\(\int \frac {e^x ((-6 x^3+21 x^4-12 x^5-12 x^6+e^8 (-6 x-3 x^2)+e^4 (12 x^2-18 x^3-12 x^4)) \log (4)+(2 x^4-2 e^4 x^4-6 x^5-4 x^6) \log ^2(4))}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 (-18 x+36 x^2)+(12 e^4 x^2-12 x^3+24 x^4) \log (4)+4 x^4 \log ^2(4)} \, dx\) [1121]

Optimal result

Integrand size = 158, antiderivative size = 33 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\frac {e^x x^2}{\frac {2}{-2+\frac {-e^4+x}{x^2}}-\frac {3}{\log (4)}} \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 5.15 (sec) , antiderivative size = 1873, normalized size of antiderivative = 56.76, number of steps used = 33, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6, 6820, 12, 6874, 2207, 2225, 2302, 2209, 2208} \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx =\text {Too large to display} \]

[In]

Int[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(12*x^2 - 18*x^3 - 12*x^4))*Log[4] + (
2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^6)*Log[4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) + (12*E^
4*x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2),x]

[Out]

(-2*E^x*Log[4])/(3 + Log[4]) + (2*E^x*x*Log[4])/(3 + Log[4]) - (E^x*x^2*Log[4])/(3 + Log[4]) + (3*E^x*Log[4]*(
4 + Log[4]))/(2*(3 + Log[4])^2) - (3*E^x*x*Log[4]*(4 + Log[4]))/(2*(3 + Log[4])^2) + (E^x*Log[4]^2*(9 - 2*E^4*
(3 + Log[4]) + Log[16]))/(4*(3 + Log[4])^3) - ((I/8)*Sqrt[3]*E^((3 - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 +
Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(
16*E^8*Log[4]*(3 + Log[4])^2 + 9*Log[16] - 12*E^4*(3 + Log[4])*Log[256]))/((3 + Log[4])^3*(-3 + 8*E^4*(3 + Log
[4]))^(3/2)) + ((I/8)*Sqrt[3]*E^((3 + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(
3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(16*E^8*Log[4]*(3 + Log[4])^2 +
9*Log[16] - 12*E^4*(3 + Log[4])*Log[256]))/((3 + Log[4])^3*(-3 + 8*E^4*(3 + Log[4]))^(3/2)) - (E^((3 - I*Sqrt[
-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3
+ Log[4])])/(3 + Log[4])]*Log[4]*(3 - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])*(16*E^8*Log[4]*(3 + Log[4])^2 + 9*Log[
16] - 12*E^4*(3 + Log[4])*Log[256]))/(32*(3 + Log[4])^4*(3 - 8*E^4*(3 + Log[4]))) - (E^x*Log[4]*(3 - I*Sqrt[-9
+ 24*E^4*(3 + Log[4])])*(16*E^8*Log[4]*(3 + Log[4])^2 + 9*Log[16] - 12*E^4*(3 + Log[4])*Log[256]))/(8*(3 + Lo
g[4])^3*(3 - 8*E^4*(3 + Log[4]))*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])) - (E^((3 + I*Sqrt[
-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3
+ Log[4])])/(3 + Log[4])]*Log[4]*(3 + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])*(16*E^8*Log[4]*(3 + Log[4])^2 + 9*Log[
16] - 12*E^4*(3 + Log[4])*Log[256]))/(32*(3 + Log[4])^4*(3 - 8*E^4*(3 + Log[4]))) - (E^x*Log[4]*(3 + I*Sqrt[-9
+ 24*E^4*(3 + Log[4])])*(16*E^8*Log[4]*(3 + Log[4])^2 + 9*Log[16] - 12*E^4*(3 + Log[4])*Log[256]))/(8*(3 + Lo
g[4])^3*(3 - 8*E^4*(3 + Log[4]))*(3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])) + (3*E^((51 + 16*L
og[4] - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) - I*Sqrt[
-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(1 - 2*E^4*(3 + Log[4]))*Log[4096])/(8*(3 + Log[4])^3*(3 - 8*E
^4*(3 + Log[4]))) + (3*E^((51 + 16*Log[4] + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[
-1/4*(3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(1 - 2*E^4*(3 + Log[4]))*L
og[4096])/(8*(3 + Log[4])^3*(3 - 8*E^4*(3 + Log[4]))) + ((I/2)*Sqrt[3]*E^((51 + 16*Log[4] - I*Sqrt[-9 + 24*E^4
*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])
/(3 + Log[4])]*Log[4]*(1 - 2*E^4*(3 + Log[4]))*Log[4096])/((3 + Log[4])^2*(-3 + 8*E^4*(3 + Log[4]))^(3/2)) - (
(I/2)*Sqrt[3]*E^((51 + 16*Log[4] + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 -
4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(1 - 2*E^4*(3 + Log[4]))*Log[4096])
/((3 + Log[4])^2*(-3 + 8*E^4*(3 + Log[4]))^(3/2)) + (3*E^(4 + x)*Log[4]*(1 - 2*E^4*(3 + Log[4]))*Log[4096])/(2
*(3 + Log[4])^2*(3 - 8*E^4*(3 + Log[4]))*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])) + (3*E^(4
+ x)*Log[4]*(1 - 2*E^4*(3 + Log[4]))*Log[4096])/(2*(3 + Log[4])^2*(3 - 8*E^4*(3 + Log[4]))*(3 - 4*x*(3 + Log[4
]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])) - (3*E^((3 - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIn
tegralEi[-1/4*(3 - 4*x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(4*Log[4]^2 + 4*E
^4*Log[4]*(3 + Log[4]) - Log[16]^2 - Log[64] - (I*(8*E^8*Log[4]*(3 + Log[4])^2 + 4*E^4*(3 + Log[4])*(8*Log[4]^
2 - Log[16]^2 + Log[4096]) - 3*(4*Log[4]^2 + Log[262144])))/Sqrt[-9 + 24*E^4*(3 + Log[4])]))/(16*(3 + Log[4])^
4) - (3*E^((3 + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) +
I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]*(4*Log[4]^2 + 4*E^4*Log[4]*(3 + Log[4]) - Log[16]^2 -
Log[64] + (I*(8*E^8*Log[4]*(3 + Log[4])^2 + 4*E^4*(3 + Log[4])*(8*Log[4]^2 - Log[16]^2 + Log[4096]) - 3*(4*Log
[4]^2 + Log[262144])))/Sqrt[-9 + 24*E^4*(3 + Log[4])]))/(16*(3 + Log[4])^4)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 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 2302

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+x^4 \left (36+4 \log ^2(4)\right )} \, dx \\ & = \int \frac {e^x x \log (4) \left (-3 e^8 (2+x)-2 e^4 x \left (-6+9 x+x^2 (6+\log (4))\right )-x^2 \left (6+6 x^2 (2+\log (4))+4 x^3 (3+\log (4))-x (21+\log (16))\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx \\ & = \log (4) \int \frac {e^x x \left (-3 e^8 (2+x)-2 e^4 x \left (-6+9 x+x^2 (6+\log (4))\right )-x^2 \left (6+6 x^2 (2+\log (4))+4 x^3 (3+\log (4))-x (21+\log (16))\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx \\ & = \log (4) \int \left (-\frac {e^x x^2}{3+\log (4)}-\frac {3 e^x x (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \left (-2 e^4 \log (4) (3+\log (4))+\log (4) \log (16)+\log (262144)\right )}{4 (3+\log (4))^3}+\frac {3 e^x \left (4 e^8 \log (4) (3+\log (4))-3 \log (16)-2 x \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+2 e^4 \left (8 \log ^2(4)-\log ^2(16)+\log (262144)\right )\right )}{8 (3+\log (4))^3 \left (3 e^4-3 x+2 x^2 (3+\log (4))\right )}+\frac {3 e^x \left (-x \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )+e^4 \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )\right )}{8 (3+\log (4))^3 \left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}\right ) \, dx \\ & = \frac {(3 \log (4)) \int \frac {e^x \left (4 e^8 \log (4) (3+\log (4))-3 \log (16)-2 x \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+2 e^4 \left (8 \log ^2(4)-\log ^2(16)+\log (262144)\right )\right )}{3 e^4-3 x+2 x^2 (3+\log (4))} \, dx}{8 (3+\log (4))^3}+\frac {(3 \log (4)) \int \frac {e^x \left (-x \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )+e^4 \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}-\frac {\log (4) \int e^x x^2 \, dx}{3+\log (4)}-\frac {(3 \log (4) (4+\log (4))) \int e^x x \, dx}{2 (3+\log (4))^2}+\frac {\left (\log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )\right ) \int e^x \, dx}{4 (3+\log (4))^3} \\ & = -\frac {e^x x^2 \log (4)}{3+\log (4)}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}+\frac {(3 \log (4)) \int \left (\frac {e^x \left (-2 \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )-\frac {2 i \left (72 e^8 \log (4)-12 \log ^2(4)+96 e^4 \log ^2(4)+48 e^8 \log ^2(4)+32 e^4 \log ^3(4)+8 e^8 \log ^3(4)-12 e^4 \log ^2(16)-4 e^4 \log (4) \log ^2(16)+12 e^4 \log (4096)+4 e^4 \log (4) \log (4096)-3 \log (262144)\right )}{\sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right )}{-3+4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}+\frac {e^x \left (-2 \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)\right )+\frac {2 i \left (72 e^8 \log (4)-12 \log ^2(4)+96 e^4 \log ^2(4)+48 e^8 \log ^2(4)+32 e^4 \log ^3(4)+8 e^8 \log ^3(4)-12 e^4 \log ^2(16)-4 e^4 \log (4) \log ^2(16)+12 e^4 \log (4096)+4 e^4 \log (4) \log (4096)-3 \log (262144)\right )}{\sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right )}{-3+4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}}\right ) \, dx}{8 (3+\log (4))^3}+\frac {(3 \log (4)) \int \left (\frac {e^x x \left (-16 e^8 \log (4) (3+\log (4))^2-9 \log (16)+12 e^4 (3+\log (4)) \log (256)\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}+\frac {e^{4+x} \left (3 \log (4096)-2 e^4 (3+\log (4)) \log (68719476736)\right )}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2}\right ) \, dx}{8 (3+\log (4))^3}+\frac {(2 \log (4)) \int e^x x \, dx}{3+\log (4)}+\frac {(3 \log (4) (4+\log (4))) \int e^x \, dx}{2 (3+\log (4))^2} \\ & = \frac {2 e^x x \log (4)}{3+\log (4)}-\frac {e^x x^2 \log (4)}{3+\log (4)}+\frac {3 e^x \log (4) (4+\log (4))}{2 (3+\log (4))^2}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}-\frac {(2 \log (4)) \int e^x \, dx}{3+\log (4)}-\frac {\left (3 \log (4) \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )\right ) \int \frac {e^x x}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}+\frac {\left (9 \log (4) \left (1-2 e^4 (3+\log (4))\right ) \log (4096)\right ) \int \frac {e^{4+x}}{\left (3 e^4-3 x+2 x^2 (3+\log (4))\right )^2} \, dx}{8 (3+\log (4))^3}-\frac {\left (3 \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)-\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )\right ) \int \frac {e^x}{-3+4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}} \, dx}{4 (3+\log (4))^3}-\frac {\left (3 \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)+\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )\right ) \int \frac {e^x}{-3+4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}} \, dx}{4 (3+\log (4))^3} \\ & = -\frac {2 e^x \log (4)}{3+\log (4)}+\frac {2 e^x x \log (4)}{3+\log (4)}-\frac {e^x x^2 \log (4)}{3+\log (4)}+\frac {3 e^x \log (4) (4+\log (4))}{2 (3+\log (4))^2}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}-\frac {3 e^{\frac {3-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {3-4 x (3+\log (4))-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}\right ) \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)-\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )}{16 (3+\log (4))^4}-\frac {3 e^{\frac {3+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {3-4 x (3+\log (4))+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}\right ) \log (4) \left (4 \log ^2(4)+4 e^4 \log (4) (3+\log (4))-\log ^2(16)-\log (64)+\frac {i \left (8 e^8 \log (4) (3+\log (4))^2+4 e^4 (3+\log (4)) \left (8 \log ^2(4)-\log ^2(16)+\log (4096)\right )-3 \left (4 \log ^2(4)+\log (262144)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )}{16 (3+\log (4))^4}-\frac {\left (3 \log (4) \left (16 e^8 \log (4) (3+\log (4))^2+9 \log (16)-12 e^4 (3+\log (4)) \log (256)\right )\right ) \int \left (-\frac {4 e^x (3+\log (4)) \left (3-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}{3 \left (-3+24 e^4+8 e^4 \log (4)\right ) \left (3-4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )^2}+\frac {4 i e^x (3+\log (4))^2}{\sqrt {3} (-3-\log (4)) \left (-3+24 e^4+8 e^4 \log (4)\right )^{3/2} \left (3-4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}-\frac {4 e^x (3+\log (4)) \left (3+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}{3 \left (-3+24 e^4+8 e^4 \log (4)\right ) \left (3-4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )^2}+\frac {4 i e^x (3+\log (4))}{\sqrt {3} \left (-3+24 e^4+8 e^4 \log (4)\right )^{3/2} \left (3-4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}\right ) \, dx}{8 (3+\log (4))^3}+\frac {\left (9 \log (4) \left (1-2 e^4 (3+\log (4))\right ) \log (4096)\right ) \int \left (-\frac {16 e^{4+x} (3+\log (4))^2}{3 \left (-3+24 e^4+8 e^4 \log (4)\right ) \left (3-4 x (3+\log (4))-i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )^2}-\frac {16 e^{4+x} (3+\log (4))^2}{3 \left (-3+24 e^4+8 e^4 \log (4)\right ) \left (3-4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )^2}+\frac {16 i e^{4+x} (3+\log (4))^2}{3 \sqrt {3} \left (-3+24 e^4+8 e^4 \log (4)\right )^{3/2} \left (3-4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}+\frac {16 i e^{4+x} (3+\log (4))^2}{3 \sqrt {3} \left (-3+24 e^4+8 e^4 \log (4)\right )^{3/2} \left (-3+4 x (3+\log (4))+i \sqrt {3 \left (-3+24 e^4+8 e^4 \log (4)\right )}\right )}\right ) \, dx}{8 (3+\log (4))^3} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx \]

[In]

Integrate[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(12*x^2 - 18*x^3 - 12*x^4))*Log[
4] + (2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^6)*Log[4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) +
(12*E^4*x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2),x]

[Out]

Integrate[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(12*x^2 - 18*x^3 - 12*x^4))*Log[
4] + (2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^6)*Log[4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) +
(12*E^4*x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2), x]

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30

method	result	size
gosper	\(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\)	\(43\)
risch	\(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\)	\(43\)
norman	\(\frac {2 x^{3} \ln \left (2\right ) {\mathrm e}^{x}-4 x^{4} \ln \left (2\right ) {\mathrm e}^{x}-2 x^{2} {\mathrm e}^{4} \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\)	\(54\)
parallelrisch	\(-\frac {12 x^{4} \ln \left (2\right ) {\mathrm e}^{x}+6 x^{2} {\mathrm e}^{4} \ln \left (2\right ) {\mathrm e}^{x}-6 x^{3} \ln \left (2\right ) {\mathrm e}^{x}}{3 \left (4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x \right )}\)	\(55\)
default	\(\text {Expression too large to display}\)	\(42483\)

[In]

int((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*ln(2)^2+2*((-3*x^2-6*x)*exp(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4)-12*x^
6-12*x^5+21*x^4-6*x^3)*ln(2))*exp(x)/(16*x^4*ln(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*ln(2)+9*exp(4)^2+(36*x^2-
18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

-2*x^2*(2*x^2+exp(4)-x)*ln(2)*exp(x)/(4*x^2*ln(2)+6*x^2+3*exp(4)-3*x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {2 \, {\left (2 \, x^{4} - x^{3} + x^{2} e^{4}\right )} e^{x} \log \left (2\right )}{4 \, x^{2} \log \left (2\right ) + 6 \, x^{2} - 3 \, x + 3 \, e^{4}} \]

[In]

integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*exp(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4
)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))*exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^
2+(36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm="fricas")

[Out]

-2*(2*x^4 - x^3 + x^2*e^4)*e^x*log(2)/(4*x^2*log(2) + 6*x^2 - 3*x + 3*e^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\frac {\left (- 4 x^{4} \log {\left (2 \right )} + 2 x^{3} \log {\left (2 \right )} - 2 x^{2} e^{4} \log {\left (2 \right )}\right ) e^{x}}{4 x^{2} \log {\left (2 \right )} + 6 x^{2} - 3 x + 3 e^{4}} \]

[In]

integrate((4*(-2*x**4*exp(4)-4*x**6-6*x**5+2*x**4)*ln(2)**2+2*((-3*x**2-6*x)*exp(4)**2+(-12*x**4-18*x**3+12*x*
*2)*exp(4)-12*x**6-12*x**5+21*x**4-6*x**3)*ln(2))*exp(x)/(16*x**4*ln(2)**2+2*(12*x**2*exp(4)+24*x**4-12*x**3)*
ln(2)+9*exp(4)**2+(36*x**2-18*x)*exp(4)+36*x**4-36*x**3+9*x**2),x)

[Out]

(-4*x**4*log(2) + 2*x**3*log(2) - 2*x**2*exp(4)*log(2))*exp(x)/(4*x**2*log(2) + 6*x**2 - 3*x + 3*exp(4))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {2 \, {\left (2 \, x^{4} \log \left (2\right ) - x^{3} \log \left (2\right ) + x^{2} e^{4} \log \left (2\right )\right )} e^{x}}{2 \, x^{2} {\left (2 \, \log \left (2\right ) + 3\right )} - 3 \, x + 3 \, e^{4}} \]

[In]

integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*exp(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4
)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))*exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^
2+(36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm="maxima")

[Out]

-2*(2*x^4*log(2) - x^3*log(2) + x^2*e^4*log(2))*e^x/(2*x^2*(2*log(2) + 3) - 3*x + 3*e^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (32) = 64\).

Time = 0.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 8.73 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {32 \, x^{4} e^{x} \log \left (2\right )^{4} + 144 \, x^{4} e^{x} \log \left (2\right )^{3} - 16 \, x^{3} e^{x} \log \left (2\right )^{4} + 216 \, x^{4} e^{x} \log \left (2\right )^{2} - 72 \, x^{3} e^{x} \log \left (2\right )^{3} + 16 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{4} + 108 \, x^{4} e^{x} \log \left (2\right ) - 108 \, x^{3} e^{x} \log \left (2\right )^{2} + 72 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{3} - 54 \, x^{3} e^{x} \log \left (2\right ) + 108 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{2} + 24 \, x e^{\left (x + 4\right )} \log \left (2\right )^{3} + 54 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right ) + 36 \, x e^{\left (x + 4\right )} \log \left (2\right )^{2} - 9 \, x e^{x} \log \left (2\right )^{2} - 12 \, e^{\left (x + 8\right )} \log \left (2\right )^{3} - 18 \, e^{\left (x + 8\right )} \log \left (2\right )^{2} + 9 \, e^{\left (x + 4\right )} \log \left (2\right )^{2}}{32 \, x^{2} \log \left (2\right )^{4} + 192 \, x^{2} \log \left (2\right )^{3} + 432 \, x^{2} \log \left (2\right )^{2} - 24 \, x \log \left (2\right )^{3} + 24 \, e^{4} \log \left (2\right )^{3} + 432 \, x^{2} \log \left (2\right ) - 108 \, x \log \left (2\right )^{2} + 108 \, e^{4} \log \left (2\right )^{2} + 162 \, x^{2} - 162 \, x \log \left (2\right ) + 162 \, e^{4} \log \left (2\right ) - 81 \, x + 81 \, e^{4}} \]

[In]

integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*exp(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4
)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))*exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^
2+(36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm="giac")

[Out]

-(32*x^4*e^x*log(2)^4 + 144*x^4*e^x*log(2)^3 - 16*x^3*e^x*log(2)^4 + 216*x^4*e^x*log(2)^2 - 72*x^3*e^x*log(2)^
3 + 16*x^2*e^(x + 4)*log(2)^4 + 108*x^4*e^x*log(2) - 108*x^3*e^x*log(2)^2 + 72*x^2*e^(x + 4)*log(2)^3 - 54*x^3
*e^x*log(2) + 108*x^2*e^(x + 4)*log(2)^2 + 24*x*e^(x + 4)*log(2)^3 + 54*x^2*e^(x + 4)*log(2) + 36*x*e^(x + 4)*
log(2)^2 - 9*x*e^x*log(2)^2 - 12*e^(x + 8)*log(2)^3 - 18*e^(x + 8)*log(2)^2 + 9*e^(x + 4)*log(2)^2)/(32*x^2*lo
g(2)^4 + 192*x^2*log(2)^3 + 432*x^2*log(2)^2 - 24*x*log(2)^3 + 24*e^4*log(2)^3 + 432*x^2*log(2) - 108*x*log(2)
^2 + 108*e^4*log(2)^2 + 162*x^2 - 162*x*log(2) + 162*e^4*log(2) - 81*x + 81*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left ({\mathrm {e}}^8\,\left (3\,x^2+6\,x\right )+{\mathrm {e}}^4\,\left (12\,x^4+18\,x^3-12\,x^2\right )+6\,x^3-21\,x^4+12\,x^5+12\,x^6\right )+4\,{\ln \left (2\right )}^2\,\left (2\,x^4\,{\mathrm {e}}^4-2\,x^4+6\,x^5+4\,x^6\right )\right )}{9\,{\mathrm {e}}^8+16\,x^4\,{\ln \left (2\right )}^2-{\mathrm {e}}^4\,\left (18\,x-36\,x^2\right )+2\,\ln \left (2\right )\,\left (24\,x^4-12\,x^3+12\,{\mathrm {e}}^4\,x^2\right )+9\,x^2-36\,x^3+36\,x^4} \,d x \]

[In]

int(-(exp(x)*(2*log(2)*(exp(8)*(6*x + 3*x^2) + exp(4)*(18*x^3 - 12*x^2 + 12*x^4) + 6*x^3 - 21*x^4 + 12*x^5 + 1
2*x^6) + 4*log(2)^2*(2*x^4*exp(4) - 2*x^4 + 6*x^5 + 4*x^6)))/(9*exp(8) + 16*x^4*log(2)^2 - exp(4)*(18*x - 36*x
^2) + 2*log(2)*(12*x^2*exp(4) - 12*x^3 + 24*x^4) + 9*x^2 - 36*x^3 + 36*x^4),x)

[Out]

int(-(exp(x)*(2*log(2)*(exp(8)*(6*x + 3*x^2) + exp(4)*(18*x^3 - 12*x^2 + 12*x^4) + 6*x^3 - 21*x^4 + 12*x^5 + 1
2*x^6) + 4*log(2)^2*(2*x^4*exp(4) - 2*x^4 + 6*x^5 + 4*x^6)))/(9*exp(8) + 16*x^4*log(2)^2 - exp(4)*(18*x - 36*x
^2) + 2*log(2)*(12*x^2*exp(4) - 12*x^3 + 24*x^4) + 9*x^2 - 36*x^3 + 36*x^4), x)