\(\int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 (-108 x^3+162 x^4-36 x^5+16 x^6)+e^{\frac {72}{e^2 x^2}} (-31104+1728 x-1152 x^2+e^2 (-12 x^3+16 x^4))+e^{\frac {36}{e^2 x^2}} (-31104+5184 x-3456 x^2+e^2 (-72 x^3+96 x^4))}{e^2 x^3} \, dx\) [6447]

Optimal result

Integrand size = 129, antiderivative size = 27 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=\left (x-2 \left (\left (3+e^{\frac {36}{e^2 x^2}}\right )^2-x+x^2\right )\right )^2 \]

[Out]

(3*x-2*x^2-2*(exp(36/x^2/exp(2))+3)^2)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(27)=54\).

Time = 0.16 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {12, 14, 2240, 643, 2326} \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=\left (2 x^2-3 x+18\right )^2+24 e^{\frac {36}{e^2 x^2}} \left (2 x^2-3 x+18\right )+48 e^{\frac {108}{e^2 x^2}}+4 e^{\frac {144}{e^2 x^2}}+4 e^{\frac {72}{e^2 x^2}} \left (2 x^2-3 x+54\right ) \]

[In]

Int[(-10368*E^(108/(E^2*x^2)) - 1152*E^(144/(E^2*x^2)) + E^2*(-108*x^3 + 162*x^4 - 36*x^5 + 16*x^6) + E^(72/(E
^2*x^2))*(-31104 + 1728*x - 1152*x^2 + E^2*(-12*x^3 + 16*x^4)) + E^(36/(E^2*x^2))*(-31104 + 5184*x - 3456*x^2
+ E^2*(-72*x^3 + 96*x^4)))/(E^2*x^3),x]

[Out]

48*E^(108/(E^2*x^2)) + 4*E^(144/(E^2*x^2)) + 24*E^(36/(E^2*x^2))*(18 - 3*x + 2*x^2) + (18 - 3*x + 2*x^2)^2 + 4
*E^(72/(E^2*x^2))*(54 - 3*x + 2*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{x^3} \, dx}{e^2} \\ & = \frac {\int \left (-\frac {10368 e^{\frac {108}{e^2 x^2}}}{x^3}-\frac {1152 e^{\frac {144}{e^2 x^2}}}{x^3}+2 e^2 (-3+4 x) \left (18-3 x+2 x^2\right )+\frac {4 e^{\frac {72}{e^2 x^2}} \left (-7776+432 x-288 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3}+\frac {24 e^{\frac {36}{e^2 x^2}} \left (-1296+216 x-144 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3}\right ) \, dx}{e^2} \\ & = 2 \int (-3+4 x) \left (18-3 x+2 x^2\right ) \, dx+\frac {4 \int \frac {e^{\frac {72}{e^2 x^2}} \left (-7776+432 x-288 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3} \, dx}{e^2}+\frac {24 \int \frac {e^{\frac {36}{e^2 x^2}} \left (-1296+216 x-144 x^2-3 e^2 x^3+4 e^2 x^4\right )}{x^3} \, dx}{e^2}-\frac {1152 \int \frac {e^{\frac {144}{e^2 x^2}}}{x^3} \, dx}{e^2}-\frac {10368 \int \frac {e^{\frac {108}{e^2 x^2}}}{x^3} \, dx}{e^2} \\ & = 48 e^{\frac {108}{e^2 x^2}}+4 e^{\frac {144}{e^2 x^2}}+24 e^{\frac {36}{e^2 x^2}} \left (18-3 x+2 x^2\right )+\left (18-3 x+2 x^2\right )^2+4 e^{\frac {72}{e^2 x^2}} \left (54-3 x+2 x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=\left (18+12 e^{\frac {36}{e^2 x^2}}+2 e^{\frac {72}{e^2 x^2}}-3 x+2 x^2\right )^2 \]

[In]

Integrate[(-10368*E^(108/(E^2*x^2)) - 1152*E^(144/(E^2*x^2)) + E^2*(-108*x^3 + 162*x^4 - 36*x^5 + 16*x^6) + E^
(72/(E^2*x^2))*(-31104 + 1728*x - 1152*x^2 + E^2*(-12*x^3 + 16*x^4)) + E^(36/(E^2*x^2))*(-31104 + 5184*x - 345
6*x^2 + E^2*(-72*x^3 + 96*x^4)))/(E^2*x^3),x]

[Out]

(18 + 12*E^(36/(E^2*x^2)) + 2*E^(72/(E^2*x^2)) - 3*x + 2*x^2)^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(101\) vs. \(2(27)=54\).

Time = 0.56 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.78

method	result	size
risch	\(\left (2 x^{2}-3 x +18\right )^{2}+4 \,{\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}+48 \,{\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}+\left (8 x^{2} {\mathrm e}^{2}-12 \,{\mathrm e}^{2} x +216 \,{\mathrm e}^{2}\right ) {\mathrm e}^{\frac {-2 x^{2}+72 \,{\mathrm e}^{-2}}{x^{2}}}+\left (48 x^{2} {\mathrm e}^{2}-72 \,{\mathrm e}^{2} x +432 \,{\mathrm e}^{2}\right ) {\mathrm e}^{\frac {-2 x^{2}+36 \,{\mathrm e}^{-2}}{x^{2}}}\)	\(102\)
parallelrisch	\({\mathrm e}^{-2} \left (4 x^{4} {\mathrm e}^{2}+8 \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2} x^{2}+4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}-12 x^{3} {\mathrm e}^{2}+48 \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}} {\mathrm e}^{2} x^{2}-12 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}+48 \,{\mathrm e}^{2} {\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}+81 x^{2} {\mathrm e}^{2}-72 x \,{\mathrm e}^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}+216 \,{\mathrm e}^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-108 \,{\mathrm e}^{2} x +432 \,{\mathrm e}^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}\right )\)	\(163\)
parts	\(4 x^{4}-12 x^{3}+81 x^{2}-108 x +48 \,{\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}+4 \,{\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}+4 \,{\mathrm e}^{-2} \left (18 i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )-144 \,\operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )+54 \,{\mathrm e}^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-4 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-36 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+3 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )\right )\right )+24 \,{\mathrm e}^{-2} \left (18 i \sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )-72 \,\operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )+18 \,{\mathrm e}^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}-4 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-18 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+3 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )\right )\right )\)	\(324\)
default	\({\mathrm e}^{-2} \left (432 i \sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )+72 i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )-108 \,{\mathrm e}^{2} x +432 \,{\mathrm e}^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}+216 \,{\mathrm e}^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}+48 \,{\mathrm e}^{2} {\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}+4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}+4 x^{4} {\mathrm e}^{2}-12 x^{3} {\mathrm e}^{2}+81 x^{2} {\mathrm e}^{2}-1728 \,\operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )-576 \,\operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )-96 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-18 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+72 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )\right )-16 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-36 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )+12 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )\right )\right )\)	\(325\)
derivativedivides	\(-{\mathrm e}^{-2} \left (-432 i \sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )-72 i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e} \,\operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )+108 \,{\mathrm e}^{2} x -432 \,{\mathrm e}^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}-216 \,{\mathrm e}^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-48 \,{\mathrm e}^{2} {\mathrm e}^{\frac {108 \,{\mathrm e}^{-2}}{x^{2}}}-4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {144 \,{\mathrm e}^{-2}}{x^{2}}}-4 x^{4} {\mathrm e}^{2}+12 x^{3} {\mathrm e}^{2}-81 x^{2} {\mathrm e}^{2}+1728 \,\operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )+576 \,\operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )+96 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-18 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {36 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )-72 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {36 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \operatorname {erf}\left (\frac {6 i {\mathrm e}^{-1}}{x}\right )\right )+16 \,{\mathrm e}^{2} \left (-\frac {x^{2} {\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}}{2}-36 \,{\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-\frac {72 \,{\mathrm e}^{-2}}{x^{2}}\right )\right )-12 \,{\mathrm e}^{2} \left (-x \,{\mathrm e}^{\frac {72 \,{\mathrm e}^{-2}}{x^{2}}}-6 i {\mathrm e}^{-1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {6 i \sqrt {2}\, {\mathrm e}^{-1}}{x}\right )\right )\right )\)	\(326\)

[In]

int((-1152*exp(36/x^2/exp(2))^4-10368*exp(36/x^2/exp(2))^3+((16*x^4-12*x^3)*exp(2)-1152*x^2+1728*x-31104)*exp(
36/x^2/exp(2))^2+((96*x^4-72*x^3)*exp(2)-3456*x^2+5184*x-31104)*exp(36/x^2/exp(2))+(16*x^6-36*x^5+162*x^4-108*
x^3)*exp(2))/x^3/exp(2),x,method=_RETURNVERBOSE)

[Out]

(2*x^2-3*x+18)^2+4*exp(144/x^2*exp(-2))+48*exp(108/x^2*exp(-2))+(8*x^2*exp(2)-12*exp(2)*x+216*exp(2))*exp(2*(-
x^2+36*exp(-2))/x^2)+(48*x^2*exp(2)-72*exp(2)*x+432*exp(2))*exp(2*(-x^2+18*exp(-2))/x^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=4 \, x^{4} - 12 \, x^{3} + 81 \, x^{2} + 4 \, {\left (2 \, x^{2} - 3 \, x + 54\right )} e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right )} + 24 \, {\left (2 \, x^{2} - 3 \, x + 18\right )} e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right )} - 108 \, x + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}}\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}}\right )} \]

[In]

integrate((-1152*exp(36/x^2/exp(2))^4-10368*exp(36/x^2/exp(2))^3+((16*x^4-12*x^3)*exp(2)-1152*x^2+1728*x-31104
)*exp(36/x^2/exp(2))^2+((96*x^4-72*x^3)*exp(2)-3456*x^2+5184*x-31104)*exp(36/x^2/exp(2))+(16*x^6-36*x^5+162*x^
4-108*x^3)*exp(2))/x^3/exp(2),x, algorithm="fricas")

[Out]

4*x^4 - 12*x^3 + 81*x^2 + 4*(2*x^2 - 3*x + 54)*e^(72*e^(-2)/x^2) + 24*(2*x^2 - 3*x + 18)*e^(36*e^(-2)/x^2) - 1
08*x + 4*e^(144*e^(-2)/x^2) + 48*e^(108*e^(-2)/x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=4 x^{4} - 12 x^{3} + 81 x^{2} - 108 x + \left (8 x^{2} - 12 x + 216\right ) e^{\frac {72}{x^{2} e^{2}}} + \left (48 x^{2} - 72 x + 432\right ) e^{\frac {36}{x^{2} e^{2}}} + 4 e^{\frac {144}{x^{2} e^{2}}} + 48 e^{\frac {108}{x^{2} e^{2}}} \]

[In]

integrate((-1152*exp(36/x**2/exp(2))**4-10368*exp(36/x**2/exp(2))**3+((16*x**4-12*x**3)*exp(2)-1152*x**2+1728*
x-31104)*exp(36/x**2/exp(2))**2+((96*x**4-72*x**3)*exp(2)-3456*x**2+5184*x-31104)*exp(36/x**2/exp(2))+(16*x**6
-36*x**5+162*x**4-108*x**3)*exp(2))/x**3/exp(2),x)

[Out]

4*x**4 - 12*x**3 + 81*x**2 - 108*x + (8*x**2 - 12*x + 216)*exp(72*exp(-2)/x**2) + (48*x**2 - 72*x + 432)*exp(3
6*exp(-2)/x**2) + 4*exp(144*exp(-2)/x**2) + 48*exp(108*exp(-2)/x**2)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 237, normalized size of antiderivative = 8.78 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx={\left (4 \, x^{4} e^{2} - 12 \, x^{3} e^{2} - 36 \, \sqrt {2} x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}} e^{2} \Gamma \left (-\frac {1}{2}, -\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right ) - 216 \, x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}} e^{2} \Gamma \left (-\frac {1}{2}, -\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) + 81 \, x^{2} e^{2} - 108 \, x e^{2} - \frac {72 \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (6 \, \sqrt {2} \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}} - \frac {432 \, \sqrt {\pi } {\left (\operatorname {erf}\left (6 \, \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {e^{\left (-2\right )}}{x^{2}}}} + 576 \, {\rm Ei}\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right ) + 1728 \, {\rm Ei}\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 216 \, e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 432 \, e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 1728 \, \Gamma \left (-1, -\frac {36 \, e^{\left (-2\right )}}{x^{2}}\right ) - 576 \, \Gamma \left (-1, -\frac {72 \, e^{\left (-2\right )}}{x^{2}}\right )\right )} e^{\left (-2\right )} \]

[In]

integrate((-1152*exp(36/x^2/exp(2))^4-10368*exp(36/x^2/exp(2))^3+((16*x^4-12*x^3)*exp(2)-1152*x^2+1728*x-31104
)*exp(36/x^2/exp(2))^2+((96*x^4-72*x^3)*exp(2)-3456*x^2+5184*x-31104)*exp(36/x^2/exp(2))+(16*x^6-36*x^5+162*x^
4-108*x^3)*exp(2))/x^3/exp(2),x, algorithm="maxima")

[Out]

(4*x^4*e^2 - 12*x^3*e^2 - 36*sqrt(2)*x*sqrt(-e^(-2)/x^2)*e^2*gamma(-1/2, -72*e^(-2)/x^2) - 216*x*sqrt(-e^(-2)/
x^2)*e^2*gamma(-1/2, -36*e^(-2)/x^2) + 81*x^2*e^2 - 108*x*e^2 - 72*sqrt(2)*sqrt(pi)*(erf(6*sqrt(2)*sqrt(-e^(-2
)/x^2)) - 1)/(x*sqrt(-e^(-2)/x^2)) - 432*sqrt(pi)*(erf(6*sqrt(-e^(-2)/x^2)) - 1)/(x*sqrt(-e^(-2)/x^2)) + 576*E
i(72*e^(-2)/x^2) + 1728*Ei(36*e^(-2)/x^2) + 4*e^(144*e^(-2)/x^2 + 2) + 48*e^(108*e^(-2)/x^2 + 2) + 216*e^(72*e
^(-2)/x^2 + 2) + 432*e^(36*e^(-2)/x^2 + 2) - 1728*gamma(-1, -36*e^(-2)/x^2) - 576*gamma(-1, -72*e^(-2)/x^2))*e
^(-2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.96 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx={\left (4 \, x^{4} e^{2} - 12 \, x^{3} e^{2} + 81 \, x^{2} e^{2} + 8 \, x^{2} e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, x^{2} e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 108 \, x e^{2} - 12 \, x e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} - 72 \, x e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 4 \, e^{\left (\frac {144 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 48 \, e^{\left (\frac {108 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 216 \, e^{\left (\frac {72 \, e^{\left (-2\right )}}{x^{2}} + 2\right )} + 432 \, e^{\left (\frac {36 \, e^{\left (-2\right )}}{x^{2}} + 2\right )}\right )} e^{\left (-2\right )} \]

[In]

integrate((-1152*exp(36/x^2/exp(2))^4-10368*exp(36/x^2/exp(2))^3+((16*x^4-12*x^3)*exp(2)-1152*x^2+1728*x-31104
)*exp(36/x^2/exp(2))^2+((96*x^4-72*x^3)*exp(2)-3456*x^2+5184*x-31104)*exp(36/x^2/exp(2))+(16*x^6-36*x^5+162*x^
4-108*x^3)*exp(2))/x^3/exp(2),x, algorithm="giac")

[Out]

(4*x^4*e^2 - 12*x^3*e^2 + 81*x^2*e^2 + 8*x^2*e^(72*e^(-2)/x^2 + 2) + 48*x^2*e^(36*e^(-2)/x^2 + 2) - 108*x*e^2
- 12*x*e^(72*e^(-2)/x^2 + 2) - 72*x*e^(36*e^(-2)/x^2 + 2) + 4*e^(144*e^(-2)/x^2 + 2) + 48*e^(108*e^(-2)/x^2 +
2) + 216*e^(72*e^(-2)/x^2 + 2) + 432*e^(36*e^(-2)/x^2 + 2))*e^(-2)

Mupad [B] (verification not implemented)

Time = 11.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {-10368 e^{\frac {108}{e^2 x^2}}-1152 e^{\frac {144}{e^2 x^2}}+e^2 \left (-108 x^3+162 x^4-36 x^5+16 x^6\right )+e^{\frac {72}{e^2 x^2}} \left (-31104+1728 x-1152 x^2+e^2 \left (-12 x^3+16 x^4\right )\right )+e^{\frac {36}{e^2 x^2}} \left (-31104+5184 x-3456 x^2+e^2 \left (-72 x^3+96 x^4\right )\right )}{e^2 x^3} \, dx=432\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}-108\,x+216\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}}+48\,{\mathrm {e}}^{\frac {108\,{\mathrm {e}}^{-2}}{x^2}}+4\,{\mathrm {e}}^{\frac {144\,{\mathrm {e}}^{-2}}{x^2}}+48\,x^2\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}+8\,x^2\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}}+81\,x^2-12\,x^3+4\,x^4-72\,x\,{\mathrm {e}}^{\frac {36\,{\mathrm {e}}^{-2}}{x^2}}-12\,x\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^{-2}}{x^2}} \]

[In]

int(-(exp(-2)*(10368*exp((108*exp(-2))/x^2) + 1152*exp((144*exp(-2))/x^2) + exp((72*exp(-2))/x^2)*(exp(2)*(12*
x^3 - 16*x^4) - 1728*x + 1152*x^2 + 31104) + exp((36*exp(-2))/x^2)*(exp(2)*(72*x^3 - 96*x^4) - 5184*x + 3456*x
^2 + 31104) + exp(2)*(108*x^3 - 162*x^4 + 36*x^5 - 16*x^6)))/x^3,x)

[Out]

432*exp((36*exp(-2))/x^2) - 108*x + 216*exp((72*exp(-2))/x^2) + 48*exp((108*exp(-2))/x^2) + 4*exp((144*exp(-2)
)/x^2) + 48*x^2*exp((36*exp(-2))/x^2) + 8*x^2*exp((72*exp(-2))/x^2) + 81*x^2 - 12*x^3 + 4*x^4 - 72*x*exp((36*e
xp(-2))/x^2) - 12*x*exp((72*exp(-2))/x^2)