\(\int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 (2 x+3 x^2)}{4 \log ^2(4)}} (-256 x-768 x^2+512 \log ^2(4))+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} (-384 x^2-1152 x^3+768 x \log ^2(4))+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} (-192 x^3-576 x^4+384 x^2 \log ^2(4)))}{\log ^2(4)} \, dx\) [9883]

Optimal result

Integrand size = 177, antiderivative size = 27 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-x+\left (4+2 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} x\right )^4 \]

[Out]

(2*x/exp(1/16*x*(2+3*x)/ln(2)^2)+4)^4-x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(27)=54\).

Time = 8.52 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.74, number of steps used = 65, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6873, 6874, 2276, 2273, 2272, 2266, 2236, 2326} \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=\frac {256 \left (3 x^2+x\right ) x^2 e^{-\frac {3 x (3 x+2)}{4 \log ^2(4)}}}{6 x+2}+\frac {768 \left (3 x^2+x\right ) x e^{-\frac {x (3 x+2)}{2 \log ^2(4)}}}{6 x+2}+\frac {1024 \left (3 x^2+x\right ) e^{-\frac {x (3 x+2)}{4 \log ^2(4)}}}{6 x+2}+16 x^4 e^{-\frac {3 x^2}{\log ^2(4)}-\frac {2 x}{\log ^2(4)}}-x \]

[In]

Int[(-32*x^4 - 96*x^5 - E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2 + 64*x^3*Log[4]^2 + E^((3*(2*x + 3*x^2))/(4*Log[4]
^2))*(-256*x - 768*x^2 + 512*Log[4]^2) + E^((2*x + 3*x^2)/(2*Log[4]^2))*(-384*x^2 - 1152*x^3 + 768*x*Log[4]^2)
+ E^((2*x + 3*x^2)/(4*Log[4]^2))*(-192*x^3 - 576*x^4 + 384*x^2*Log[4]^2))/(E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^
2),x]

[Out]

-x + 16*E^((-2*x)/Log[4]^2 - (3*x^2)/Log[4]^2)*x^4 + (1024*(x + 3*x^2))/(E^((x*(2 + 3*x))/(4*Log[4]^2))*(2 + 6
*x)) + (768*x*(x + 3*x^2))/(E^((x*(2 + 3*x))/(2*Log[4]^2))*(2 + 6*x)) + (256*x^2*(x + 3*x^2))/(E^((3*x*(2 + 3*
x))/(4*Log[4]^2))*(2 + 6*x))

Rule 12

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

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2272

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

Rule 2273

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + (-Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2)
, x], x] - Dist[(m - 1)*(e^2/(2*c*Log[F])), Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a,
b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]

Rule 2276

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

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]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)} \\ & = \frac {\int e^{-\frac {x (2+3 x)}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)} \\ & = \frac {\int \left (-32 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^4-96 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^5-\log ^2(4)+64 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^3 \log ^2(4)-256 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2-2 \log ^2(4)\right )-384 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2-2 \log ^2(4)\right )-192 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2-2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)} \\ & = -x+64 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^3 \, dx-\frac {32 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}-\frac {96 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^5 \, dx}{\log ^2(4)}-\frac {192 \int e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)}-\frac {256 \int e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)}-\frac {384 \int e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)} \\ & = -x+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}+64 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}-\frac {96 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^5 \, dx}{\log ^2(4)} \\ & = -x+\frac {16}{3} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {32}{3} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \log ^2(4)-16 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-\frac {64}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-64 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx}{3 \log ^2(4)}+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}+\frac {1}{3} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx \\ & = -x-\frac {16}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}+\frac {56}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \log ^2(4)-\frac {32}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \log ^4(4)+\frac {32}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+\frac {16}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+\frac {64}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+16 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx+\frac {64}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx}{9 \log ^2(4)}-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx}{3 \log ^2(4)}-\frac {1}{3} \left (8 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{9} \left (32 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{9} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{3} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx \\ & = -x+\frac {16}{27} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {8}{3} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \log ^2(4)-\frac {16}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {32}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {16}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {64}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {32}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx-\frac {16}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx-\frac {64}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx}{27 \log ^2(4)}+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx}{9 \log ^2(4)}+\frac {1}{3} \left (8 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx+\frac {1}{9} \left (32 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx+\frac {1}{9} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{3} \left (8 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {1}{9} \left (32 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {1}{9} \left (64 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx \\ & = -\frac {16}{81} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}}-x+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {20}{3} e^{\frac {1}{3 \log ^2(4)}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {1+3 x}{\sqrt {3} \log (4)}\right ) \log ^3(4)+\frac {16}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx+\frac {32}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx+\frac {16}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx+\frac {64}{27} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{27} \left (16 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {1}{27} \left (32 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {1}{9} \left (16 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {1}{27} \left (64 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx}{81 \log ^2(4)}-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx}{27 \log ^2(4)}+\frac {1}{3} \left (8 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {1}{9} \left (32 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {1}{9} \left (64 e^{\frac {1}{3 \log ^2(4)}} \log ^2(4)\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx \\ & = -x+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {80}{27} e^{\frac {1}{3 \log ^2(4)}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {1+3 x}{\sqrt {3} \log (4)}\right ) \log (4)+\frac {1}{27} \left (16 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {1}{27} \left (32 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {1}{9} \left (16 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {1}{27} \left (64 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx}{81 \log ^2(4)}-\frac {\left (32 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx}{81 \log ^2(4)} \\ & = -x+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {16 e^{\frac {1}{3 \log ^2(4)}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {1+3 x}{\sqrt {3} \log (4)}\right )}{81 \log (4)}+\frac {\left (32 e^{\frac {1}{3 \log ^2(4)}}\right ) \int \exp \left (-\frac {1}{12} \left (-\frac {2}{\log ^2(4)}-\frac {6 x}{\log ^2(4)}\right )^2 \log ^2(4)\right ) \, dx}{81 \log ^2(4)} \\ & = -x+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(27)=54\).

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-e^{-\frac {x (2+3 x)}{\log ^2(4)}} x \left (-512 e^{\frac {3 x (2+3 x)}{4 \log ^2(4)}}+e^{\frac {x (2+3 x)}{\log ^2(4)}}-384 e^{\frac {x (2+3 x)}{2 \log ^2(4)}} x-128 e^{\frac {x (2+3 x)}{4 \log ^2(4)}} x^2-16 x^3\right ) \]

[In]

Integrate[(-32*x^4 - 96*x^5 - E^((2*x + 3*x^2)/Log[4]^2)*Log[4]^2 + 64*x^3*Log[4]^2 + E^((3*(2*x + 3*x^2))/(4*
Log[4]^2))*(-256*x - 768*x^2 + 512*Log[4]^2) + E^((2*x + 3*x^2)/(2*Log[4]^2))*(-384*x^2 - 1152*x^3 + 768*x*Log
[4]^2) + E^((2*x + 3*x^2)/(4*Log[4]^2))*(-192*x^3 - 576*x^4 + 384*x^2*Log[4]^2))/(E^((2*x + 3*x^2)/Log[4]^2)*L
og[4]^2),x]

[Out]

-((x*(-512*E^((3*x*(2 + 3*x))/(4*Log[4]^2)) + E^((x*(2 + 3*x))/Log[4]^2) - 384*E^((x*(2 + 3*x))/(2*Log[4]^2))*
x - 128*E^((x*(2 + 3*x))/(4*Log[4]^2))*x^2 - 16*x^3))/E^((x*(2 + 3*x))/Log[4]^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(26)=52\).

Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78

method	result	size
risch	\(-x +512 x \,{\mathrm e}^{-\frac {x \left (2+3 x \right )}{16 \ln \left (2\right )^{2}}}+384 x^{2} {\mathrm e}^{-\frac {x \left (2+3 x \right )}{8 \ln \left (2\right )^{2}}}+128 x^{3} {\mathrm e}^{-\frac {3 x \left (2+3 x \right )}{16 \ln \left (2\right )^{2}}}+16 x^{4} {\mathrm e}^{-\frac {x \left (2+3 x \right )}{4 \ln \left (2\right )^{2}}}\)	\(75\)
parts	\(-x +512 x \,{\mathrm e}^{-\frac {3 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {x}{8 \ln \left (2\right )^{2}}}+384 x^{2} {\mathrm e}^{-\frac {3 x^{2}}{8 \ln \left (2\right )^{2}}-\frac {x}{4 \ln \left (2\right )^{2}}}+128 x^{3} {\mathrm e}^{-\frac {9 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {3 x}{8 \ln \left (2\right )^{2}}}+16 x^{4} {\mathrm e}^{-\frac {3 x^{2}}{4 \ln \left (2\right )^{2}}-\frac {x}{2 \ln \left (2\right )^{2}}}\)	\(95\)
default	\(\frac {-4 x \ln \left (2\right )^{2}+64 \ln \left (2\right )^{2} x^{4} {\mathrm e}^{-\frac {3 x^{2}}{4 \ln \left (2\right )^{2}}-\frac {x}{2 \ln \left (2\right )^{2}}}+512 \ln \left (2\right )^{2} x^{3} {\mathrm e}^{-\frac {9 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {3 x}{8 \ln \left (2\right )^{2}}}+1536 \ln \left (2\right )^{2} x^{2} {\mathrm e}^{-\frac {3 x^{2}}{8 \ln \left (2\right )^{2}}-\frac {x}{4 \ln \left (2\right )^{2}}}+2048 \ln \left (2\right )^{2} x \,{\mathrm e}^{-\frac {3 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {x}{8 \ln \left (2\right )^{2}}}}{4 \ln \left (2\right )^{2}}\)	\(121\)
parallelrisch	\(-\frac {\left (12 \ln \left (2\right )^{2} {\mathrm e}^{\frac {3 x^{2}+2 x}{4 \ln \left (2\right )^{2}}} x -192 x^{4} \ln \left (2\right )^{2}-1536 \ln \left (2\right )^{2} x^{3} {\mathrm e}^{\frac {3 x^{2}+2 x}{16 \ln \left (2\right )^{2}}}-4608 \ln \left (2\right )^{2} x^{2} {\mathrm e}^{\frac {3 x^{2}+2 x}{8 \ln \left (2\right )^{2}}}-6144 \ln \left (2\right )^{2} x \,{\mathrm e}^{\frac {\frac {9}{16} x^{2}+\frac {3}{8} x}{\ln \left (2\right )^{2}}}\right ) {\mathrm e}^{-\frac {x \left (2+3 x \right )}{4 \ln \left (2\right )^{2}}}}{12 \ln \left (2\right )^{2}}\)	\(134\)

[In]

int(1/4*(-4*ln(2)^2*exp(1/16*(3*x^2+2*x)/ln(2)^2)^4+(2048*ln(2)^2-768*x^2-256*x)*exp(1/16*(3*x^2+2*x)/ln(2)^2)
^3+(3072*x*ln(2)^2-1152*x^3-384*x^2)*exp(1/16*(3*x^2+2*x)/ln(2)^2)^2+(1536*x^2*ln(2)^2-576*x^4-192*x^3)*exp(1/
16*(3*x^2+2*x)/ln(2)^2)+256*x^3*ln(2)^2-96*x^5-32*x^4)/ln(2)^2/exp(1/16*(3*x^2+2*x)/ln(2)^2)^4,x,method=_RETUR
NVERBOSE)

[Out]

-x+512*x*exp(-1/16*x*(2+3*x)/ln(2)^2)+384*x^2*exp(-1/8*x*(2+3*x)/ln(2)^2)+128*x^3*exp(-3/16*x*(2+3*x)/ln(2)^2)
+16*x^4*exp(-1/4*x*(2+3*x)/ln(2)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.81 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx={\left (16 \, x^{4} + 128 \, x^{3} e^{\left (\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} + 384 \, x^{2} e^{\left (\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - x e^{\left (\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} + 512 \, x e^{\left (\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} \]

[In]

integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2-768*x^2-256*x)*exp(1/16*(3*x^2+2*x)
/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3-384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-19
2*x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2)^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^
4,x, algorithm="fricas")

[Out]

(16*x^4 + 128*x^3*e^(1/16*(3*x^2 + 2*x)/log(2)^2) + 384*x^2*e^(1/8*(3*x^2 + 2*x)/log(2)^2) - x*e^(1/4*(3*x^2 +
2*x)/log(2)^2) + 512*x*e^(3/16*(3*x^2 + 2*x)/log(2)^2))*e^(-1/4*(3*x^2 + 2*x)/log(2)^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=16 x^{4} e^{- \frac {4 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 128 x^{3} e^{- \frac {3 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 384 x^{2} e^{- \frac {2 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} - x + 512 x e^{- \frac {\frac {3 x^{2}}{16} + \frac {x}{8}}{\log {\left (2 \right )}^{2}}} \]

[In]

integrate(1/4*(-4*ln(2)**2*exp(1/16*(3*x**2+2*x)/ln(2)**2)**4+(2048*ln(2)**2-768*x**2-256*x)*exp(1/16*(3*x**2+
2*x)/ln(2)**2)**3+(3072*x*ln(2)**2-1152*x**3-384*x**2)*exp(1/16*(3*x**2+2*x)/ln(2)**2)**2+(1536*x**2*ln(2)**2-
576*x**4-192*x**3)*exp(1/16*(3*x**2+2*x)/ln(2)**2)+256*x**3*ln(2)**2-96*x**5-32*x**4)/ln(2)**2/exp(1/16*(3*x**
2+2*x)/ln(2)**2)**4,x)

[Out]

16*x**4*exp(-4*(3*x**2/16 + x/8)/log(2)**2) + 128*x**3*exp(-3*(3*x**2/16 + x/8)/log(2)**2) + 384*x**2*exp(-2*(
3*x**2/16 + x/8)/log(2)**2) - x + 512*x*exp(-(3*x**2/16 + x/8)/log(2)**2)

Maxima [C] (verification not implemented)

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

Time = 0.54 (sec) , antiderivative size = 2622, normalized size of antiderivative = 97.11 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=\text {Too large to display} \]

[In]

integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2-768*x^2-256*x)*exp(1/16*(3*x^2+2*x)
/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3-384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-19
2*x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2)^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^
4,x, algorithm="maxima")

[Out]

1/243*(82944*sqrt(3)*sqrt(pi)*erf(1/4*sqrt(3)*x/log(2) + 1/12*sqrt(3)/log(2))*e^(1/48/log(2)^2)*log(2)^3 + 192
*sqrt(3)*(36*sqrt(3)*sqrt(1/3)*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log
(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(7/2)*log(2)^5) - 24*sqrt(3)*gamma(2, 1/12*(3*x/lo
g(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(7/2)*log(2)^4) - sqrt(3)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 + 1
/log(2)^2)*(erf(1/2*sqrt(1/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^
2)^2)*(-1/log(2)^2)^(7/2)*log(2)^7) - 6*sqrt(3)*e^(-1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2
)^(7/2)*log(2)^6))*e^(1/12/log(2)^2)*log(2)^2/sqrt(-1/log(2)^2) - 82944*sqrt(3/2)*(sqrt(3/2)*sqrt(1/6)*sqrt(pi
)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/2*sqrt(1/6)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/l
og(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(3/2)*log(2)^3) + 2*sqrt(3/2)*e^(-1/24*(3*x/log(2)^2 + 1/log(2)^2)^2*lo
g(2)^2)/((-1/log(2)^2)^(3/2)*log(2)^2))*e^(1/24/log(2)^2)*log(2)^2/sqrt(-1/log(2)^2) + 6912*(16*(3*x/log(2)^2
+ 1/log(2)^2)^3*gamma(3/2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)
*(-1/log(2)^2)^(5/2)*log(2)^3) - sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/4*sqrt((3*x/log(2)^2 + 1/log(2)^2
)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(5/2)*log(2)^5) - 8*e^(-1/16*(3*x/log(2)^
2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(5/2)*log(2)^4))*e^(1/16/log(2)^2)*log(2)^2/sqrt(-1/log(2)^2) - 243
*x*log(2)^2 - 8*sqrt(3)*(144*sqrt(3)*sqrt(1/3)*(3*x/log(2)^2 + 1/log(2)^2)^5*gamma(5/2, 1/12*(3*x/log(2)^2 + 1
/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(5/2)*(-1/log(2)^2)^(9/2)*log(2)^5) + 72*sqrt(3)*sqrt(
1/3)*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1
/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(9/2)*log(2)^7) - 96*sqrt(3)*gamma(2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log
(2)^2)/((-1/log(2)^2)^(9/2)*log(2)^6) - sqrt(3)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/2*sqrt(1
/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(9/2)*
log(2)^9) - 8*sqrt(3)*e^(-1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(9/2)*log(2)^8))*e^(1/12
/log(2)^2)/sqrt(-1/log(2)^2) - 8*sqrt(3)*(720*sqrt(3)*sqrt(1/3)*(3*x/log(2)^2 + 1/log(2)^2)^5*gamma(5/2, 1/12*
(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(5/2)*(-1/log(2)^2)^(11/2)*log(2)^7)
- 288*sqrt(3)*gamma(3, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(11/2)*log(2)^6) + 120*sqrt
(3)*sqrt(1/3)*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log
(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(11/2)*log(2)^9) - 240*sqrt(3)*gamma(2, 1/12*(3*x/log(2)^2 + 1/log(
2)^2)^2*log(2)^2)/((-1/log(2)^2)^(11/2)*log(2)^8) - sqrt(3)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(er
f(1/2*sqrt(1/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(
2)^2)^(11/2)*log(2)^11) - 10*sqrt(3)*e^(-1/12*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(11/2)*lo
g(2)^10))*e^(1/12/log(2)^2)/sqrt(-1/log(2)^2) - 3456*sqrt(3/2)*(24*sqrt(3/2)*sqrt(1/6)*(3*x/log(2)^2 + 1/log(2
)^2)^3*gamma(3/2, 1/24*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(
2)^2)^(5/2)*log(2)^3) - sqrt(3/2)*sqrt(1/6)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/2*sqrt(1/6)*sqrt((3*x/
log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(5/2)*log(2)^5) - 4*
sqrt(3/2)*e^(-1/24*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(5/2)*log(2)^4))*e^(1/24/log(2)^2)/s
qrt(-1/log(2)^2) - 3456*sqrt(3/2)*(72*sqrt(3/2)*sqrt(1/6)*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/24*(3*x/l
og(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(7/2)*log(2)^5) - 48*sq
rt(3/2)*gamma(2, 1/24*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(7/2)*log(2)^4) - sqrt(3/2)*sqrt(
1/6)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/2*sqrt(1/6)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/
(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(7/2)*log(2)^7) - 6*sqrt(3/2)*e^(-1/24*(3*x/log(2)^2 + 1/lo
g(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(7/2)*log(2)^6))*e^(1/24/log(2)^2)/sqrt(-1/log(2)^2) + 3456*sqrt(3)*(sqrt(3
)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/4*sqrt(1/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)
) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(3/2)*log(2)^3) + 4*sqrt(3)*e^(-1/48*(3*x/log(2)^2 +
1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(3/2)*log(2)^2))*e^(1/48/log(2)^2)/sqrt(-1/log(2)^2) - 3456*sqrt(3)*(4
8*sqrt(3)*sqrt(1/3)*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/48*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3
*x/log(2)^2 + 1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(5/2)*log(2)^3) - sqrt(3)*sqrt(1/3)*sqrt(pi)*(3*x/log(2)^2 +
1/log(2)^2)*(erf(1/4*sqrt(1/3)*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)
^2)^2)*(-1/log(2)^2)^(5/2)*log(2)^5) - 8*sqrt(3)*e^(-1/48*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^
2)^(5/2)*log(2)^4))*e^(1/48/log(2)^2)/sqrt(-1/log(2)^2) - 288*(256*(3*x/log(2)^2 + 1/log(2)^2)^5*gamma(5/2, 1/
16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2)^2)^2)^(5/2)*(-1/log(2)^2)^(9/2)*log(2)^5
) + 96*(3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 +
1/log(2)^2)^2)^(3/2)*(-1/log(2)^2)^(9/2)*log(2)^7) - 256*gamma(2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2
)/((-1/log(2)^2)^(9/2)*log(2)^6) - sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/4*sqrt((3*x/log(2)^2 + 1/log(2)
^2)^2)*log(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(9/2)*log(2)^9) - 16*e^(-1/16*(3*x/log(
2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(9/2)*log(2)^8))*e^(1/16/log(2)^2)/sqrt(-1/log(2)^2) - 288*(48*(
3*x/log(2)^2 + 1/log(2)^2)^3*gamma(3/2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/(((3*x/log(2)^2 + 1/log(2
)^2)^2)^(3/2)*(-1/log(2)^2)^(7/2)*log(2)^5) - 64*gamma(2, 1/16*(3*x/log(2)^2 + 1/log(2)^2)^2*log(2)^2)/((-1/lo
g(2)^2)^(7/2)*log(2)^4) - sqrt(pi)*(3*x/log(2)^2 + 1/log(2)^2)*(erf(1/4*sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*lo
g(2)) - 1)/(sqrt((3*x/log(2)^2 + 1/log(2)^2)^2)*(-1/log(2)^2)^(7/2)*log(2)^7) - 12*e^(-1/16*(3*x/log(2)^2 + 1/
log(2)^2)^2*log(2)^2)/((-1/log(2)^2)^(7/2)*log(2)^6))*e^(1/16/log(2)^2)/sqrt(-1/log(2)^2))/log(2)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (25) = 50\).

Time = 0.31 (sec) , antiderivative size = 228, normalized size of antiderivative = 8.44 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=-\frac {81 \, x \log \left (2\right )^{2} - 13824 \, {\left ({\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} - 3456 \, {\left ({\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 2 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - 384 \, {\left ({\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} - 3 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} + 3 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )} - 16 \, {\left ({\left (3 \, x + 1\right )}^{4} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} + 6 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )}}{81 \, \log \left (2\right )^{2}} \]

[In]

integrate(1/4*(-4*log(2)^2*exp(1/16*(3*x^2+2*x)/log(2)^2)^4+(2048*log(2)^2-768*x^2-256*x)*exp(1/16*(3*x^2+2*x)
/log(2)^2)^3+(3072*x*log(2)^2-1152*x^3-384*x^2)*exp(1/16*(3*x^2+2*x)/log(2)^2)^2+(1536*x^2*log(2)^2-576*x^4-19
2*x^3)*exp(1/16*(3*x^2+2*x)/log(2)^2)+256*x^3*log(2)^2-96*x^5-32*x^4)/log(2)^2/exp(1/16*(3*x^2+2*x)/log(2)^2)^
4,x, algorithm="giac")

[Out]

-1/81*(81*x*log(2)^2 - 13824*((3*x + 1)*log(2)^2 - log(2)^2)*e^(-1/16*(3*x^2 + 2*x)/log(2)^2) - 3456*((3*x + 1
)^2*log(2)^2 - 2*(3*x + 1)*log(2)^2 + log(2)^2)*e^(-1/8*(3*x^2 + 2*x)/log(2)^2) - 384*((3*x + 1)^3*log(2)^2 -
3*(3*x + 1)^2*log(2)^2 + 3*(3*x + 1)*log(2)^2 - log(2)^2)*e^(-3/16*(3*x^2 + 2*x)/log(2)^2) - 16*((3*x + 1)^4*l
og(2)^2 - 4*(3*x + 1)^3*log(2)^2 + 6*(3*x + 1)^2*log(2)^2 - 4*(3*x + 1)*log(2)^2 + log(2)^2)*e^(-1/4*(3*x^2 +
2*x)/log(2)^2))/log(2)^2

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right )}{\log ^2(4)} \, dx=512\,x\,{\mathrm {e}}^{-\frac {3\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {x}{8\,{\ln \left (2\right )}^2}}-x+16\,x^4\,{\mathrm {e}}^{-\frac {3\,x^2}{4\,{\ln \left (2\right )}^2}-\frac {x}{2\,{\ln \left (2\right )}^2}}+384\,x^2\,{\mathrm {e}}^{-\frac {3\,x^2}{8\,{\ln \left (2\right )}^2}-\frac {x}{4\,{\ln \left (2\right )}^2}}+128\,x^3\,{\mathrm {e}}^{-\frac {9\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {3\,x}{8\,{\ln \left (2\right )}^2}} \]

[In]

int(-(exp(-(4*(x/8 + (3*x^2)/16))/log(2)^2)*(exp((4*(x/8 + (3*x^2)/16))/log(2)^2)*log(2)^2 - 64*x^3*log(2)^2 +
(exp((x/8 + (3*x^2)/16)/log(2)^2)*(192*x^3 - 1536*x^2*log(2)^2 + 576*x^4))/4 + (exp((2*(x/8 + (3*x^2)/16))/lo
g(2)^2)*(384*x^2 - 3072*x*log(2)^2 + 1152*x^3))/4 + 8*x^4 + 24*x^5 + (exp((3*(x/8 + (3*x^2)/16))/log(2)^2)*(25
6*x - 2048*log(2)^2 + 768*x^2))/4))/log(2)^2,x)

[Out]

512*x*exp(- (3*x^2)/(16*log(2)^2) - x/(8*log(2)^2)) - x + 16*x^4*exp(- (3*x^2)/(4*log(2)^2) - x/(2*log(2)^2))
+ 384*x^2*exp(- (3*x^2)/(8*log(2)^2) - x/(4*log(2)^2)) + 128*x^3*exp(- (9*x^2)/(16*log(2)^2) - (3*x)/(8*log(2)
^2))