3.40.7 \(\int \frac {48+e^{2 e^{\frac {x^2}{2}}} (4+e^{3+x} (-1+x))+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} (4 x^2-e^{3+x} x^2))}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} (28 x-7 e^{3+x} x)+e^{2 e^{\frac {x^2}{2}}} (-4 x+e^{3+x} x)} \, dx\)

Optimal. Leaf size=36 \[ -2+\log \left (\frac {4-e^{3+x}}{x+\frac {x}{3-e^{e^{\frac {x^2}{2}}}}}\right ) \]

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Rubi [F]  time = 20.88, 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 {48+e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )+e^{3+x} (-12+12 x)+e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{-48 x+12 e^{3+x} x+e^{e^{\frac {x^2}{2}}} \left (28 x-7 e^{3+x} x\right )+e^{2 e^{\frac {x^2}{2}}} \left (-4 x+e^{3+x} x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48 + E^(2*E^(x^2/2))*(4 + E^(3 + x)*(-1 + x)) + E^(3 + x)*(-12 + 12*x) + E^E^(x^2/2)*(-28 + E^(3 + x)*(7
- 7*x) + E^(x^2/2)*(4*x^2 - E^(3 + x)*x^2)))/(-48*x + 12*E^(3 + x)*x + E^E^(x^2/2)*(28*x - 7*E^(3 + x)*x) + E^
(2*E^(x^2/2))*(-4*x + E^(3 + x)*x)),x]

[Out]

x + Log[3 - E^E^(x^2/2)] - Log[4 - E^E^(x^2/2)] + 16*Defer[Int][(-4 + E^E^(x^2/2))^(-1), x] - 9*Defer[Int][(-3
+ E^E^(x^2/2))^(-1), x] + 4*Defer[Int][E^(2*E^(x^2/2))/((-4 + E^E^(x^2/2))*(-4 + E^(3 + x))), x] + 12*Defer[I
nt][E^(3 + x)/((-4 + E^E^(x^2/2))*(-4 + E^(3 + x))), x] - 7*Defer[Int][E^(3 + E^(x^2/2) + x)/((-4 + E^E^(x^2/2
))*(-4 + E^(3 + x))), x] - 4*Defer[Int][E^(2*E^(x^2/2))/((-3 + E^E^(x^2/2))*(-4 + E^(3 + x))), x] - 12*Defer[I
nt][E^(3 + x)/((-3 + E^E^(x^2/2))*(-4 + E^(3 + x))), x] + 7*Defer[Int][E^(3 + E^(x^2/2) + x)/((-3 + E^E^(x^2/2
))*(-4 + E^(3 + x))), x] + 48*Defer[Int][1/((3 - E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] - 28*Defer[Int][E^E^(x^2
/2)/((3 - E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] + 48*Defer[Int][1/((-4 + E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] -
28*Defer[Int][E^E^(x^2/2)/((-4 + E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] - 12*Defer[Int][E^(3 + x)/((-4 + E^E^(x
^2/2))*(-4 + E^(3 + x))*x), x] + 7*Defer[Int][E^(3 + E^(x^2/2) + x)/((-4 + E^E^(x^2/2))*(-4 + E^(3 + x))*x), x
] + 12*Defer[Int][E^(3 + x)/((-3 + E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] - 7*Defer[Int][E^(3 + E^(x^2/2) + x)/(
(-3 + E^E^(x^2/2))*(-4 + E^(3 + x))*x), x] - Defer[Subst][Defer[Int][E^(2*E^(x/2))/((-4 + E^E^(x/2))*x), x], x
, x^2]/2 + Defer[Subst][Defer[Int][E^(2*E^(x/2))/((-3 + E^E^(x/2))*x), x], x, x^2]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-48-e^{2 e^{\frac {x^2}{2}}} \left (4+e^{3+x} (-1+x)\right )-e^{3+x} (-12+12 x)-e^{e^{\frac {x^2}{2}}} \left (-28+e^{3+x} (7-7 x)+e^{\frac {x^2}{2}} \left (4 x^2-e^{3+x} x^2\right )\right )}{\left (12-7 e^{e^{\frac {x^2}{2}}}+e^{2 e^{\frac {x^2}{2}}}\right ) \left (4-e^{3+x}\right ) x} \, dx\\ &=\int \left (\frac {48}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}-\frac {28 e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {12 e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}-\frac {7 e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{\frac {1}{2} \left (2 e^{\frac {x^2}{2}}+x^2\right )} x}{-12+7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}}+\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx\\ &=-\left (7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\right )+12 \int \frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{\frac {1}{2} \left (2 e^{\frac {x^2}{2}}+x^2\right )} x}{-12+7 e^{e^{\frac {x^2}{2}}}-e^{2 e^{\frac {x^2}{2}}}} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{\frac {1}{2} \left (2 e^{x/2}+x\right )}}{-12+7 e^{e^{x/2}}-e^{2 e^{x/2}}} \, dx,x,x^2\right )-7 \int \left (\frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (\frac {e^{3+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx-28 \int \left (\frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+48 \int \left (\frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+\int \left (\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}+\frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx\\ &=-\left (7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\right )-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x} (-1+x)}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} \left (4-e^{3+x}+e^{3+x} x\right )}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\operatorname {Subst}\left (\int \frac {e^{e^x+x}}{-12+7 e^{e^x}-e^{2 e^x}} \, dx,x,\frac {x^2}{2}\right )\\ &=-\left (7 \int \left (\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx\right )-7 \int \left (-\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (\frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx+12 \int \left (-\frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x}\right ) \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}+\frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx+\int \left (-\frac {4 e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )}-\frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx+\operatorname {Subst}\left (\int \frac {e^x}{-12+7 e^x-e^{2 x}} \, dx,x,e^{\frac {x^2}{2}}\right )\\ &=4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}} (-1+x)}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx+\operatorname {Subst}\left (\int \frac {1}{-12+7 x-x^2} \, dx,x,e^{e^{\frac {x^2}{2}}}\right )\\ &=4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (\frac {e^{2 e^{\frac {x^2}{2}}}}{-4+e^{e^{\frac {x^2}{2}}}}-\frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx-\int \left (\frac {e^{2 e^{\frac {x^2}{2}}}}{-3+e^{e^{\frac {x^2}{2}}}}-\frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{3-x} \, dx,x,e^{e^{\frac {x^2}{2}}}\right )+\operatorname {Subst}\left (\int \frac {1}{4-x} \, dx,x,e^{e^{\frac {x^2}{2}}}\right )\\ &=\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}}}{-4+e^{e^{\frac {x^2}{2}}}} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}}}{-3+e^{e^{\frac {x^2}{2}}}} \, dx-\int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx+\int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) x} \, dx\\ &=\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-4+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-3+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+\int \left (4+e^{e^{\frac {x^2}{2}}}+\frac {16}{-4+e^{e^{\frac {x^2}{2}}}}\right ) \, dx-\int \left (3+e^{e^{\frac {x^2}{2}}}+\frac {9}{-3+e^{e^{\frac {x^2}{2}}}}\right ) \, dx\\ &=x+\log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-4+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{2 e^{x/2}}}{\left (-3+e^{e^{x/2}}\right ) x} \, dx,x,x^2\right )+4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-4 \int \frac {e^{2 e^{\frac {x^2}{2}}}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx+7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-7 \int \frac {e^{3+e^{\frac {x^2}{2}}+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-9 \int \frac {1}{-3+e^{e^{\frac {x^2}{2}}}} \, dx+12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right )} \, dx-12 \int \frac {e^{3+x}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+12 \int \frac {e^{3+x}}{\left (-3+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+16 \int \frac {1}{-4+e^{e^{\frac {x^2}{2}}}} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx-28 \int \frac {e^{e^{\frac {x^2}{2}}}}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (3-e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx+48 \int \frac {1}{\left (-4+e^{e^{\frac {x^2}{2}}}\right ) \left (-4+e^{3+x}\right ) x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 49, normalized size = 1.36 \begin {gather*} \log \left (3-e^{e^{\frac {x^2}{2}}}\right )-\log \left (4-e^{e^{\frac {x^2}{2}}}\right )+\log \left (4-e^{3+x}\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 + E^(2*E^(x^2/2))*(4 + E^(3 + x)*(-1 + x)) + E^(3 + x)*(-12 + 12*x) + E^E^(x^2/2)*(-28 + E^(3 +
x)*(7 - 7*x) + E^(x^2/2)*(4*x^2 - E^(3 + x)*x^2)))/(-48*x + 12*E^(3 + x)*x + E^E^(x^2/2)*(28*x - 7*E^(3 + x)*x
) + E^(2*E^(x^2/2))*(-4*x + E^(3 + x)*x)),x]

[Out]

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

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fricas [A]  time = 0.75, size = 34, normalized size = 0.94 \begin {gather*} -\log \relax (x) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*ex
p(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*
x^2))+12*exp(3+x)*x-48*x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 9.03, size = 34, normalized size = 0.94 \begin {gather*} -\log \relax (x) + \log \left (e^{\left (x + 3\right )} - 4\right ) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*ex
p(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*
x^2))+12*exp(3+x)*x-48*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.35, size = 36, normalized size = 1.00

method	result	size
risch	\(-\ln \relax (x )-3+\ln \left ({\mathrm e}^{3+x}-4\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-3\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}}{2}}}-4\right )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-1)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*exp(exp(
1/2*x^2))+(12*x-12)*exp(3+x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*x^2))+
12*exp(3+x)*x-48*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 37, normalized size = 1.03 \begin {gather*} \log \left ({\left (e^{\left (x + 3\right )} - 4\right )} e^{\left (-3\right )}\right ) - \log \relax (x) + \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 3\right ) - \log \left (e^{\left (e^{\left (\frac {1}{2} \, x^{2}\right )}\right )} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(3+x)+4)*exp(exp(1/2*x^2))^2+((-x^2*exp(3+x)+4*x^2)*exp(1/2*x^2)+(-7*x+7)*exp(3+x)-28)*ex
p(exp(1/2*x^2))+(12*x-12)*exp(3+x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x^2))^2+(-7*exp(3+x)*x+28*x)*exp(exp(1/2*
x^2))+12*exp(3+x)*x-48*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.28, size = 69, normalized size = 1.92 \begin {gather*} \ln \left ({\mathrm {e}}^3\,{\mathrm {e}}^x-4\right )-\ln \left (24\,x\,\sqrt {{\mathrm {e}}^{x^2}}-6\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}\right )+\ln \left (8\,x\,\sqrt {{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^{\sqrt {{\mathrm {e}}^{x^2}}}-24\,x\,\sqrt {{\mathrm {e}}^{x^2}}\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*exp(x^2/2))*(exp(x + 3)*(x - 1) + 4) - exp(exp(x^2/2))*(exp(x^2/2)*(x^2*exp(x + 3) - 4*x^2) + exp(
x + 3)*(7*x - 7) + 28) + exp(x + 3)*(12*x - 12) + 48)/(48*x + exp(2*exp(x^2/2))*(4*x - x*exp(x + 3)) - 12*x*ex
p(x + 3) - exp(exp(x^2/2))*(28*x - 7*x*exp(x + 3))),x)

[Out]

log(exp(3)*exp(x) - 4) - log(24*x*exp(x^2)^(1/2) - 6*x*exp(x^2)^(1/2)*exp(exp(x^2)^(1/2))) + log(8*x*exp(x^2)^
(1/2)*exp(exp(x^2)^(1/2)) - 24*x*exp(x^2)^(1/2)) - log(x)

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sympy [A]  time = 0.32, size = 34, normalized size = 0.94 \begin {gather*} - \log {\relax (x )} + \log {\left (e^{x + 3} - 4 \right )} - \log {\left (e^{e^{\frac {x^{2}}{2}}} - 4 \right )} + \log {\left (e^{e^{\frac {x^{2}}{2}}} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(3+x)+4)*exp(exp(1/2*x**2))**2+((-x**2*exp(3+x)+4*x**2)*exp(1/2*x**2)+(-7*x+7)*exp(3+x)-2
8)*exp(exp(1/2*x**2))+(12*x-12)*exp(3+x)+48)/((exp(3+x)*x-4*x)*exp(exp(1/2*x**2))**2+(-7*exp(3+x)*x+28*x)*exp(
exp(1/2*x**2))+12*exp(3+x)*x-48*x),x)

[Out]

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

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