3.17.10 \(\int \frac {-e^{e^2+\frac {e^{e^2}}{10 x^2}}-240 x^3+160 x^4+e^{3 x} (-80 x^3+220 x^4-120 x^5)+e^{6 x} (20 x^3-50 x^4+30 x^5)+e^{\frac {e^{e^2}}{20 x^2}} (e^{e^2} (6+e^{3 x} (-1+x)-4 x)+40 x^3+e^{3 x} (20 x^3-30 x^4))}{5 x^3} \, dx\)

Optimal. Leaf size=31 \[ \left (-2+e^{\frac {e^{e^2}}{20 x^2}}+\left (4-e^{3 x}\right ) (-1+x)\right )^2 \]

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Rubi [F]  time = 3.24, 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^{e^2+\frac {e^{e^2}}{10 x^2}}-240 x^3+160 x^4+e^{3 x} \left (-80 x^3+220 x^4-120 x^5\right )+e^{6 x} \left (20 x^3-50 x^4+30 x^5\right )+e^{\frac {e^{e^2}}{20 x^2}} \left (e^{e^2} \left (6+e^{3 x} (-1+x)-4 x\right )+40 x^3+e^{3 x} \left (20 x^3-30 x^4\right )\right )}{5 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-E^(E^2 + E^E^2/(10*x^2)) - 240*x^3 + 160*x^4 + E^(3*x)*(-80*x^3 + 220*x^4 - 120*x^5) + E^(6*x)*(20*x^3 -
50*x^4 + 30*x^5) + E^(E^E^2/(20*x^2))*(E^E^2*(6 + E^(3*x)*(-1 + x) - 4*x) + 40*x^3 + E^(3*x)*(20*x^3 - 30*x^4
)))/(5*x^3),x]

[Out]

-12*E^(3*x) + E^(6*x) + (6 - E^(E^E^2/(20*x^2)) - 4*x)^2 + 20*E^(3*x)*x - 2*E^(6*x)*x - 8*E^(3*x)*x^2 + E^(6*x
)*x^2 + 4*Defer[Int][E^((E^E^2 + 60*x^3)/(20*x^2)), x] - Defer[Int][E^(E^2 + (E^E^2 + 60*x^3)/(20*x^2))/x^3, x
]/5 + Defer[Int][E^(E^2 + (E^E^2 + 60*x^3)/(20*x^2))/x^2, x]/5 - 6*Defer[Int][E^((E^E^2 + 60*x^3)/(20*x^2))*x,
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-e^{e^2+\frac {e^{e^2}}{10 x^2}}-240 x^3+160 x^4+e^{3 x} \left (-80 x^3+220 x^4-120 x^5\right )+e^{6 x} \left (20 x^3-50 x^4+30 x^5\right )+e^{\frac {e^{e^2}}{20 x^2}} \left (e^{e^2} \left (6+e^{3 x} (-1+x)-4 x\right )+40 x^3+e^{3 x} \left (20 x^3-30 x^4\right )\right )}{x^3} \, dx\\ &=\frac {1}{5} \int \left (10 e^{6 x} (-1+x) (-2+3 x)+\frac {\left (-6+e^{\frac {e^{e^2}}{20 x^2}}+4 x\right ) \left (-e^{e^2+\frac {e^{e^2}}{20 x^2}}+40 x^3\right )}{x^3}-\frac {e^{3 x} \left (e^{e^2+\frac {e^{e^2}}{20 x^2}}-e^{e^2+\frac {e^{e^2}}{20 x^2}} x+80 x^3-20 e^{\frac {e^{e^2}}{20 x^2}} x^3-220 x^4+30 e^{\frac {e^{e^2}}{20 x^2}} x^4+120 x^5\right )}{x^3}\right ) \, dx\\ &=\frac {1}{5} \int \frac {\left (-6+e^{\frac {e^{e^2}}{20 x^2}}+4 x\right ) \left (-e^{e^2+\frac {e^{e^2}}{20 x^2}}+40 x^3\right )}{x^3} \, dx-\frac {1}{5} \int \frac {e^{3 x} \left (e^{e^2+\frac {e^{e^2}}{20 x^2}}-e^{e^2+\frac {e^{e^2}}{20 x^2}} x+80 x^3-20 e^{\frac {e^{e^2}}{20 x^2}} x^3-220 x^4+30 e^{\frac {e^{e^2}}{20 x^2}} x^4+120 x^5\right )}{x^3} \, dx+2 \int e^{6 x} (-1+x) (-2+3 x) \, dx\\ &=\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2-\frac {1}{5} \int \frac {e^{3 x} \left (-e^{e^2+\frac {e^{e^2}}{20 x^2}} (-1+x)+10 e^{\frac {e^{e^2}}{20 x^2}} x^3 (-2+3 x)+20 x^3 \left (4-11 x+6 x^2\right )\right )}{x^3} \, dx+2 \int \left (2 e^{6 x}-5 e^{6 x} x+3 e^{6 x} x^2\right ) \, dx\\ &=\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2-\frac {1}{5} \int \left (20 e^{3 x} (-1+2 x) (-4+3 x)+\frac {e^{\frac {e^{e^2}}{20 x^2}+3 x} \left (e^{e^2}-e^{e^2} x-20 x^3+30 x^4\right )}{x^3}\right ) \, dx+4 \int e^{6 x} \, dx+6 \int e^{6 x} x^2 \, dx-10 \int e^{6 x} x \, dx\\ &=\frac {2 e^{6 x}}{3}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2-\frac {5}{3} e^{6 x} x+e^{6 x} x^2-\frac {1}{5} \int \frac {e^{\frac {e^{e^2}}{20 x^2}+3 x} \left (e^{e^2}-e^{e^2} x-20 x^3+30 x^4\right )}{x^3} \, dx+\frac {5}{3} \int e^{6 x} \, dx-2 \int e^{6 x} x \, dx-4 \int e^{3 x} (-1+2 x) (-4+3 x) \, dx\\ &=\frac {17 e^{6 x}}{18}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2-2 e^{6 x} x+e^{6 x} x^2-\frac {1}{5} \int \frac {e^{\frac {e^{e^2}+60 x^3}{20 x^2}} \left (e^{e^2}-e^{e^2} x-20 x^3+30 x^4\right )}{x^3} \, dx+\frac {1}{3} \int e^{6 x} \, dx-4 \int \left (4 e^{3 x}-11 e^{3 x} x+6 e^{3 x} x^2\right ) \, dx\\ &=e^{6 x}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2-2 e^{6 x} x+e^{6 x} x^2-\frac {1}{5} \int \left (-20 e^{\frac {e^{e^2}+60 x^3}{20 x^2}}+\frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^3}-\frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^2}+30 e^{\frac {e^{e^2}+60 x^3}{20 x^2}} x\right ) \, dx-16 \int e^{3 x} \, dx-24 \int e^{3 x} x^2 \, dx+44 \int e^{3 x} x \, dx\\ &=-\frac {16 e^{3 x}}{3}+e^{6 x}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2+\frac {44}{3} e^{3 x} x-2 e^{6 x} x-8 e^{3 x} x^2+e^{6 x} x^2-\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^3} \, dx+\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^2} \, dx+4 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} \, dx-6 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} x \, dx-\frac {44}{3} \int e^{3 x} \, dx+16 \int e^{3 x} x \, dx\\ &=-\frac {92 e^{3 x}}{9}+e^{6 x}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2+20 e^{3 x} x-2 e^{6 x} x-8 e^{3 x} x^2+e^{6 x} x^2-\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^3} \, dx+\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^2} \, dx+4 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} \, dx-\frac {16}{3} \int e^{3 x} \, dx-6 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} x \, dx\\ &=-12 e^{3 x}+e^{6 x}+\left (6-e^{\frac {e^{e^2}}{20 x^2}}-4 x\right )^2+20 e^{3 x} x-2 e^{6 x} x-8 e^{3 x} x^2+e^{6 x} x^2-\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^3} \, dx+\frac {1}{5} \int \frac {e^{e^2+\frac {e^{e^2}+60 x^3}{20 x^2}}}{x^2} \, dx+4 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} \, dx-6 \int e^{\frac {e^{e^2}+60 x^3}{20 x^2}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.76, size = 91, normalized size = 2.94 \begin {gather*} \frac {1}{5} \left (5 e^{\frac {e^{e^2}}{10 x^2}}-e^{\frac {e^{e^2}}{20 x^2}} (60-40 x)+5 e^{6 x} (-1+x)^2-240 x+80 x^2-10 e^{3 x} (-1+x) \left (-6+e^{\frac {e^{e^2}}{20 x^2}}+4 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^(E^2 + E^E^2/(10*x^2)) - 240*x^3 + 160*x^4 + E^(3*x)*(-80*x^3 + 220*x^4 - 120*x^5) + E^(6*x)*(20
*x^3 - 50*x^4 + 30*x^5) + E^(E^E^2/(20*x^2))*(E^E^2*(6 + E^(3*x)*(-1 + x) - 4*x) + 40*x^3 + E^(3*x)*(20*x^3 -
30*x^4)))/(5*x^3),x]

[Out]

(5*E^(E^E^2/(10*x^2)) - E^(E^E^2/(20*x^2))*(60 - 40*x) + 5*E^(6*x)*(-1 + x)^2 - 240*x + 80*x^2 - 10*E^(3*x)*(-
1 + x)*(-6 + E^(E^E^2/(20*x^2)) + 4*x))/5

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fricas [B]  time = 0.74, size = 71, normalized size = 2.29 \begin {gather*} 16 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\left (6 \, x\right )} - 4 \, {\left (2 \, x^{2} - 5 \, x + 3\right )} e^{\left (3 \, x\right )} - 2 \, {\left ({\left (x - 1\right )} e^{\left (3 \, x\right )} - 4 \, x + 6\right )} e^{\left (\frac {e^{\left (e^{2}\right )}}{20 \, x^{2}}\right )} - 48 \, x + e^{\left (\frac {e^{\left (e^{2}\right )}}{10 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-exp(exp(2))*exp(1/20*exp(exp(2))/x^2)^2+(((x-1)*exp(3*x)+6-4*x)*exp(exp(2))+(-30*x^4+20*x^3)*e
xp(3*x)+40*x^3)*exp(1/20*exp(exp(2))/x^2)+(30*x^5-50*x^4+20*x^3)*exp(3*x)^2+(-120*x^5+220*x^4-80*x^3)*exp(3*x)
+160*x^4-240*x^3)/x^3,x, algorithm="fricas")

[Out]

16*x^2 + (x^2 - 2*x + 1)*e^(6*x) - 4*(2*x^2 - 5*x + 3)*e^(3*x) - 2*((x - 1)*e^(3*x) - 4*x + 6)*e^(1/20*e^(e^2)
/x^2) - 48*x + e^(1/10*e^(e^2)/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {160 \, x^{4} - 240 \, x^{3} + 10 \, {\left (3 \, x^{5} - 5 \, x^{4} + 2 \, x^{3}\right )} e^{\left (6 \, x\right )} - 20 \, {\left (6 \, x^{5} - 11 \, x^{4} + 4 \, x^{3}\right )} e^{\left (3 \, x\right )} + {\left (40 \, x^{3} - 10 \, {\left (3 \, x^{4} - 2 \, x^{3}\right )} e^{\left (3 \, x\right )} + {\left ({\left (x - 1\right )} e^{\left (3 \, x\right )} - 4 \, x + 6\right )} e^{\left (e^{2}\right )}\right )} e^{\left (\frac {e^{\left (e^{2}\right )}}{20 \, x^{2}}\right )} - e^{\left (\frac {e^{\left (e^{2}\right )}}{10 \, x^{2}} + e^{2}\right )}}{5 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-exp(exp(2))*exp(1/20*exp(exp(2))/x^2)^2+(((x-1)*exp(3*x)+6-4*x)*exp(exp(2))+(-30*x^4+20*x^3)*e
xp(3*x)+40*x^3)*exp(1/20*exp(exp(2))/x^2)+(30*x^5-50*x^4+20*x^3)*exp(3*x)^2+(-120*x^5+220*x^4-80*x^3)*exp(3*x)
+160*x^4-240*x^3)/x^3,x, algorithm="giac")

[Out]

integrate(1/5*(160*x^4 - 240*x^3 + 10*(3*x^5 - 5*x^4 + 2*x^3)*e^(6*x) - 20*(6*x^5 - 11*x^4 + 4*x^3)*e^(3*x) +
(40*x^3 - 10*(3*x^4 - 2*x^3)*e^(3*x) + ((x - 1)*e^(3*x) - 4*x + 6)*e^(e^2))*e^(1/20*e^(e^2)/x^2) - e^(1/10*e^(
e^2)/x^2 + e^2))/x^3, x)

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maple [B]  time = 0.14, size = 80, normalized size = 2.58

method	result	size
risch	\(16 x^{2}-48 x +\frac {\left (5 x^{2}-10 x +5\right ) {\mathrm e}^{6 x}}{5}+\frac {\left (-40 x^{2}+100 x -60\right ) {\mathrm e}^{3 x}}{5}+{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{2}}}{10 x^{2}}}+\frac {\left (-10 x \,{\mathrm e}^{3 x}+40 x +10 \,{\mathrm e}^{3 x}-60\right ) {\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{2}}}{20 x^{2}}}}{5}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-exp(exp(2))*exp(1/20*exp(exp(2))/x^2)^2+(((x-1)*exp(3*x)+6-4*x)*exp(exp(2))+(-30*x^4+20*x^3)*exp(3*x
)+40*x^3)*exp(1/20*exp(exp(2))/x^2)+(30*x^5-50*x^4+20*x^3)*exp(3*x)^2+(-120*x^5+220*x^4-80*x^3)*exp(3*x)+160*x
^4-240*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

16*x^2-48*x+1/5*(5*x^2-10*x+5)*exp(6*x)+1/5*(-40*x^2+100*x-60)*exp(3*x)+exp(1/10*exp(exp(2))/x^2)+1/5*(-10*x*e
xp(3*x)+40*x+10*exp(3*x)-60)*exp(1/20*exp(exp(2))/x^2)

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maxima [C]  time = 0.83, size = 181, normalized size = 5.84 \begin {gather*} 2 \, \sqrt {\frac {1}{5}} x \sqrt {-\frac {e^{\left (e^{2}\right )}}{x^{2}}} \Gamma \left (-\frac {1}{2}, -\frac {e^{\left (e^{2}\right )}}{20 \, x^{2}}\right ) + 16 \, x^{2} + \frac {1}{18} \, {\left (18 \, x^{2} - 6 \, x + 1\right )} e^{\left (6 \, x\right )} - \frac {5}{18} \, {\left (6 \, x - 1\right )} e^{\left (6 \, x\right )} - \frac {8}{9} \, {\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} + \frac {44}{9} \, {\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} - 2 \, {\left (x - 1\right )} e^{\left (3 \, x + \frac {e^{\left (e^{2}\right )}}{20 \, x^{2}}\right )} + \frac {4 \, \sqrt {\frac {1}{5}} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {1}{5}} \sqrt {-\frac {e^{\left (e^{2}\right )}}{x^{2}}}\right ) - 1\right )} e^{\left (e^{2}\right )}}{x \sqrt {-\frac {e^{\left (e^{2}\right )}}{x^{2}}}} - 48 \, x + \frac {2}{3} \, e^{\left (6 \, x\right )} - \frac {16}{3} \, e^{\left (3 \, x\right )} + e^{\left (\frac {e^{\left (e^{2}\right )}}{10 \, x^{2}}\right )} - 12 \, e^{\left (\frac {e^{\left (e^{2}\right )}}{20 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-exp(exp(2))*exp(1/20*exp(exp(2))/x^2)^2+(((x-1)*exp(3*x)+6-4*x)*exp(exp(2))+(-30*x^4+20*x^3)*e
xp(3*x)+40*x^3)*exp(1/20*exp(exp(2))/x^2)+(30*x^5-50*x^4+20*x^3)*exp(3*x)^2+(-120*x^5+220*x^4-80*x^3)*exp(3*x)
+160*x^4-240*x^3)/x^3,x, algorithm="maxima")

[Out]

2*sqrt(1/5)*x*sqrt(-e^(e^2)/x^2)*gamma(-1/2, -1/20*e^(e^2)/x^2) + 16*x^2 + 1/18*(18*x^2 - 6*x + 1)*e^(6*x) - 5
/18*(6*x - 1)*e^(6*x) - 8/9*(9*x^2 - 6*x + 2)*e^(3*x) + 44/9*(3*x - 1)*e^(3*x) - 2*(x - 1)*e^(3*x + 1/20*e^(e^
2)/x^2) + 4*sqrt(1/5)*sqrt(pi)*(erf(1/2*sqrt(1/5)*sqrt(-e^(e^2)/x^2)) - 1)*e^(e^2)/(x*sqrt(-e^(e^2)/x^2)) - 48
*x + 2/3*e^(6*x) - 16/3*e^(3*x) + e^(1/10*e^(e^2)/x^2) - 12*e^(1/20*e^(e^2)/x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^2}}{10\,x^2}}\,{\mathrm {e}}^{{\mathrm {e}}^2}}{5}-\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^2}}{20\,x^2}}\,\left ({\mathrm {e}}^{3\,x}\,\left (20\,x^3-30\,x^4\right )+{\mathrm {e}}^{{\mathrm {e}}^2}\,\left ({\mathrm {e}}^{3\,x}\,\left (x-1\right )-4\,x+6\right )+40\,x^3\right )}{5}-\frac {{\mathrm {e}}^{6\,x}\,\left (30\,x^5-50\,x^4+20\,x^3\right )}{5}+\frac {{\mathrm {e}}^{3\,x}\,\left (120\,x^5-220\,x^4+80\,x^3\right )}{5}+48\,x^3-32\,x^4}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(exp(exp(2))/(10*x^2))*exp(exp(2)))/5 - (exp(exp(exp(2))/(20*x^2))*(exp(3*x)*(20*x^3 - 30*x^4) + exp
(exp(2))*(exp(3*x)*(x - 1) - 4*x + 6) + 40*x^3))/5 - (exp(6*x)*(20*x^3 - 50*x^4 + 30*x^5))/5 + (exp(3*x)*(80*x
^3 - 220*x^4 + 120*x^5))/5 + 48*x^3 - 32*x^4)/x^3,x)

[Out]

int(-((exp(exp(exp(2))/(10*x^2))*exp(exp(2)))/5 - (exp(exp(exp(2))/(20*x^2))*(exp(3*x)*(20*x^3 - 30*x^4) + exp
(exp(2))*(exp(3*x)*(x - 1) - 4*x + 6) + 40*x^3))/5 - (exp(6*x)*(20*x^3 - 50*x^4 + 30*x^5))/5 + (exp(3*x)*(80*x
^3 - 220*x^4 + 120*x^5))/5 + 48*x^3 - 32*x^4)/x^3, x)

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sympy [B]  time = 7.67, size = 80, normalized size = 2.58 \begin {gather*} 16 x^{2} - 48 x + \left (- 8 x^{2} + 20 x - 12\right ) e^{3 x} + \left (x^{2} - 2 x + 1\right ) e^{6 x} + \left (- 2 x e^{3 x} + 8 x + 2 e^{3 x} - 12\right ) e^{\frac {e^{e^{2}}}{20 x^{2}}} + e^{\frac {e^{e^{2}}}{10 x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-exp(exp(2))*exp(1/20*exp(exp(2))/x**2)**2+(((x-1)*exp(3*x)+6-4*x)*exp(exp(2))+(-30*x**4+20*x**
3)*exp(3*x)+40*x**3)*exp(1/20*exp(exp(2))/x**2)+(30*x**5-50*x**4+20*x**3)*exp(3*x)**2+(-120*x**5+220*x**4-80*x
**3)*exp(3*x)+160*x**4-240*x**3)/x**3,x)

[Out]

16*x**2 - 48*x + (-8*x**2 + 20*x - 12)*exp(3*x) + (x**2 - 2*x + 1)*exp(6*x) + (-2*x*exp(3*x) + 8*x + 2*exp(3*x
) - 12)*exp(exp(exp(2))/(20*x**2)) + exp(exp(exp(2))/(10*x**2))

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