3.4.17 \(\int \frac {e^{-x+\frac {e^{-x} (-4 x^3 \log (3)+e^x (12+9 x^2) \log (3)-4 e^x x \log (3) \log (\frac {4}{x}))}{3 x^2}} (e^x (-24+4 x) \log (3)+(-4 x^3+4 x^4) \log (3)+4 e^x x \log (3) \log (\frac {4}{x}))}{3 x^3} \, dx\)

Optimal. Leaf size=35 \[ 3^{3+\frac {4 \left (\frac {3}{x}-e^{-x} x^2-\log \left (\frac {4}{x}\right )\right )}{3 x}} \]

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Rubi [F]  time = 8.89, 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 {\exp \left (-x+\frac {e^{-x} \left (-4 x^3 \log (3)+e^x \left (12+9 x^2\right ) \log (3)-4 e^x x \log (3) \log \left (\frac {4}{x}\right )\right )}{3 x^2}\right ) \left (e^x (-24+4 x) \log (3)+\left (-4 x^3+4 x^4\right ) \log (3)+4 e^x x \log (3) \log \left (\frac {4}{x}\right )\right )}{3 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-x + (-4*x^3*Log[3] + E^x*(12 + 9*x^2)*Log[3] - 4*E^x*x*Log[3]*Log[4/x])/(3*E^x*x^2))*(E^x*(-24 + 4*x)
*Log[3] + (-4*x^3 + 4*x^4)*Log[3] + 4*E^x*x*Log[3]*Log[4/x]))/(3*x^3),x]

[Out]

(-4*Log[3]*Defer[Int][3^(3 + 4/x^2 - (4*x)/(3*E^x))*E^(-x - (4*Log[3]*Log[4/x])/(3*x)), x])/3 - (8*Log[3]*Defe
r[Int][3^(4 + 4/x^2 - (4*x)/(3*E^x))/(E^((4*Log[3]*Log[4/x])/(3*x))*x^3), x])/3 + (4*Log[3]*Defer[Int][3^(3 +
4/x^2 - (4*x)/(3*E^x))/(E^((4*Log[3]*Log[4/x])/(3*x))*x^2), x])/3 + (4*Log[3]*Defer[Int][3^(3 + 4/x^2 - (4*x)/
(3*E^x))*E^(-x - (4*Log[3]*Log[4/x])/(3*x))*x, x])/3 + (4*Log[3]*Defer[Int][(3^(3 + 4/x^2 - (4*x)/(3*E^x))*Log
[4/x])/(E^((4*Log[3]*Log[4/x])/(3*x))*x^2), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {\exp \left (-x+\frac {e^{-x} \left (-4 x^3 \log (3)+e^x \left (12+9 x^2\right ) \log (3)-4 e^x x \log (3) \log \left (\frac {4}{x}\right )\right )}{3 x^2}\right ) \left (e^x (-24+4 x) \log (3)+\left (-4 x^3+4 x^4\right ) \log (3)+4 e^x x \log (3) \log \left (\frac {4}{x}\right )\right )}{x^3} \, dx\\ &=\frac {1}{3} \int \frac {4\ 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \log (3) \left (e^x (-6+x)+(-1+x) x^3+e^x x \log \left (\frac {4}{x}\right )\right )}{x^3} \, dx\\ &=\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \left (e^x (-6+x)+(-1+x) x^3+e^x x \log \left (\frac {4}{x}\right )\right )}{x^3} \, dx\\ &=\frac {1}{3} (4 \log (3)) \int \left (-3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}}+3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x+\frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \left (-6+x+x \log \left (\frac {4}{x}\right )\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \, dx\right )+\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x \, dx+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \left (-6+x+x \log \left (\frac {4}{x}\right )\right )}{x^3} \, dx\\ &=-\left (\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \, dx\right )+\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x \, dx+\frac {1}{3} (4 \log (3)) \int \left (\frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} (-6+x)}{x^3}+\frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \log \left (\frac {4}{x}\right )}{x^2}\right ) \, dx\\ &=-\left (\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \, dx\right )+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} (-6+x)}{x^3} \, dx+\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x \, dx+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \log \left (\frac {4}{x}\right )}{x^2} \, dx\\ &=-\left (\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \, dx\right )+\frac {1}{3} (4 \log (3)) \int \left (-\frac {2\ 3^{4+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}}}{x^3}+\frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}}}{x^2}\right ) \, dx+\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x \, dx+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \log \left (\frac {4}{x}\right )}{x^2} \, dx\\ &=-\left (\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \, dx\right )+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}}}{x^2} \, dx+\frac {1}{3} (4 \log (3)) \int 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-x-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} x \, dx+\frac {1}{3} (4 \log (3)) \int \frac {3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}} \log \left (\frac {4}{x}\right )}{x^2} \, dx-\frac {1}{3} (8 \log (3)) \int \frac {3^{4+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}} e^{-\frac {4 \log (3) \log \left (\frac {4}{x}\right )}{3 x}}}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.37, size = 40, normalized size = 1.14 \begin {gather*} \frac {4\ 3^{3+\frac {4}{x^2}-\frac {4 e^{-x} x}{3}-\frac {4 \log \left (\frac {4}{x}\right )}{3 x}} \log (3)}{\log (81)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x + (-4*x^3*Log[3] + E^x*(12 + 9*x^2)*Log[3] - 4*E^x*x*Log[3]*Log[4/x])/(3*E^x*x^2))*(E^x*(-24
+ 4*x)*Log[3] + (-4*x^3 + 4*x^4)*Log[3] + 4*E^x*x*Log[3]*Log[4/x]))/(3*x^3),x]

[Out]

(4*3^(3 + 4/x^2 - (4*x)/(3*E^x) - (4*Log[4/x])/(3*x))*Log[3])/Log[81]

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fricas [A]  time = 0.74, size = 52, normalized size = 1.49 \begin {gather*} e^{\left (x - \frac {{\left (4 \, x^{3} \log \relax (3) + 4 \, x e^{x} \log \relax (3) \log \left (\frac {4}{x}\right ) + 3 \, {\left (x^{3} - {\left (3 \, x^{2} + 4\right )} \log \relax (3)\right )} e^{x}\right )} e^{\left (-x\right )}}{3 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x*log(3)*exp(x)*log(4/x)+(4*x-24)*log(3)*exp(x)+(4*x^4-4*x^3)*log(3))*exp(1/3*(-4*x*log(3)*ex
p(x)*log(4/x)+(9*x^2+12)*log(3)*exp(x)-4*x^3*log(3))/exp(x)/x^2)/exp(x)/x^3,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left (x e^{x} \log \relax (3) \log \left (\frac {4}{x}\right ) + {\left (x - 6\right )} e^{x} \log \relax (3) + {\left (x^{4} - x^{3}\right )} \log \relax (3)\right )} e^{\left (-x - \frac {{\left (4 \, x^{3} \log \relax (3) + 4 \, x e^{x} \log \relax (3) \log \left (\frac {4}{x}\right ) - 3 \, {\left (3 \, x^{2} + 4\right )} e^{x} \log \relax (3)\right )} e^{\left (-x\right )}}{3 \, x^{2}}\right )}}{3 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x*log(3)*exp(x)*log(4/x)+(4*x-24)*log(3)*exp(x)+(4*x^4-4*x^3)*log(3))*exp(1/3*(-4*x*log(3)*ex
p(x)*log(4/x)+(9*x^2+12)*log(3)*exp(x)-4*x^3*log(3))/exp(x)/x^2)/exp(x)/x^3,x, algorithm="giac")

[Out]

integrate(4/3*(x*e^x*log(3)*log(4/x) + (x - 6)*e^x*log(3) + (x^4 - x^3)*log(3))*e^(-x - 1/3*(4*x^3*log(3) + 4*
x*e^x*log(3)*log(4/x) - 3*(3*x^2 + 4)*e^x*log(3))*e^(-x)/x^2)/x^3, x)

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maple [A]  time = 1.63, size = 34, normalized size = 0.97

method	result	size
risch	\(3^{\frac {9 x^{2}+4 x \ln \relax (x )-8 x \ln \relax (2)-4 x^{3} {\mathrm e}^{-x}+12}{3 x^{2}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(4*x*ln(3)*exp(x)*ln(4/x)+(4*x-24)*ln(3)*exp(x)+(4*x^4-4*x^3)*ln(3))*exp(1/3*(-4*x*ln(3)*exp(x)*ln(4/x
)+(9*x^2+12)*ln(3)*exp(x)-4*x^3*ln(3))/exp(x)/x^2)/exp(x)/x^3,x,method=_RETURNVERBOSE)

[Out]

3^(1/3*(9*x^2+4*x*ln(x)-8*x*ln(2)-4*x^3*exp(-x)+12)/x^2)

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maxima [A]  time = 1.18, size = 38, normalized size = 1.09 \begin {gather*} 27 \, e^{\left (-\frac {4}{3} \, x e^{\left (-x\right )} \log \relax (3) - \frac {8 \, \log \relax (3) \log \relax (2)}{3 \, x} + \frac {4 \, \log \relax (3) \log \relax (x)}{3 \, x} + \frac {4 \, \log \relax (3)}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x*log(3)*exp(x)*log(4/x)+(4*x-24)*log(3)*exp(x)+(4*x^4-4*x^3)*log(3))*exp(1/3*(-4*x*log(3)*ex
p(x)*log(4/x)+(9*x^2+12)*log(3)*exp(x)-4*x^3*log(3))/exp(x)/x^2)/exp(x)/x^3,x, algorithm="maxima")

[Out]

27*e^(-4/3*x*e^(-x)*log(3) - 8/3*log(3)*log(2)/x + 4/3*log(3)*log(x)/x + 4*log(3)/x^2)

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mupad [B]  time = 0.68, size = 43, normalized size = 1.23 \begin {gather*} \frac {27\,3^{\frac {4}{x^2}}\,{\mathrm {e}}^{-\frac {4\,\ln \left (\frac {1}{x}\right )\,\ln \relax (3)}{3\,x}}}{2^{\frac {8\,\ln \relax (3)}{3\,x}}\,3^{\frac {4\,x\,{\mathrm {e}}^{-x}}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*exp(-(exp(-x)*((4*x^3*log(3))/3 - (exp(x)*log(3)*(9*x^2 + 12))/3 + (4*x*exp(x)*log(3)*log(4/x))/3
))/x^2)*(exp(x)*log(3)*(4*x - 24) - log(3)*(4*x^3 - 4*x^4) + 4*x*exp(x)*log(3)*log(4/x)))/(3*x^3),x)

[Out]

(27*3^(4/x^2)*exp(-(4*log(1/x)*log(3))/(3*x)))/(2^((8*log(3))/(3*x))*3^((4*x*exp(-x))/3))

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sympy [A]  time = 0.65, size = 49, normalized size = 1.40 \begin {gather*} e^{\frac {\left (- \frac {4 x^{3} \log {\relax (3 )}}{3} - \frac {4 x e^{x} \log {\relax (3 )} \log {\left (\frac {4}{x} \right )}}{3} + \frac {\left (9 x^{2} + 12\right ) e^{x} \log {\relax (3 )}}{3}\right ) e^{- x}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x*ln(3)*exp(x)*ln(4/x)+(4*x-24)*ln(3)*exp(x)+(4*x**4-4*x**3)*ln(3))*exp(1/3*(-4*x*ln(3)*exp(x
)*ln(4/x)+(9*x**2+12)*ln(3)*exp(x)-4*x**3*ln(3))/exp(x)/x**2)/exp(x)/x**3,x)

[Out]

exp((-4*x**3*log(3)/3 - 4*x*exp(x)*log(3)*log(4/x)/3 + (9*x**2 + 12)*exp(x)*log(3)/3)*exp(-x)/x**2)

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