\(\int \frac {e^x (36 x-60 x^2+42 x^3-24 x^4+6 x^5+(-12+48 x^2-48 x^3+12 x^4) \log (2))}{x^4+(-4 x^2+4 x^3) \log (2)+(4-8 x+4 x^2) \log ^2(2)} \, dx\) [5890]

Optimal result

Integrand size = 83, antiderivative size = 30 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {3 e^x x \left (-4+\frac {3}{x}+x\right )}{\frac {x^2}{-2+2 x}+\log (2)} \]

[Out]

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

Rubi [C] (verified)

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

Time = 4.16 (sec) , antiderivative size = 1114, normalized size of antiderivative = 37.13, number of steps used = 30, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6873, 12, 6820, 6874, 2207, 2225, 6860, 2209, 2208} \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx =\text {Too large to display} \]

[In]

Int[(E^x*(36*x - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5 + (-12 + 48*x^2 - 48*x^3 + 12*x^4)*Log[2]))/(x^4 + (-4*x^2 +
4*x^3)*Log[2] + (4 - 8*x + 4*x^2)*Log[2]^2),x]

[Out]

-6*E^x + 6*E^x*x - 12*E^x*(2 + Log[2]) + (3*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*ExpIntegralEi[(2*x + Log[4] - Sqrt
[Log[4]*(4 + Log[4])])/2]*(6 + 4*Log[4]^3 + Log[4]*(17 - 4*Log[8]) + 2*Log[4]^2*(7 - Log[8])))/(Sqrt[Log[4]]*(
4 + Log[4])^(3/2)) - (3*ExpIntegralEi[(2*x + Log[4] + Sqrt[Log[4]*(4 + Log[4])])/2]*(6 + 4*Log[4]^3 + Log[4]*(
17 - 4*Log[8]) + 2*Log[4]^2*(7 - Log[8])))/(E^(Sqrt[Log[4]*(4 + Log[4])]/2)*Sqrt[Log[4]]*(4 + Log[4])^(3/2)) -
(3*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*ExpIntegralEi[(2*x + Log[4] - Sqrt[Log[4]*(4 + Log[4])])/2]*(Log[4] - Sqrt
[Log[4]*(4 + Log[4])])*(6 + 4*Log[4]^3 + Log[4]*(17 - 4*Log[8]) + 2*Log[4]^2*(7 - Log[8])))/(2*Log[4]*(4 + Log
[4])) + (6*E^x*(Log[4] - Sqrt[Log[4]*(4 + Log[4])])*(6 + 4*Log[4]^3 + Log[4]*(17 - 4*Log[8]) + 2*Log[4]^2*(7 -
Log[8])))/(Log[4]*(4 + Log[4])*(2*x + Log[4] - Sqrt[Log[4]*(4 + Log[4])])) - (3*ExpIntegralEi[(2*x + Log[4] +
Sqrt[Log[4]*(4 + Log[4])])/2]*(Log[4] + Sqrt[Log[4]*(4 + Log[4])])*(6 + 4*Log[4]^3 + Log[4]*(17 - 4*Log[8]) +
2*Log[4]^2*(7 - Log[8])))/(2*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*Log[4]*(4 + Log[4])) + (6*E^x*(Log[4] + Sqrt[Log
[4]*(4 + Log[4])])*(6 + 4*Log[4]^3 + Log[4]*(17 - 4*Log[8]) + 2*Log[4]^2*(7 - Log[8])))/(Log[4]*(4 + Log[4])*(
2*x + Log[4] + Sqrt[Log[4]*(4 + Log[4])])) + (3*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*ExpIntegralEi[(2*x + Log[4] -
Sqrt[Log[4]*(4 + Log[4])])/2]*(7 + Log[4]^2 + Log[4096] - (10*Log[4]^2 + Log[4]^3 + 2*(10 - Log[8]*Log[16] - L
og[64]) + Log[4]*(35 + Log[4096]))/Sqrt[Log[4]*(4 + Log[4])]))/2 + (3*ExpIntegralEi[(2*x + Log[4] + Sqrt[Log[4
]*(4 + Log[4])])/2]*(7 + Log[4]^2 + Log[4096] + (10*Log[4]^2 + Log[4]^3 + 2*(10 - Log[8]*Log[16] - Log[64]) +
Log[4]*(35 + Log[4096]))/Sqrt[Log[4]*(4 + Log[4])]))/(2*E^(Sqrt[Log[4]*(4 + Log[4])]/2)) + (6*E^(Sqrt[Log[4]*(
4 + Log[4])]/2)*ExpIntegralEi[(2*x + Log[4] - Sqrt[Log[4]*(4 + Log[4])])/2]*(11 + 4*Log[4]^2 - Log[8]*Log[16]
+ Log[16384]))/(Sqrt[Log[4]]*(4 + Log[4])^(3/2)) - (6*ExpIntegralEi[(2*x + Log[4] + Sqrt[Log[4]*(4 + Log[4])])
/2]*(11 + 4*Log[4]^2 - Log[8]*Log[16] + Log[16384]))/(E^(Sqrt[Log[4]*(4 + Log[4])]/2)*Sqrt[Log[4]]*(4 + Log[4]
)^(3/2)) - (3*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*ExpIntegralEi[(2*x + Log[4] - Sqrt[Log[4]*(4 + Log[4])])/2]*(11
+ 4*Log[4]^2 - Log[8]*Log[16] + Log[16384]))/(4 + Log[4]) - (3*ExpIntegralEi[(2*x + Log[4] + Sqrt[Log[4]*(4 +
Log[4])])/2]*(11 + 4*Log[4]^2 - Log[8]*Log[16] + Log[16384]))/(E^(Sqrt[Log[4]*(4 + Log[4])]/2)*(4 + Log[4])) +
(12*E^x*Log[4]*(11 + 4*Log[4]^2 - Log[8]*Log[16] + Log[16384]))/((2*x + Log[4] - Sqrt[Log[4]*(4 + Log[4])])*(
Log[4]^2 + Log[256])) + (12*E^x*Log[4]*(11 + 4*Log[4]^2 - Log[8]*Log[16] + Log[16384]))/((2*x + Log[4] + Sqrt[
Log[4]*(4 + Log[4])])*(Log[4]^2 + Log[256]))

Rule 12

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

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6820

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

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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}& = \int \frac {6 e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{x^4+4 x^3 \log (2)-4 x^2 (1-\log (2)) \log (2)+4 \log ^2(2)-8 x \log ^2(2)} \, dx \\ & = 6 \int \frac {e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{x^4+4 x^3 \log (2)-4 x^2 (1-\log (2)) \log (2)+4 \log ^2(2)-8 x \log ^2(2)} \, dx \\ & = 6 \int \frac {e^x (1-x) \left (-x^4+x^3 (3-\log (4))+x (6-\log (4))-\log (4)-2 x^2 (2-\log (8))\right )}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx \\ & = 6 \int \left (e^x x-4 e^x \left (1+\frac {\log (2)}{2}\right )+\frac {e^x \left (-10-14 \log (4)-5 \log ^2(4)+\log (8) \log (16)+\log (64)+x \left (7+\log ^2(4)+\log (4096)\right )\right )}{x^2-\log (4)+x \log (4)}+\frac {e^x \left (x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )-\log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}\right ) \, dx \\ & = 6 \int e^x x \, dx+6 \int \frac {e^x \left (-10-14 \log (4)-5 \log ^2(4)+\log (8) \log (16)+\log (64)+x \left (7+\log ^2(4)+\log (4096)\right )\right )}{x^2-\log (4)+x \log (4)} \, dx+6 \int \frac {e^x \left (x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )-\log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right )}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx-(12 (2+\log (2))) \int e^x \, dx \\ & = 6 e^x x-12 e^x (2+\log (2))-6 \int e^x \, dx+6 \int \left (\frac {e^x \left (7+\log ^2(4)+\log (4096)-\frac {-20-35 \log (4)-10 \log ^2(4)-\log ^3(4)+2 \log (8) \log (16)+2 \log (64)-\log (4) \log (4096)}{\sqrt {\log (4) (4+\log (4))}}\right )}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}}+\frac {e^x \left (7+\log ^2(4)+\log (4096)+\frac {-20-35 \log (4)-10 \log ^2(4)-\log ^3(4)+2 \log (8) \log (16)+2 \log (64)-\log (4) \log (4096)}{\sqrt {\log (4) (4+\log (4))}}\right )}{2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}}\right ) \, dx+6 \int \left (\frac {e^x x \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}-\frac {e^x \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )}{\left (x^2-\log (4)+x \log (4)\right )^2}\right ) \, dx \\ & = -6 e^x+6 e^x x-12 e^x (2+\log (2))+\left (6 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x x}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx+\left (6 \left (7+\log ^2(4)+\log (4096)-\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )\right ) \int \frac {e^x}{2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}} \, dx+\left (6 \left (7+\log ^2(4)+\log (4096)+\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )\right ) \int \frac {e^x}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx-\left (6 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{\left (x^2-\log (4)+x \log (4)\right )^2} \, dx \\ & = -6 e^x+6 e^x x-12 e^x (2+\log (2))+\frac {3}{2} e^{\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)-\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )+\frac {3}{2} e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)+\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )+\left (6 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \left (\frac {2 e^x \left (-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}{\log (4) (4+\log (4)) \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}-\frac {2 e^x}{\sqrt {\log (4)} (4+\log (4))^{3/2} \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}+\frac {2 e^x \left (-\log (4)-\sqrt {\log (4) (4+\log (4))}\right )}{\log (4) (4+\log (4)) \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}-\frac {2 e^x}{\sqrt {\log (4)} (4+\log (4))^{3/2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}\right ) \, dx-\left (6 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \left (\frac {4 e^x}{\log (4) (4+\log (4)) \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}+\frac {4 e^x}{(\log (4) (4+\log (4)))^{3/2} \left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}+\frac {4 e^x}{\log (4) (4+\log (4)) \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2}+\frac {4 e^x}{(\log (4) (4+\log (4)))^{3/2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )}\right ) \, dx \\ & = -6 e^x+6 e^x x-12 e^x (2+\log (2))+\frac {3}{2} e^{\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)-\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)-\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )+\frac {3}{2} e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \text {Ei}\left (\frac {1}{2} \left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (7+\log ^2(4)+\log (4096)+\frac {10 \log ^2(4)+\log ^3(4)+2 (10-\log (8) \log (16)-\log (64))+\log (4) (35+\log (4096))}{\sqrt {\log (4) (4+\log (4))}}\right )-\frac {\left (12 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x}{-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx}{\sqrt {\log (4)} (4+\log (4))^{3/2}}-\frac {\left (12 \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx}{\sqrt {\log (4)} (4+\log (4))^{3/2}}-\frac {\left (12 \left (\log (4)-\sqrt {\log (4) (4+\log (4))}\right ) \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x}{\left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2} \, dx}{\log (4) (4+\log (4))}-\frac {\left (12 \left (\log (4)+\sqrt {\log (4) (4+\log (4))}\right ) \left (6+4 \log ^3(4)+\log (4) (17-4 \log (8))+2 \log ^2(4) (7-\log (8))\right )\right ) \int \frac {e^x}{\left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2} \, dx}{\log (4) (4+\log (4))}-\frac {\left (24 \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx}{\sqrt {\log (4)} (4+\log (4))^{3/2}}-\frac {\left (24 \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}} \, dx}{\sqrt {\log (4)} (4+\log (4))^{3/2}}-\frac {\left (24 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{\left (-2 x-\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2} \, dx}{\log ^2(4)+\log (256)}-\frac {\left (24 \log (4) \left (11+4 \log ^2(4)-\log (8) \log (16)+\log (16384)\right )\right ) \int \frac {e^x}{\left (2 x+\log (4)+\sqrt {\log (4) (4+\log (4))}\right )^2} \, dx}{\log ^2(4)+\log (256)} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 11.88 (sec) , antiderivative size = 629, normalized size of antiderivative = 20.97 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {3 e^{-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (4 e^{x+\frac {1}{2} \sqrt {\log (4) (4+\log (4))}} \left (-3 \log ^{\frac {7}{2}}(4) \sqrt {4+\log (4)}-12 \sqrt {\log (4) (4+\log (4))}-5 x^2 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )+x^3 \left (\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))}\right )-x \left ((-13+24 \log (2)) \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}-3 \log ^{\frac {7}{2}}(4) \sqrt {4+\log (4)}+4 \sqrt {\log (4) (4+\log (4))} (-7+\log (8))+\log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)} (-12+\log (64))\right )+\log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)} (-9+\log (64))+3 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (-1+\log (64))\right )+3 e^{\sqrt {\log (4) (4+\log (4))}} \operatorname {ExpIntegralEi}\left (x+\log (2)-\frac {1}{2} \sqrt {\log (4) (4+\log (4))}\right ) \left (x^2-\log (4)+x \log (4)\right ) \left (5 \log ^3(4)-3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)+\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))-\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )-3 \operatorname {ExpIntegralEi}\left (x+\frac {1}{2} \left (\log (4)+\sqrt {\log (4) (4+\log (4))}\right )\right ) \left (x^2 \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+x \log (4) \left (5 \log ^3(4)+3 \log ^{\frac {5}{2}}(4) \sqrt {4+\log (4)}-\log (4) \log (16)-\log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)} (2+\log (64))+\log (16) \sqrt {\log ^2(4)+\log (256)}-\log ^2(4) (-2+\log (1024))\right )+\log ^2(4) \left (-5 \log ^2(4)-3 \log ^{\frac {3}{2}}(4) \sqrt {4+\log (4)}+\log (64) \sqrt {\log ^2(4)+\log (256)}+\log (4) \log (1024)\right )\right )\right )}{2 \sqrt {\log (4)} (4+\log (4))^{3/2} \left (x^2-\log (4)+x \log (4)\right )} \]

[In]

Integrate[(E^x*(36*x - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5 + (-12 + 48*x^2 - 48*x^3 + 12*x^4)*Log[2]))/(x^4 + (-4
*x^2 + 4*x^3)*Log[2] + (4 - 8*x + 4*x^2)*Log[2]^2),x]

[Out]

(3*(4*E^(x + Sqrt[Log[4]*(4 + Log[4])]/2)*(-3*Log[4]^(7/2)*Sqrt[4 + Log[4]] - 12*Sqrt[Log[4]*(4 + Log[4])] - 5
*x^2*(Log[4]^(3/2)*Sqrt[4 + Log[4]] + 4*Sqrt[Log[4]*(4 + Log[4])]) + x^3*(Log[4]^(3/2)*Sqrt[4 + Log[4]] + 4*Sq
rt[Log[4]*(4 + Log[4])]) - x*((-13 + 24*Log[2])*Log[4]^(3/2)*Sqrt[4 + Log[4]] - 3*Log[4]^(7/2)*Sqrt[4 + Log[4]
] + 4*Sqrt[Log[4]*(4 + Log[4])]*(-7 + Log[8]) + Log[4]^(5/2)*Sqrt[4 + Log[4]]*(-12 + Log[64])) + Log[4]^(5/2)*
Sqrt[4 + Log[4]]*(-9 + Log[64]) + 3*Log[4]^(3/2)*Sqrt[4 + Log[4]]*(-1 + Log[64])) + 3*E^Sqrt[Log[4]*(4 + Log[4
])]*ExpIntegralEi[x + Log[2] - Sqrt[Log[4]*(4 + Log[4])]/2]*(x^2 - Log[4] + x*Log[4])*(5*Log[4]^3 - 3*Log[4]^(
5/2)*Sqrt[4 + Log[4]] - Log[4]*Log[16] + Log[4]^(3/2)*Sqrt[4 + Log[4]]*(2 + Log[64]) - Log[16]*Sqrt[Log[4]^2 +
Log[256]] - Log[4]^2*(-2 + Log[1024])) - 3*ExpIntegralEi[x + (Log[4] + Sqrt[Log[4]*(4 + Log[4])])/2]*(x^2*(5*
Log[4]^3 + 3*Log[4]^(5/2)*Sqrt[4 + Log[4]] - Log[4]*Log[16] - Log[4]^(3/2)*Sqrt[4 + Log[4]]*(2 + Log[64]) + Lo
g[16]*Sqrt[Log[4]^2 + Log[256]] - Log[4]^2*(-2 + Log[1024])) + x*Log[4]*(5*Log[4]^3 + 3*Log[4]^(5/2)*Sqrt[4 +
Log[4]] - Log[4]*Log[16] - Log[4]^(3/2)*Sqrt[4 + Log[4]]*(2 + Log[64]) + Log[16]*Sqrt[Log[4]^2 + Log[256]] - L
og[4]^2*(-2 + Log[1024])) + Log[4]^2*(-5*Log[4]^2 - 3*Log[4]^(3/2)*Sqrt[4 + Log[4]] + Log[64]*Sqrt[Log[4]^2 +
Log[256]] + Log[4]*Log[1024]))))/(2*E^(Sqrt[Log[4]*(4 + Log[4])]/2)*Sqrt[Log[4]]*(4 + Log[4])^(3/2)*(x^2 - Log
[4] + x*Log[4]))

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10

method	result	size
gosper	\(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\)	\(33\)
risch	\(\frac {6 \left (x^{3}-5 x^{2}+7 x -3\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\)	\(33\)
norman	\(\frac {42 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\)	\(41\)
parallelrisch	\(\frac {42 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x}}{2 x \ln \left (2\right )+x^{2}-2 \ln \left (2\right )}\)	\(41\)
default	\(\text {Expression too large to display}\)	\(5402\)

[In]

int(((12*x^4-48*x^3+48*x^2-12)*ln(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp(x)/((4*x^2-8*x+4)*ln(2)^2+(4*x^3-4*x
^2)*ln(2)+x^4),x,method=_RETURNVERBOSE)

[Out]

6*(x^3-5*x^2+7*x-3)*exp(x)/(2*x*ln(2)+x^2-2*ln(2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, {\left (x - 1\right )} \log \left (2\right )} \]

[In]

integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4
*x^3-4*x^2)*log(2)+x^4),x, algorithm="fricas")

[Out]

6*(x^3 - 5*x^2 + 7*x - 3)*e^x/(x^2 + 2*(x - 1)*log(2))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {\left (6 x^{3} - 30 x^{2} + 42 x - 18\right ) e^{x}}{x^{2} + 2 x \log {\left (2 \right )} - 2 \log {\left (2 \right )}} \]

[In]

integrate(((12*x**4-48*x**3+48*x**2-12)*ln(2)+6*x**5-24*x**4+42*x**3-60*x**2+36*x)*exp(x)/((4*x**2-8*x+4)*ln(2
)**2+(4*x**3-4*x**2)*ln(2)+x**4),x)

[Out]

(6*x**3 - 30*x**2 + 42*x - 18)*exp(x)/(x**2 + 2*x*log(2) - 2*log(2))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} - 5 \, x^{2} + 7 \, x - 3\right )} e^{x}}{x^{2} + 2 \, x \log \left (2\right ) - 2 \, \log \left (2\right )} \]

[In]

integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4
*x^3-4*x^2)*log(2)+x^4),x, algorithm="maxima")

[Out]

6*(x^3 - 5*x^2 + 7*x - 3)*e^x/(x^2 + 2*x*log(2) - 2*log(2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\frac {6 \, {\left (x^{3} e^{x} - 5 \, x^{2} e^{x} + 7 \, x e^{x} - 3 \, e^{x}\right )}}{x^{2} + 2 \, x \log \left (2\right ) - 2 \, \log \left (2\right )} \]

[In]

integrate(((12*x^4-48*x^3+48*x^2-12)*log(2)+6*x^5-24*x^4+42*x^3-60*x^2+36*x)*exp(x)/((4*x^2-8*x+4)*log(2)^2+(4
*x^3-4*x^2)*log(2)+x^4),x, algorithm="giac")

[Out]

6*(x^3*e^x - 5*x^2*e^x + 7*x*e^x - 3*e^x)/(x^2 + 2*x*log(2) - 2*log(2))

Mupad [F(-1)]

Timed out. \[ \int \frac {e^x \left (36 x-60 x^2+42 x^3-24 x^4+6 x^5+\left (-12+48 x^2-48 x^3+12 x^4\right ) \log (2)\right )}{x^4+\left (-4 x^2+4 x^3\right ) \log (2)+\left (4-8 x+4 x^2\right ) \log ^2(2)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (36\,x+\ln \left (2\right )\,\left (12\,x^4-48\,x^3+48\,x^2-12\right )-60\,x^2+42\,x^3-24\,x^4+6\,x^5\right )}{{\ln \left (2\right )}^2\,\left (4\,x^2-8\,x+4\right )-\ln \left (2\right )\,\left (4\,x^2-4\,x^3\right )+x^4} \,d x \]

[In]

int((exp(x)*(36*x + log(2)*(48*x^2 - 48*x^3 + 12*x^4 - 12) - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5))/(log(2)^2*(4*x
^2 - 8*x + 4) - log(2)*(4*x^2 - 4*x^3) + x^4),x)

[Out]

int((exp(x)*(36*x + log(2)*(48*x^2 - 48*x^3 + 12*x^4 - 12) - 60*x^2 + 42*x^3 - 24*x^4 + 6*x^5))/(log(2)^2*(4*x
^2 - 8*x + 4) - log(2)*(4*x^2 - 4*x^3) + x^4), x)