3.53.46 \(\int \frac {-240 e^2+e^{6 x} (240 x-2 x^3)+e^3 (-240 x+2 x^3)+e^{2 x} (e^2 (720 x-6 x^3)+e (480+4 x^3))+e^{4 x} (-240-4 x^3+e (-720 x+6 x^3))}{e^{\frac {120}{x^2}+6 x} x^6+e^{\frac {120}{x^2}+4 x} (-3 x^5-3 e x^6)+e^{\frac {120}{x^2}+2 x} (3 x^4+6 e x^5+3 e^2 x^6)+e^{\frac {120}{x^2}} (-x^3-3 e x^4-3 e^2 x^5-e^3 x^6)} \, dx\)

Optimal. Leaf size=23 \[ \frac {e^{-\frac {120}{x^2}}}{\left (\frac {1}{e-e^{2 x}}+x\right )^2} \]

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Rubi [F]  time = 9.62, 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 {-240 e^2+e^{6 x} \left (240 x-2 x^3\right )+e^3 \left (-240 x+2 x^3\right )+e^{2 x} \left (e^2 \left (720 x-6 x^3\right )+e \left (480+4 x^3\right )\right )+e^{4 x} \left (-240-4 x^3+e \left (-720 x+6 x^3\right )\right )}{e^{\frac {120}{x^2}+6 x} x^6+e^{\frac {120}{x^2}+4 x} \left (-3 x^5-3 e x^6\right )+e^{\frac {120}{x^2}+2 x} \left (3 x^4+6 e x^5+3 e^2 x^6\right )+e^{\frac {120}{x^2}} \left (-x^3-3 e x^4-3 e^2 x^5-e^3 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-240*E^2 + E^(6*x)*(240*x - 2*x^3) + E^3*(-240*x + 2*x^3) + E^(2*x)*(E^2*(720*x - 6*x^3) + E*(480 + 4*x^3
)) + E^(4*x)*(-240 - 4*x^3 + E*(-720*x + 6*x^3)))/(E^(120/x^2 + 6*x)*x^6 + E^(120/x^2 + 4*x)*(-3*x^5 - 3*E*x^6
) + E^(120/x^2 + 2*x)*(3*x^4 + 6*E*x^5 + 3*E^2*x^6) + E^(120/x^2)*(-x^3 - 3*E*x^4 - 3*E^2*x^5 - E^3*x^6)),x]

[Out]

1/(E^(120/x^2)*x^2) + 4*Defer[Int][E^(1 - 120/x^2)/(x*(1 + E*x - E^(2*x)*x)^3), x] - 4*Defer[Int][E^(1 - 120/x
^2)/(x*(1 + E*x - E^(2*x)*x)^2), x] - 2*Defer[Int][1/(E^(120/x^2)*x^3*(-1 - E*x + E^(2*x)*x)^3), x] - 4*Defer[
Int][1/(E^(120/x^2)*x^2*(-1 - E*x + E^(2*x)*x)^3), x] + 240*Defer[Int][1/(E^(120/x^2)*x^5*(-1 - E*x + E^(2*x)*
x)^2), x] - 6*Defer[Int][1/(E^(120/x^2)*x^3*(-1 - E*x + E^(2*x)*x)^2), x] - 8*Defer[Int][1/(E^(120/x^2)*x^2*(-
1 - E*x + E^(2*x)*x)^2), x] + 480*Defer[Int][1/(E^(120/x^2)*x^5*(-1 - E*x + E^(2*x)*x)), x] - 6*Defer[Int][1/(
E^(120/x^2)*x^3*(-1 - E*x + E^(2*x)*x)), x] - 4*Defer[Int][1/(E^(120/x^2)*x^2*(-1 - E*x + E^(2*x)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-\frac {120}{x^2}} \left (e-e^{2 x}\right ) \left (120 e-e^2 x \left (-120+x^2\right )-e^{4 x} x \left (-120+x^2\right )+2 e^{1+2 x} x \left (-120+x^2\right )-2 e^{2 x} \left (60+x^3\right )\right )}{x^3 \left (1+e x-e^{2 x} x\right )^3} \, dx\\ &=2 \int \frac {e^{-\frac {120}{x^2}} \left (e-e^{2 x}\right ) \left (120 e-e^2 x \left (-120+x^2\right )-e^{4 x} x \left (-120+x^2\right )+2 e^{1+2 x} x \left (-120+x^2\right )-2 e^{2 x} \left (60+x^3\right )\right )}{x^3 \left (1+e x-e^{2 x} x\right )^3} \, dx\\ &=2 \int \left (\frac {e^{-\frac {120}{x^2}} \left (120-x^2\right )}{x^5}-\frac {e^{-\frac {120}{x^2}} \left (1+2 x+2 e x^2\right )}{x^3 \left (-1-e x+e^{2 x} x\right )^3}-\frac {e^{-\frac {120}{x^2}} \left (-240+3 x^2+2 x^3\right )}{x^5 \left (-1-e x+e^{2 x} x\right )}-\frac {e^{-\frac {120}{x^2}} \left (-120+3 x^2+4 x^3+2 e x^4\right )}{x^5 \left (-1-e x+e^{2 x} x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{-\frac {120}{x^2}} \left (120-x^2\right )}{x^5} \, dx-2 \int \frac {e^{-\frac {120}{x^2}} \left (1+2 x+2 e x^2\right )}{x^3 \left (-1-e x+e^{2 x} x\right )^3} \, dx-2 \int \frac {e^{-\frac {120}{x^2}} \left (-240+3 x^2+2 x^3\right )}{x^5 \left (-1-e x+e^{2 x} x\right )} \, dx-2 \int \frac {e^{-\frac {120}{x^2}} \left (-120+3 x^2+4 x^3+2 e x^4\right )}{x^5 \left (-1-e x+e^{2 x} x\right )^2} \, dx\\ &=\frac {e^{-\frac {120}{x^2}}}{x^2}-2 \int \left (-\frac {2 e^{1-\frac {120}{x^2}}}{x \left (1+e x-e^{2 x} x\right )^3}+\frac {e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )^3}+\frac {2 e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )^3}\right ) \, dx-2 \int \left (\frac {2 e^{1-\frac {120}{x^2}}}{x \left (1+e x-e^{2 x} x\right )^2}-\frac {120 e^{-\frac {120}{x^2}}}{x^5 \left (-1-e x+e^{2 x} x\right )^2}+\frac {3 e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )^2}+\frac {4 e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )^2}\right ) \, dx-2 \int \left (-\frac {240 e^{-\frac {120}{x^2}}}{x^5 \left (-1-e x+e^{2 x} x\right )}+\frac {3 e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )}+\frac {2 e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )}\right ) \, dx\\ &=\frac {e^{-\frac {120}{x^2}}}{x^2}-2 \int \frac {e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )^3} \, dx+4 \int \frac {e^{1-\frac {120}{x^2}}}{x \left (1+e x-e^{2 x} x\right )^3} \, dx-4 \int \frac {e^{1-\frac {120}{x^2}}}{x \left (1+e x-e^{2 x} x\right )^2} \, dx-4 \int \frac {e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )^3} \, dx-4 \int \frac {e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )} \, dx-6 \int \frac {e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )^2} \, dx-6 \int \frac {e^{-\frac {120}{x^2}}}{x^3 \left (-1-e x+e^{2 x} x\right )} \, dx-8 \int \frac {e^{-\frac {120}{x^2}}}{x^2 \left (-1-e x+e^{2 x} x\right )^2} \, dx+240 \int \frac {e^{-\frac {120}{x^2}}}{x^5 \left (-1-e x+e^{2 x} x\right )^2} \, dx+480 \int \frac {e^{-\frac {120}{x^2}}}{x^5 \left (-1-e x+e^{2 x} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 34, normalized size = 1.48 \begin {gather*} \frac {e^{-\frac {120}{x^2}} \left (e-e^{2 x}\right )^2}{\left (1+e x-e^{2 x} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-240*E^2 + E^(6*x)*(240*x - 2*x^3) + E^3*(-240*x + 2*x^3) + E^(2*x)*(E^2*(720*x - 6*x^3) + E*(480 +
4*x^3)) + E^(4*x)*(-240 - 4*x^3 + E*(-720*x + 6*x^3)))/(E^(120/x^2 + 6*x)*x^6 + E^(120/x^2 + 4*x)*(-3*x^5 - 3
*E*x^6) + E^(120/x^2 + 2*x)*(3*x^4 + 6*E*x^5 + 3*E^2*x^6) + E^(120/x^2)*(-x^3 - 3*E*x^4 - 3*E^2*x^5 - E^3*x^6)
),x]

[Out]

(E - E^(2*x))^2/(E^(120/x^2)*(1 + E*x - E^(2*x)*x)^2)

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fricas [B]  time = 0.49, size = 147, normalized size = 6.39 \begin {gather*} \frac {e^{\left (\frac {8 \, {\left (x^{3} + 30\right )}}{x^{2}} + \frac {12 \, {\left (x^{3} + 20\right )}}{x^{2}} + 2\right )} - 2 \, e^{\left (\frac {4 \, {\left (x^{3} + 30\right )}}{x^{2}} + \frac {18 \, {\left (x^{3} + 20\right )}}{x^{2}} + 1\right )} + e^{\left (\frac {24 \, {\left (x^{3} + 20\right )}}{x^{2}}\right )}}{x^{2} e^{\left (\frac {12 \, {\left (x^{3} + 30\right )}}{x^{2}} + \frac {12 \, {\left (x^{3} + 20\right )}}{x^{2}}\right )} + {\left (x^{2} e^{2} + 2 \, x e + 1\right )} e^{\left (\frac {20 \, {\left (x^{3} + 30\right )}}{x^{2}}\right )} - 2 \, {\left (x^{2} e + x\right )} e^{\left (\frac {16 \, {\left (x^{3} + 30\right )}}{x^{2}} + \frac {6 \, {\left (x^{3} + 20\right )}}{x^{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+240*x)*exp(x)^6+((6*x^3-720*x)*exp(1)-4*x^3-240)*exp(x)^4+((-6*x^3+720*x)*exp(1)^2+(4*x^3+4
80)*exp(1))*exp(x)^2+(2*x^3-240*x)*exp(1)^3-240*exp(1)^2)/(x^6*exp(60/x^2)^2*exp(x)^6+(-3*x^6*exp(1)-3*x^5)*ex
p(60/x^2)^2*exp(x)^4+(3*x^6*exp(1)^2+6*x^5*exp(1)+3*x^4)*exp(60/x^2)^2*exp(x)^2+(-x^6*exp(1)^3-3*x^5*exp(1)^2-
3*x^4*exp(1)-x^3)*exp(60/x^2)^2),x, algorithm="fricas")

[Out]

(e^(8*(x^3 + 30)/x^2 + 12*(x^3 + 20)/x^2 + 2) - 2*e^(4*(x^3 + 30)/x^2 + 18*(x^3 + 20)/x^2 + 1) + e^(24*(x^3 +
20)/x^2))/(x^2*e^(12*(x^3 + 30)/x^2 + 12*(x^3 + 20)/x^2) + (x^2*e^2 + 2*x*e + 1)*e^(20*(x^3 + 30)/x^2) - 2*(x^
2*e + x)*e^(16*(x^3 + 30)/x^2 + 6*(x^3 + 20)/x^2))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+240*x)*exp(x)^6+((6*x^3-720*x)*exp(1)-4*x^3-240)*exp(x)^4+((-6*x^3+720*x)*exp(1)^2+(4*x^3+4
80)*exp(1))*exp(x)^2+(2*x^3-240*x)*exp(1)^3-240*exp(1)^2)/(x^6*exp(60/x^2)^2*exp(x)^6+(-3*x^6*exp(1)-3*x^5)*ex
p(60/x^2)^2*exp(x)^4+(3*x^6*exp(1)^2+6*x^5*exp(1)+3*x^4)*exp(60/x^2)^2*exp(x)^2+(-x^6*exp(1)^3-3*x^5*exp(1)^2-
3*x^4*exp(1)-x^3)*exp(60/x^2)^2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x^{3}+240 x \right ) {\mathrm e}^{6 x}+\left (\left (6 x^{3}-720 x \right ) {\mathrm e}-4 x^{3}-240\right ) {\mathrm e}^{4 x}+\left (\left (-6 x^{3}+720 x \right ) {\mathrm e}^{2}+\left (4 x^{3}+480\right ) {\mathrm e}\right ) {\mathrm e}^{2 x}+\left (2 x^{3}-240 x \right ) {\mathrm e}^{3}-240 \,{\mathrm e}^{2}}{x^{6} {\mathrm e}^{\frac {120}{x^{2}}} {\mathrm e}^{6 x}+\left (-3 x^{6} {\mathrm e}-3 x^{5}\right ) {\mathrm e}^{\frac {120}{x^{2}}} {\mathrm e}^{4 x}+\left (3 x^{6} {\mathrm e}^{2}+6 x^{5} {\mathrm e}+3 x^{4}\right ) {\mathrm e}^{\frac {120}{x^{2}}} {\mathrm e}^{2 x}+\left (-x^{6} {\mathrm e}^{3}-3 \,{\mathrm e}^{2} x^{5}-3 x^{4} {\mathrm e}-x^{3}\right ) {\mathrm e}^{\frac {120}{x^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+240*x)*exp(x)^6+((6*x^3-720*x)*exp(1)-4*x^3-240)*exp(x)^4+((-6*x^3+720*x)*exp(1)^2+(4*x^3+480)*ex
p(1))*exp(x)^2+(2*x^3-240*x)*exp(1)^3-240*exp(1)^2)/(x^6*exp(60/x^2)^2*exp(x)^6+(-3*x^6*exp(1)-3*x^5)*exp(60/x
^2)^2*exp(x)^4+(3*x^6*exp(1)^2+6*x^5*exp(1)+3*x^4)*exp(60/x^2)^2*exp(x)^2+(-x^6*exp(1)^3-3*x^5*exp(1)^2-3*x^4*
exp(1)-x^3)*exp(60/x^2)^2),x)

[Out]

int(((-2*x^3+240*x)*exp(x)^6+((6*x^3-720*x)*exp(1)-4*x^3-240)*exp(x)^4+((-6*x^3+720*x)*exp(1)^2+(4*x^3+480)*ex
p(1))*exp(x)^2+(2*x^3-240*x)*exp(1)^3-240*exp(1)^2)/(x^6*exp(60/x^2)^2*exp(x)^6+(-3*x^6*exp(1)-3*x^5)*exp(60/x
^2)^2*exp(x)^4+(3*x^6*exp(1)^2+6*x^5*exp(1)+3*x^4)*exp(60/x^2)^2*exp(x)^2+(-x^6*exp(1)^3-3*x^5*exp(1)^2-3*x^4*
exp(1)-x^3)*exp(60/x^2)^2),x)

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maxima [B]  time = 0.45, size = 59, normalized size = 2.57 \begin {gather*} \frac {{\left (e^{2} + e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x + 1\right )}\right )} e^{\left (-\frac {120}{x^{2}}\right )}}{x^{2} e^{2} + x^{2} e^{\left (4 \, x\right )} + 2 \, x e - 2 \, {\left (x^{2} e + x\right )} e^{\left (2 \, x\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+240*x)*exp(x)^6+((6*x^3-720*x)*exp(1)-4*x^3-240)*exp(x)^4+((-6*x^3+720*x)*exp(1)^2+(4*x^3+4
80)*exp(1))*exp(x)^2+(2*x^3-240*x)*exp(1)^3-240*exp(1)^2)/(x^6*exp(60/x^2)^2*exp(x)^6+(-3*x^6*exp(1)-3*x^5)*ex
p(60/x^2)^2*exp(x)^4+(3*x^6*exp(1)^2+6*x^5*exp(1)+3*x^4)*exp(60/x^2)^2*exp(x)^2+(-x^6*exp(1)^3-3*x^5*exp(1)^2-
3*x^4*exp(1)-x^3)*exp(60/x^2)^2),x, algorithm="maxima")

[Out]

(e^2 + e^(4*x) - 2*e^(2*x + 1))*e^(-120/x^2)/(x^2*e^2 + x^2*e^(4*x) + 2*x*e - 2*(x^2*e + x)*e^(2*x) + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {240\,{\mathrm {e}}^2-{\mathrm {e}}^{6\,x}\,\left (240\,x-2\,x^3\right )+{\mathrm {e}}^3\,\left (240\,x-2\,x^3\right )+{\mathrm {e}}^{4\,x}\,\left (\mathrm {e}\,\left (720\,x-6\,x^3\right )+4\,x^3+240\right )-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^2\,\left (720\,x-6\,x^3\right )+\mathrm {e}\,\left (4\,x^3+480\right )\right )}{{\mathrm {e}}^{\frac {120}{x^2}}\,\left ({\mathrm {e}}^3\,x^6+3\,{\mathrm {e}}^2\,x^5+3\,\mathrm {e}\,x^4+x^3\right )-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\frac {120}{x^2}}\,\left (3\,{\mathrm {e}}^2\,x^6+6\,\mathrm {e}\,x^5+3\,x^4\right )-x^6\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{\frac {120}{x^2}}+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{\frac {120}{x^2}}\,\left (3\,\mathrm {e}\,x^6+3\,x^5\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((240*exp(2) - exp(6*x)*(240*x - 2*x^3) + exp(3)*(240*x - 2*x^3) + exp(4*x)*(exp(1)*(720*x - 6*x^3) + 4*x^3
+ 240) - exp(2*x)*(exp(2)*(720*x - 6*x^3) + exp(1)*(4*x^3 + 480)))/(exp(120/x^2)*(3*x^4*exp(1) + 3*x^5*exp(2)
+ x^6*exp(3) + x^3) - exp(2*x)*exp(120/x^2)*(6*x^5*exp(1) + 3*x^6*exp(2) + 3*x^4) - x^6*exp(6*x)*exp(120/x^2)
+ exp(4*x)*exp(120/x^2)*(3*x^6*exp(1) + 3*x^5)),x)

[Out]

int((240*exp(2) - exp(6*x)*(240*x - 2*x^3) + exp(3)*(240*x - 2*x^3) + exp(4*x)*(exp(1)*(720*x - 6*x^3) + 4*x^3
+ 240) - exp(2*x)*(exp(2)*(720*x - 6*x^3) + exp(1)*(4*x^3 + 480)))/(exp(120/x^2)*(3*x^4*exp(1) + 3*x^5*exp(2)
+ x^6*exp(3) + x^3) - exp(2*x)*exp(120/x^2)*(6*x^5*exp(1) + 3*x^6*exp(2) + 3*x^4) - x^6*exp(6*x)*exp(120/x^2)
+ exp(4*x)*exp(120/x^2)*(3*x^6*exp(1) + 3*x^5)), x)

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sympy [B]  time = 0.50, size = 116, normalized size = 5.04 \begin {gather*} \frac {2 x e^{2 x} - 2 e x - 1}{x^{4} e^{\frac {120}{x^{2}}} e^{4 x} + x^{4} e^{2} e^{\frac {120}{x^{2}}} + 2 e x^{3} e^{\frac {120}{x^{2}}} + x^{2} e^{\frac {120}{x^{2}}} + \left (- 2 e x^{4} e^{\frac {120}{x^{2}}} - 2 x^{3} e^{\frac {120}{x^{2}}}\right ) e^{2 x}} + \frac {e^{- \frac {120}{x^{2}}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+240*x)*exp(x)**6+((6*x**3-720*x)*exp(1)-4*x**3-240)*exp(x)**4+((-6*x**3+720*x)*exp(1)**2+(
4*x**3+480)*exp(1))*exp(x)**2+(2*x**3-240*x)*exp(1)**3-240*exp(1)**2)/(x**6*exp(60/x**2)**2*exp(x)**6+(-3*x**6
*exp(1)-3*x**5)*exp(60/x**2)**2*exp(x)**4+(3*x**6*exp(1)**2+6*x**5*exp(1)+3*x**4)*exp(60/x**2)**2*exp(x)**2+(-
x**6*exp(1)**3-3*x**5*exp(1)**2-3*x**4*exp(1)-x**3)*exp(60/x**2)**2),x)

[Out]

(2*x*exp(2*x) - 2*E*x - 1)/(x**4*exp(120/x**2)*exp(4*x) + x**4*exp(2)*exp(120/x**2) + 2*E*x**3*exp(120/x**2) +
x**2*exp(120/x**2) + (-2*E*x**4*exp(120/x**2) - 2*x**3*exp(120/x**2))*exp(2*x)) + exp(-120/x**2)/x**2

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