\(\int \frac {e^{-\frac {e^x}{x}} (e^{2 x} (9-9 x)+24 e^x x+24 x^2+(-16 e^x x-16 x^2+e^{2 x} (-12+12 x)) \log (\frac {4}{x^2})+e^{2 x} (4-4 x) \log ^2(\frac {4}{x^2}))}{4 x^3} \, dx\) [7849]

Optimal result

Integrand size = 89, antiderivative size = 31 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {e^{-\frac {e^x}{x}} \left (e^x+x\right ) \left (-\frac {3}{2}+\log \left (\frac {4}{x^2}\right )\right )^2}{x} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx \]

[In]

Int[(E^(2*x)*(9 - 9*x) + 24*E^x*x + 24*x^2 + (-16*E^x*x - 16*x^2 + E^(2*x)*(-12 + 12*x))*Log[4/x^2] + E^(2*x)*
(4 - 4*x)*Log[4/x^2]^2)/(4*E^(E^x/x)*x^3),x]

[Out]

(9*Defer[Int][E^(-(E^x/x) + 2*x)/x^3, x])/4 - 3*Log[4/x^2]*Defer[Int][E^(-(E^x/x) + 2*x)/x^3, x] + 6*Defer[Int
][E^(-(E^x/x) + x)/x^2, x] - 4*Log[4/x^2]*Defer[Int][E^(-(E^x/x) + x)/x^2, x] - (9*Defer[Int][E^(-(E^x/x) + 2*
x)/x^2, x])/4 + 3*Log[4/x^2]*Defer[Int][E^(-(E^x/x) + 2*x)/x^2, x] + 6*Defer[Int][1/(E^(E^x/x)*x), x] - 4*Log[
4/x^2]*Defer[Int][1/(E^(E^x/x)*x), x] + Defer[Int][(E^(-(E^x/x) + 2*x)*Log[4/x^2]^2)/x^3, x] - Defer[Int][(E^(
-(E^x/x) + 2*x)*Log[4/x^2]^2)/x^2, x] - 6*Defer[Int][Defer[Int][E^(-(E^x/x) + 2*x)/x^3, x]/x, x] - 8*Defer[Int
][Defer[Int][E^(-(E^x/x) + x)/x^2, x]/x, x] + 6*Defer[Int][Defer[Int][E^(-(E^x/x) + 2*x)/x^2, x]/x, x] - 8*Def
er[Int][Defer[Int][1/(E^(E^x/x)*x), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{x^3} \, dx \\ & = \frac {1}{4} \int \frac {e^{-\frac {e^x}{x}} \left (3-2 \log \left (\frac {4}{x^2}\right )\right ) \left (-3 e^{2 x} (-1+x)+8 e^x x+8 x^2+2 e^{2 x} (-1+x) \log \left (\frac {4}{x^2}\right )\right )}{x^3} \, dx \\ & = \frac {1}{4} \int \left (-\frac {8 e^{-\frac {e^x}{x}+x} \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )}{x^2}-\frac {8 e^{-\frac {e^x}{x}} \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )}{x}-\frac {e^{-\frac {e^x}{x}+2 x} (-1+x) \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )^2}{x^3}\right ) \, dx \\ & = -\left (\frac {1}{4} \int \frac {e^{-\frac {e^x}{x}+2 x} (-1+x) \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )^2}{x^3} \, dx\right )-2 \int \frac {e^{-\frac {e^x}{x}+x} \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )}{x^2} \, dx-2 \int \frac {e^{-\frac {e^x}{x}} \left (-3+2 \log \left (\frac {4}{x^2}\right )\right )}{x} \, dx \\ & = -\left (\frac {1}{4} \int \left (\frac {9 e^{-\frac {e^x}{x}+2 x} (-1+x)}{x^3}-\frac {12 e^{-\frac {e^x}{x}+2 x} (-1+x) \log \left (\frac {4}{x^2}\right )}{x^3}+\frac {4 e^{-\frac {e^x}{x}+2 x} (-1+x) \log ^2\left (\frac {4}{x^2}\right )}{x^3}\right ) \, dx\right )-2 \int \left (-\frac {3 e^{-\frac {e^x}{x}+x}}{x^2}+\frac {2 e^{-\frac {e^x}{x}+x} \log \left (\frac {4}{x^2}\right )}{x^2}\right ) \, dx-2 \int \left (-\frac {3 e^{-\frac {e^x}{x}}}{x}+\frac {2 e^{-\frac {e^x}{x}} \log \left (\frac {4}{x^2}\right )}{x}\right ) \, dx \\ & = -\left (\frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x} (-1+x)}{x^3} \, dx\right )+3 \int \frac {e^{-\frac {e^x}{x}+2 x} (-1+x) \log \left (\frac {4}{x^2}\right )}{x^3} \, dx-4 \int \frac {e^{-\frac {e^x}{x}+x} \log \left (\frac {4}{x^2}\right )}{x^2} \, dx-4 \int \frac {e^{-\frac {e^x}{x}} \log \left (\frac {4}{x^2}\right )}{x} \, dx+6 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-\int \frac {e^{-\frac {e^x}{x}+2 x} (-1+x) \log ^2\left (\frac {4}{x^2}\right )}{x^3} \, dx \\ & = -\left (\frac {9}{4} \int \left (-\frac {e^{-\frac {e^x}{x}+2 x}}{x^3}+\frac {e^{-\frac {e^x}{x}+2 x}}{x^2}\right ) \, dx\right )-3 \int -\frac {2 \left (-\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx\right )}{x} \, dx+4 \int -\frac {2 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x} \, dx+4 \int -\frac {2 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx+6 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-\int \left (-\frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^3}+\frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^2}\right ) \, dx \\ & = \frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx-\frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+6 \int \frac {-\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx}{x} \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x} \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx-\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^3} \, dx-\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^2} \, dx \\ & = \frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx-\frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+6 \int \left (-\frac {\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx}{x}+\frac {\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx}{x}\right ) \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x} \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx-\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^3} \, dx-\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^2} \, dx \\ & = \frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx-\frac {9}{4} \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+6 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-6 \int \frac {\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx}{x} \, dx+6 \int \frac {\int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx}{x} \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x} \, dx-8 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx-\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^3} \, dx+\left (3 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+2 x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-\left (4 \log \left (\frac {4}{x^2}\right )\right ) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^3} \, dx-\int \frac {e^{-\frac {e^x}{x}+2 x} \log ^2\left (\frac {4}{x^2}\right )}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {e^{-\frac {e^x}{x}} \left (e^x+x\right ) \left (3-2 \log \left (\frac {4}{x^2}\right )\right )^2}{4 x} \]

[In]

Integrate[(E^(2*x)*(9 - 9*x) + 24*E^x*x + 24*x^2 + (-16*E^x*x - 16*x^2 + E^(2*x)*(-12 + 12*x))*Log[4/x^2] + E^
(2*x)*(4 - 4*x)*Log[4/x^2]^2)/(4*E^(E^x/x)*x^3),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(27)=54\).

Time = 0.63 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10

method	result	size
parallelrisch	\(\frac {\left (4 x \ln \left (\frac {4}{x^{2}}\right )^{2}+4 \ln \left (\frac {4}{x^{2}}\right )^{2} {\mathrm e}^{x}-12 x \ln \left (\frac {4}{x^{2}}\right )-12 \,{\mathrm e}^{x} \ln \left (\frac {4}{x^{2}}\right )+9 x +9 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}}{4 x}\)	\(65\)
risch	\(\frac {\left (9 x +16 \,{\mathrm e}^{x} \ln \left (x \right )^{2}-24 \,{\mathrm e}^{x} \ln \left (2\right )+16 \ln \left (2\right )^{2} {\mathrm e}^{x}+16 x \ln \left (2\right )^{2}+24 \,{\mathrm e}^{x} \ln \left (x \right )+16 x \ln \left (x \right )^{2}-24 x \ln \left (2\right )+24 x \ln \left (x \right )-32 x \ln \left (2\right ) \ln \left (x \right )+9 \,{\mathrm e}^{x}+8 i \pi \ln \left (2\right ) x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 i \pi \ln \left (2\right ) x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) \ln \left (x \right )-x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} {\mathrm e}^{x}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-6 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-6 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}+4 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-6 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} {\mathrm e}^{x}+4 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5} {\mathrm e}^{x}+8 i \pi \ln \left (2\right ) x \operatorname {csgn}\left (i x^{2}\right )^{3}+8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}-8 i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3} \ln \left (x \right )+12 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-32 \ln \left (2\right ) {\mathrm e}^{x} \ln \left (x \right )-6 i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-6 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}+12 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-8 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x} \ln \left (x \right )+8 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}-8 i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) {\mathrm e}^{x} \ln \left (x \right )+16 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x} \ln \left (x \right )+16 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} \ln \left (x \right )-16 i \pi \ln \left (2\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}}{4 x}\)	\(641\)

[In]

int(1/4*((4-4*x)*exp(x)^2*ln(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x)*x-16*x^2)*ln(4/x^2)+(-9*x+9)*exp(x)^2+24*e
xp(x)*x+24*x^2)/x^3/exp(exp(x)/x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {{\left (4 \, {\left (x + e^{x}\right )} \log \left (\frac {4}{x^{2}}\right )^{2} - 12 \, {\left (x + e^{x}\right )} \log \left (\frac {4}{x^{2}}\right ) + 9 \, x + 9 \, e^{x}\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x} \]

[In]

integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x)*x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x
)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x)/x),x, algorithm="fricas")

[Out]

1/4*(4*(x + e^x)*log(4/x^2)^2 - 12*(x + e^x)*log(4/x^2) + 9*x + 9*e^x)*e^(-e^x/x)/x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\frac {\left (4 x \log {\left (\frac {4}{x^{2}} \right )}^{2} - 12 x \log {\left (\frac {4}{x^{2}} \right )} + 9 x + 4 e^{x} \log {\left (\frac {4}{x^{2}} \right )}^{2} - 12 e^{x} \log {\left (\frac {4}{x^{2}} \right )} + 9 e^{x}\right ) e^{- \frac {e^{x}}{x}}}{4 x} \]

[In]

integrate(1/4*((4-4*x)*exp(x)**2*ln(4/x**2)**2+((12*x-12)*exp(x)**2-16*exp(x)*x-16*x**2)*ln(4/x**2)+(-9*x+9)*e
xp(x)**2+24*exp(x)*x+24*x**2)/x**3/exp(exp(x)/x),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (29) = 58\).

Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=-\frac {{\left (8 \, x {\left (4 \, \log \left (2\right ) - 3\right )} \log \left (x\right ) - 16 \, x \log \left (x\right )^{2} - {\left (16 \, \log \left (2\right )^{2} - 24 \, \log \left (2\right ) + 9\right )} x - {\left (16 \, \log \left (2\right )^{2} - 8 \, {\left (4 \, \log \left (2\right ) - 3\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 24 \, \log \left (2\right ) + 9\right )} e^{x}\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x} \]

[In]

integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x)*x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x
)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x)/x),x, algorithm="maxima")

[Out]

-1/4*(8*x*(4*log(2) - 3)*log(x) - 16*x*log(x)^2 - (16*log(2)^2 - 24*log(2) + 9)*x - (16*log(2)^2 - 8*(4*log(2)
- 3)*log(x) + 16*log(x)^2 - 24*log(2) + 9)*e^x)*e^(-e^x/x)/x

Giac [F]

\[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\int { -\frac {{\left (4 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} \log \left (\frac {4}{x^{2}}\right )^{2} - 24 \, x^{2} + 9 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} - 24 \, x e^{x} + 4 \, {\left (4 \, x^{2} - 3 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} + 4 \, x e^{x}\right )} \log \left (\frac {4}{x^{2}}\right )\right )} e^{\left (-\frac {e^{x}}{x}\right )}}{4 \, x^{3}} \,d x } \]

[In]

integrate(1/4*((4-4*x)*exp(x)^2*log(4/x^2)^2+((12*x-12)*exp(x)^2-16*exp(x)*x-16*x^2)*log(4/x^2)+(-9*x+9)*exp(x
)^2+24*exp(x)*x+24*x^2)/x^3/exp(exp(x)/x),x, algorithm="giac")

[Out]

integrate(-1/4*(4*(x - 1)*e^(2*x)*log(4/x^2)^2 - 24*x^2 + 9*(x - 1)*e^(2*x) - 24*x*e^x + 4*(4*x^2 - 3*(x - 1)*
e^(2*x) + 4*x*e^x)*log(4/x^2))*e^(-e^x/x)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {e^x}{x}} \left (e^{2 x} (9-9 x)+24 e^x x+24 x^2+\left (-16 e^x x-16 x^2+e^{2 x} (-12+12 x)\right ) \log \left (\frac {4}{x^2}\right )+e^{2 x} (4-4 x) \log ^2\left (\frac {4}{x^2}\right )\right )}{4 x^3} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{x}}\,\left (\frac {\ln \left (\frac {4}{x^2}\right )\,\left (16\,x\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,\left (12\,x-12\right )+16\,x^2\right )}{4}-6\,x\,{\mathrm {e}}^x+\frac {{\mathrm {e}}^{2\,x}\,\left (9\,x-9\right )}{4}-6\,x^2+\frac {{\mathrm {e}}^{2\,x}\,{\ln \left (\frac {4}{x^2}\right )}^2\,\left (4\,x-4\right )}{4}\right )}{x^3} \,d x \]

[In]

int(-(exp(-exp(x)/x)*((log(4/x^2)*(16*x*exp(x) - exp(2*x)*(12*x - 12) + 16*x^2))/4 - 6*x*exp(x) + (exp(2*x)*(9
*x - 9))/4 - 6*x^2 + (exp(2*x)*log(4/x^2)^2*(4*x - 4))/4))/x^3,x)

[Out]

int(-(exp(-exp(x)/x)*((log(4/x^2)*(16*x*exp(x) - exp(2*x)*(12*x - 12) + 16*x^2))/4 - 6*x*exp(x) + (exp(2*x)*(9
*x - 9))/4 - 6*x^2 + (exp(2*x)*log(4/x^2)^2*(4*x - 4))/4))/x^3, x)