3.96.76 \(\int \frac {4+e^{\frac {e^{3 x}}{x}} (-1+\frac {e^{3 x} (-1+3 x) \log (-\frac {x}{3})}{x})}{x \log ^2(-\frac {x}{3})} \, dx\) [9576]

Optimal. Leaf size=22 \[ \frac {-4+e^{\frac {e^{3 x}}{x}}}{\log \left (-\frac {x}{3}\right )} \]

[Out]

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

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Rubi [F]
time = 2.03, 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 {4+e^{\frac {e^{3 x}}{x}} \left (-1+\frac {e^{3 x} (-1+3 x) \log \left (-\frac {x}{3}\right )}{x}\right )}{x \log ^2\left (-\frac {x}{3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.21, size = 22, normalized size = 1.00 \begin {gather*} \frac {-4+e^{\frac {e^{3 x}}{x}}}{\log \left (-\frac {x}{3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4 + E^(E^(3*x)/x))/Log[-1/3*x]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.26, size = 89, normalized size = 4.05

method	result	size
risch	\(-\frac {8 i}{2 \pi \mathrm {csgn}\left (i x \right )^{2}-2 \pi \mathrm {csgn}\left (i x \right )^{3}-2 \pi -2 i \ln \left (3\right )+2 i \ln \left (x \right )}+\frac {2 i {\mathrm e}^{\frac {{\mathrm e}^{3 x}}{x}}}{2 \pi \mathrm {csgn}\left (i x \right )^{2}-2 \pi \mathrm {csgn}\left (i x \right )^{3}-2 \pi -2 i \ln \left (3\right )+2 i \ln \left (x \right )}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*I/(2*Pi*csgn(I*x)^2-2*Pi*csgn(I*x)^3-2*Pi-2*I*ln(3)+2*I*ln(x))+2*I/(2*Pi*csgn(I*x)^2-2*Pi*csgn(I*x)^3-2*Pi-
2*I*ln(3)+2*I*ln(x))*exp(1/x*exp(3*x))

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Maxima [A]
time = 0.49, size = 31, normalized size = 1.41 \begin {gather*} -\frac {e^{\left (\frac {e^{\left (3 \, x\right )}}{x}\right )}}{\log \left (3\right ) - \log \left (-x\right )} - \frac {4}{\log \left (-\frac {1}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-1+3*x)*log(-1/3*x)*exp(-log(x)+3*x)-1)*exp(exp(-log(x)+3*x))+4)/x/log(-1/3*x)^2,x, algorithm="ma
xima")

[Out]

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

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Fricas [A]
time = 0.38, size = 27, normalized size = 1.23 \begin {gather*} \frac {e^{\left (-e^{\left (3 \, x - \log \left (3\right ) - \log \left (-\frac {1}{3} \, x\right )\right )}\right )} - 4}{\log \left (-\frac {1}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-1+3*x)*log(-1/3*x)*exp(-log(x)+3*x)-1)*exp(exp(-log(x)+3*x))+4)/x/log(-1/3*x)^2,x, algorithm="fr
icas")

[Out]

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

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Sympy [A]
time = 0.14, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {e^{3 x}}{x}}}{\log {\left (- \frac {x}{3} \right )}} - \frac {4}{\log {\left (- \frac {x}{3} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(3*x)/x)/log(-x/3) - 4/log(-x/3)

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Giac [A]
time = 0.40, size = 36, normalized size = 1.64 \begin {gather*} -\frac {{\left (4 \, e^{\left (3 \, x\right )} - e^{\left (\frac {3 \, x^{2} + e^{\left (3 \, x\right )}}{x}\right )}\right )} e^{\left (-3 \, x\right )}}{\log \left (-\frac {1}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-1+3*x)*log(-1/3*x)*exp(-log(x)+3*x)-1)*exp(exp(-log(x)+3*x))+4)/x/log(-1/3*x)^2,x, algorithm="gi
ac")

[Out]

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

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Mupad [B]
time = 6.48, size = 18, normalized size = 0.82 \begin {gather*} \frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{3\,x}}{x}}-4}{\ln \left (-\frac {x}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(exp(3*x)/x) - 4)/log(-x/3)

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