3.48.43 \(\int \frac {e^{\frac {4+x-x^2}{x}} (-4-x)-x^3}{(e^{\frac {4+x-x^2}{x}} x^2-x^3) \log (\frac {e^{e^3+x} (-e^{\frac {4+x-x^2}{x}}+x)}{x})} \, dx\) [4743]

Optimal. Leaf size=31 \[ \log \left (\log \left (\frac {e^{e^3+x} \left (-e^{\frac {4+x-x^2}{x}}+x\right )}{x}\right )\right ) \]

[Out]

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

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Rubi [F]
time = 3.67, 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 {e^{\frac {4+x-x^2}{x}} (-4-x)-x^3}{\left (e^{\frac {4+x-x^2}{x}} x^2-x^3\right ) \log \left (\frac {e^{e^3+x} \left (-e^{\frac {4+x-x^2}{x}}+x\right )}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 26, normalized size = 0.84 \begin {gather*} \log \left (\log \left (e^{e^3} \left (e^x-\frac {e^{1+\frac {4}{x}}}{x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[Log[E^E^3*(E^x - E^(1 + 4/x)/x)]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.58, size = 396, normalized size = 12.77

method	result	size
risch	\(\ln \left (\ln \left ({\mathrm e}^{x}\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right )-\pi \mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right )^{3}+\pi \mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{x}\right )+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right ) {\mathrm e}^{x}}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x} \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right ) {\mathrm e}^{x}}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-\pi \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right ) {\mathrm e}^{x}}{x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right ) {\mathrm e}^{x}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 i {\mathrm e}^{3}+2 i \ln \left (x \right )-2 i \ln \left (-{\mathrm e}^{-\frac {x^{2}-x -4}{x}}+x \right )\right )}{2}\right )\)	\(396\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(ln(exp(x))+1/2*I*(Pi*csgn(I*(-exp(-(x^2-x-4)/x)+x))*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x))^2-Pi*csgn(I*(-exp
(-(x^2-x-4)/x)+x))*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x))*csgn(I*exp(x))-Pi*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x
))^3+Pi*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x))^2*csgn(I*exp(x))+Pi*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x))*csgn(I
/x*(-exp(-(x^2-x-4)/x)+x)*exp(x))^2-Pi*csgn(I*exp(x)*(-exp(-(x^2-x-4)/x)+x))*csgn(I/x*(-exp(-(x^2-x-4)/x)+x)*e
xp(x))*csgn(I/x)-Pi*csgn(I/x*(-exp(-(x^2-x-4)/x)+x)*exp(x))^3+Pi*csgn(I/x*(-exp(-(x^2-x-4)/x)+x)*exp(x))^2*csg
n(I/x)-2*I*exp(3)+2*I*ln(x)-2*I*ln(-exp(-(x^2-x-4)/x)+x)))

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Maxima [A]
time = 0.33, size = 24, normalized size = 0.77 \begin {gather*} \log \left (e^{3} + \log \left (x e^{x} - e^{\left (\frac {4}{x} + 1\right )}\right ) - \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 29, normalized size = 0.94 \begin {gather*} \log \left (\log \left (\frac {{\left (x - e^{\left (-\frac {x^{2} - x - 4}{x}\right )}\right )} e^{\left (x + e^{3}\right )}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.35, size = 24, normalized size = 0.77 \begin {gather*} \log {\left (\log {\left (\frac {\left (x - e^{\frac {- x^{2} + x + 4}{x}}\right ) e^{x} e^{e^{3}}}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
time = 0.47, size = 62, normalized size = 2.00 \begin {gather*} \log \left (\log \left (\frac {{\left (x e^{\left (\frac {x e^{3} + x + 4}{x}\right )} - e^{\left (-\frac {x^{2} - x - 4}{x} + \frac {x e^{3} + x + 4}{x}\right )}\right )} e^{\left (\frac {x^{2} - x - 4}{x}\right )}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.90, size = 25, normalized size = 0.81 \begin {gather*} \ln \left (\ln \left ({\mathrm {e}}^{{\mathrm {e}}^3}\,{\mathrm {e}}^x-\frac {\mathrm {e}\,{\mathrm {e}}^{4/x}\,{\mathrm {e}}^{{\mathrm {e}}^3}}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(exp(exp(3))*exp(x) - (exp(1)*exp(4/x)*exp(exp(3)))/x))

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