {x \ln \relax (x )}{3}+\frac {x \,{\mathrm e}^{4}}{3}-x^{2}-\frac {5 x}{3}+\ln \left (x^{-\frac {x}{3}} {\mathrm e}^{-\frac {x \left ({\mathrm e}^{4}-3 x -5\right )}{3}}-\frac {2 x}{3}\right )\)	\(42\)
norman	\(\left (-\frac {5}{3}+\frac {{\mathrm e}^{4}}{3}\right ) x -x^{2}+\frac {x \ln \relax (x )}{3}+\ln \left (-9 \,{\mathrm e}^{-\frac {x \ln \relax (x )}{3}-\frac {x \,{\mathrm e}^{4}}{3}+x^{2}+\frac {5 x}{3}}+6 x \right )\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)-2*x*exp(4)+12*x^2+8*x-6)/(9*exp(-1/3*x*ln(x)-1/3*x*exp(4)+x^2+5/3*x)-6*x),x,method=_RETURNVERB
OSE)

[Out]

1/3*x*ln(x)+1/3*x*exp(4)-x^2-5/3*x+ln(x^(-1/3*x)*exp(-1/3*x*(exp(4)-3*x-5))-2/3*x)

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maxima [B]  time = 0.41, size = 55, normalized size = 2.04 \begin {gather*} -x^{2} + \frac {1}{3} \, {\left (x + 3\right )} \log \relax (x) - \frac {5}{3} \, x + \log \left (\frac {2 \, x e^{\left (\frac {1}{3} \, x e^{4} + \frac {1}{3} \, x \log \relax (x)\right )} - 3 \, e^{\left (x^{2} + \frac {5}{3} \, x\right )}}{2 \, x x^{\frac {1}{3} \, x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)-2*x*exp(4)+12*x^2+8*x-6)/(9*exp(-1/3*x*log(x)-1/3*x*exp(4)+x^2+5/3*x)-6*x),x, algorithm
="maxima")

[Out]

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

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mupad [B]  time = 6.81, size = 46, normalized size = 1.70 \begin {gather*} \ln \left (x-\frac {3\,{\mathrm {e}}^{x^2}\,{\left ({\mathrm {e}}^x\right )}^{5/3}}{2\,x^{x/3}\,{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^4}\right )}^{1/3}}\right )+\frac {x\,\ln \relax (x)}{3}-x^2+x\,\left (\frac {{\mathrm {e}}^4}{3}-\frac {5}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(4) - 8*x + 2*x*log(x) - 12*x^2 + 6)/(6*x - 9*exp((5*x)/3 - (x*exp(4))/3 - (x*log(x))/3 + x^2)),x)

[Out]

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

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sympy [B]  time = 0.36, size = 53, normalized size = 1.96 \begin {gather*} - \frac {x^{2}}{3} + \frac {x \log {\relax (x )}}{9} + x \left (- \frac {5}{9} + \frac {e^{4}}{9}\right ) + \frac {\log {\left (- \frac {2 x}{3} + e^{x^{2} - \frac {x \log {\relax (x )}}{3} - \frac {x e^{4}}{3} + \frac {5 x}{3}} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x)-2*x*exp(4)+12*x**2+8*x-6)/(9*exp(-1/3*x*ln(x)-1/3*x*exp(4)+x**2+5/3*x)-6*x),x)

[Out]

-x**2/3 + x*log(x)/9 + x*(-5/9 + exp(4)/9) + log(-2*x/3 + exp(x**2 - x*log(x)/3 - x*exp(4)/3 + 5*x/3))/3

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