∫ 44−16x+8x2+(−8+8x) log(4)+2 log2(4)+ex(−4−6x−12x2+4x3+(−5−8x+4x2) log(4)+(−1+x) log2(4))__________________________________________________________________________ 81+36x−32x2−8x3+4x4+(−18x−4x2+4x3) log(4)+x2 log2(4) dx [7060]

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.36, antiderivative size = 1106, normalized size of antiderivative = 48.09, number of steps used = 31, number of rules used = 9, integrand size = 112, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.080, Rules used = {6, 6873, 6820, 6874, 1674, 8, 2302, 2209, 2208} \begin {gather*} \frac {e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} (2-\log (4)) \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right ) \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right )}{2 \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \left (1-\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right ) \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right )-\frac {e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \left (40-8 \log (2)+\log ^2(4)\right ) \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right )}{76-4 \log (4)+\log ^2(4)}+\frac {4 e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \left (40-8 \log (2)+\log ^2(4)\right ) \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {\sqrt {2} e^{\frac {1}{4} \left (2+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} (2-\log (4))^2 \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {2 (-2 x-\log (4)+2)}{-2 x^2+(2-\log (4)) x+9}+\frac {2 e^x (2-\log (4)) \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x-\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}-\frac {4 e^x \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x-\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}+\frac {2 e^x (2-\log (4)) \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x+\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}-\frac {4 e^x \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x+\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}+\frac {e^{\frac {1}{4} \left (2-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4)) \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{2 \sqrt {2} \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (1+\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )-\frac {e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{76-4 \log (4)+\log ^2(4)}-\frac {4 e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {\sqrt {2} e^{\frac {1}{4} \left (2-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4))^2}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}} \end {gather*}

Int[(44 - 16*x + 8*x^2 + (-8 + 8*x)*Log[4] + 2*Log[4]^2 + E^x*(-4 - 6*x - 12*x^2 + 4*x^3 + (-5 - 8*x + 4*x^2)*

Log[4] + (-1 + x)*Log[4]^2))/(81 + 36*x - 32*x^2 - 8*x^3 + 4*x^4 + (-18*x - 4*x^2 + 4*x^3)*Log[4] + x^2*Log[4]

(-2*(2 - 2*x - Log[4]))/(9 - 2*x^2 + x*(2 - Log[4])) - (Sqrt[2]*E^((2 + Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*Exp

IntegralEi[(-2 + 4*x + Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(2 - Log[4])^2)/(76 - 4*Log[4] + Log[4]^2)^

(3/2) + (Sqrt[2]*E^((2 - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log

[4] + Log[4]^2])/4]*(2 - Log[4])^2)/(76 - 4*Log[4] + Log[4]^2)^(3/2) + (4*E^((2 - Log[4] + Sqrt[76 - 4*Log[4]

+ Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log[4]^

2))/(76 - 4*Log[4] + Log[4]^2)^(3/2) - (4*E^((2 - Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-

2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log[4]^2))/(76 - 4*Log[4] + Log[4]^2)^(

3/2) - (E^((2 - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] - Sqrt[76 - 4*Log

[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log[4]^2))/(76 - 4*Log[4] + Log[4]^2) - (E^((2 - Log[4] - Sqrt[76 - 4*Log

[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log

[4]^2))/(76 - 4*Log[4] + Log[4]^2) + (E^((2 - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 +

4*x + Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(1 - (6 - Log[4])/Sqrt[76 - 4*Log[4] + Log[4]^2]))/2 + (E^((

2 - Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]

^2])/4]*(1 + (6 - Log[4])/Sqrt[76 - 4*Log[4] + Log[4]^2]))/2 + (E^((2 - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*Exp

IntegralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(2 - Log[4])*(2 - Log[4] - Sqrt[76 - 4*Log[

4] + Log[4]^2]))/(2*Sqrt[2]*(76 - 4*Log[4] + Log[4]^2)) - (4*E^x*(40 - 8*Log[2] + Log[4]^2))/((76 - 4*Log[4] +

Log[4]^2)*(2 - 4*x - Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])) + (2*E^x*(2 - Log[4])*(2 - Log[4] - Sqrt[76 -

4*Log[4] + Log[4]^2]))/((76 - 4*Log[4] + Log[4]^2)*(2 - 4*x - Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])) + (E^(

(2 - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] - Sqrt[76 - 4*Log[4] + Log[4

]^2])/4]*(2 - Log[4])*(2 - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2]))/(2*(76 - 4*Log[4] + Log[4]^2)) - (4*E^x*(

40 - 8*Log[2] + Log[4]^2))/((76 - 4*Log[4] + Log[4]^2)*(2 - 4*x - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])) +

(2*E^x*(2 - Log[4])*(2 - Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2]))/((76 - 4*Log[4] + Log[4]^2)*(2 - 4*x - Log[

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

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*

x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b

*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)

)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (

2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1

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

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg

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

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \left (-32+\log ^2(4)\right )} \, dx\\ &=\int \frac {-16 x+8 x^2+(-8+8 x) \log (4)+44 \left (1+\frac {\log ^2(4)}{22}\right )+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+4 x^4+18 x (2-\log (4))-4 x^3 (2-\log (4))-x^2 (8-\log (4)) (4+\log (4))} \, dx\\ &=\int \frac {2 \left (22+4 x^2+4 x (-2+\log (4))-4 \log (4)+\log ^2(4)\right )+e^x \left (-4+4 x^3+4 x^2 (-3+\log (4))-5 \log (4)-\log ^2(4)+x \left (-6-8 \log (4)+\log ^2(4)\right )\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx\\ &=\int \left (\frac {2 \left (22+4 x^2-4 x (2-\log (4))-4 \log (4)+\log ^2(4)\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2}+\frac {e^x \left (4 x^3-4 x^2 (3-\log (4))-(1+\log (4)) (4+\log (4))-x \left (6+8 \log (4)-\log ^2(4)\right )\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2}\right ) \, dx\\ &=2 \int \frac {22+4 x^2-4 x (2-\log (4))-4 \log (4)+\log ^2(4)}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx+\int \frac {e^x \left (4 x^3-4 x^2 (3-\log (4))-(1+\log (4)) (4+\log (4))-x \left (6+8 \log (4)-\log ^2(4)\right )\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}-\frac {2 \int 0 \, dx}{76-4 \log (4)+\log ^2(4)}+\int \left (\frac {e^x (4-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+\frac {e^x \left (-40+2 x (2-\log (4))-\log ^2(4)+\log (256)\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2}\right ) \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+\int \frac {e^x (4-2 x-\log (4))}{9-2 x^2+x (2-\log (4))} \, dx+\int \frac {e^x \left (-40+2 x (2-\log (4))-\log ^2(4)+\log (256)\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+\int \left (\frac {2 e^x x (2-\log (4))}{\left (9-2 x^2+x (2-\log (4))\right )^2}-\frac {40 e^x \left (1+\frac {1}{40} \left (-8 \log (2)+\log ^2(4)\right )\right )}{\left (9-2 x^2+x (2-\log (4))\right )^2}\right ) \, dx+\int \left (\frac {e^x \left (-2+\frac {2 (-6+\log (4))}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{2-4 x-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}}+\frac {e^x \left (-2-\frac {2 (-6+\log (4))}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}}\right ) \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+(2 (2-\log (4))) \int \frac {e^x x}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx-\left (40-8 \log (2)+\log ^2(4)\right ) \int \frac {e^x}{\left (9-2 x^2+x (2-\log (4))\right )^2} \, dx-\left (2 \left (1-\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )\right ) \int \frac {e^x}{2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx-\left (2 \left (1+\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )\right ) \int \frac {e^x}{2-4 x-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {Ei}\left (\frac {1}{4} \left (-2+4 x+\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \left (1-\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {Ei}\left (\frac {1}{4} \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \left (1+\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+(2 (2-\log (4))) \int \left (\frac {4 e^x \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2}+\frac {4 e^x (2-\log (4))}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2} \left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}+\frac {4 e^x \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2}+\frac {4 e^x (2-\log (4))}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2} \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}\right ) \, dx-\left (40-8 \log (2)+\log ^2(4)\right ) \int \left (\frac {16 e^x}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2}+\frac {16 e^x}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2} \left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}+\frac {16 e^x}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2}+\frac {16 e^x}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2} \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}\right ) \, dx\\ &=-\frac {2 (2-2 x-\log (4))}{9-2 x^2+x (2-\log (4))}+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {Ei}\left (\frac {1}{4} \left (-2+4 x+\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \left (1-\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \text {Ei}\left (\frac {1}{4} \left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \left (1+\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+\frac {\left (8 (2-\log (4))^2\right ) \int \frac {e^x}{2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {\left (8 (2-\log (4))^2\right ) \int \frac {e^x}{-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {\left (16 \left (40-8 \log (2)+\log ^2(4)\right )\right ) \int \frac {e^x}{2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {\left (16 \left (40-8 \log (2)+\log ^2(4)\right )\right ) \int \frac {e^x}{-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}} \, dx}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {\left (16 \left (40-8 \log (2)+\log ^2(4)\right )\right ) \int \frac {e^x}{\left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2} \, dx}{76-4 \log (4)+\log ^2(4)}-\frac {\left (16 \left (40-8 \log (2)+\log ^2(4)\right )\right ) \int \frac {e^x}{\left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2} \, dx}{76-4 \log (4)+\log ^2(4)}+\frac {\left (8 (2-\log (4)) \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \int \frac {e^x}{\left (-2+4 x+\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2} \, dx}{76-4 \log (4)+\log ^2(4)}+\frac {\left (8 (2-\log (4)) \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )\right ) \int \frac {e^x}{\left (2-4 x-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )^2} \, dx}{76-4 \log (4)+\log ^2(4)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.07, size = 28, normalized size = 1.22 \begin {gather*} \frac {\left (-2+e^x\right ) (-2+2 x+\log (4))}{-9+2 x^2+x (-2+\log (4))} \end {gather*}

Integrate[(44 - 16*x + 8*x^2 + (-8 + 8*x)*Log[4] + 2*Log[4]^2 + E^x*(-4 - 6*x - 12*x^2 + 4*x^3 + (-5 - 8*x + 4

*x^2)*Log[4] + (-1 + x)*Log[4]^2))/(81 + 36*x - 32*x^2 - 8*x^3 + 4*x^4 + (-18*x - 4*x^2 + 4*x^3)*Log[4] + x^2*

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. $818$ vs. $2(22)=44$.
time = 0.48, size = 819, normalized size = 35.61


method	result	size

norman	$\frac {-4 x +\left (2 \ln \left (2\right )-2\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} x +4-4 \ln \left (2\right )}{2 x \ln \left (2\right )+2 x^{2}-2 x -9}$	$42$
risch	$\frac {-2 x +2-2 \ln \left (2\right )}{x \ln \left (2\right )+x^{2}-x -\frac {9}{2}}+\frac {2 \left (x +\ln \left (2\right )-1\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+2 x^{2}-2 x -9}$	$52$
default	$\text {Expression too large to display}$	$819$

int(((4*(x-1)*ln(2)^2+2*(4*x^2-8*x-5)*ln(2)+4*x^3-12*x^2-6*x-4)*exp(x)+8*ln(2)^2+2*(8*x-8)*ln(2)+8*x^2-16*x+44

)/(4*x^2*ln(2)^2+2*(4*x^3-4*x^2-18*x)*ln(2)+4*x^4-8*x^3-32*x^2+36*x+81),x,method=_RETURNVERBOSE)

2*ln(2)^3*exp(x)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)+3*exp(x)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2

*x-9)*ln(2)^2+49*exp(x)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)*ln(2)+exp(x)*(2*x*ln(2)^3-6*x*ln(2)^2-9*l

n(2)^2+33*x*ln(2)+18*ln(2)-29*x-90)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)+3*exp(x)*(2*x*ln(2)^2-4*x*ln(

2)-9*ln(2)+20*x+9)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)-3*exp(x)*(x*ln(2)-x-9)/(ln(2)^2-2*ln(2)+19)/(2

*x*ln(2)+2*x^2-2*x-9)-32*ln(2)/(-4*ln(2)^2+8*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)*x+2*exp(x)*(ln(2)+2*x-1)/(ln(2)

^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)-72/(2*ln(2)^2-4*ln(2)+38)/(ln(2)^2-2*ln(2)+19)^(1/2)*arctanh(1/2*(4*x+2

*ln(2)-2)/(ln(2)^2-2*ln(2)+19)^(1/2))-16*((2-2*ln(2))*x+18)/(-4*ln(2)^2+8*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)+44

*(4*x+2*ln(2)-2)/(-4*ln(2)^2+8*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)-144/(-4*ln(2)^2+8*ln(2)-76)/(ln(2)^2-2*ln(2)+

19)^(1/2)*arctanh(1/2*(4*x+2*ln(2)-2)/(ln(2)^2-2*ln(2)+19)^(1/2))+2*ln(2)^2*exp(x)/(ln(2)^2-2*ln(2)+19)/(2*x*l

n(2)+2*x^2-2*x-9)*x-2*ln(2)^3*exp(x)/(ln(2)^2-2*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)*x-22*ln(2)*exp(x)/(ln(2)^2-2

*ln(2)+19)/(2*x*ln(2)+2*x^2-2*x-9)*x+16*ln(2)^3/(-4*ln(2)^2+8*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)+8*(-1/4*(ln(2)

^2-2*ln(2)+10)/(ln(2)^2-2*ln(2)+19)*x+9/8*(ln(2)-1)/(ln(2)^2-2*ln(2)+19))/(x*ln(2)+x^2-x-9/2)-48/(-4*ln(2)^2+8

*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)*ln(2)^2+320/(-4*ln(2)^2+8*ln(2)-76)/(2*x*ln(2)+2*x^2-2*x-9)*ln(2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 692 vs. $2 (21) = 42$.
time = 0.55, size = 692, normalized size = 30.09 \begin {gather*} -4 \, {\left (\frac {2 \, x + \log \left (2\right ) - 1}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171} + \frac {\log \left (\frac {2 \, x - \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}{2 \, x + \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}\right )}{{\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )}^{\frac {3}{2}}}\right )} \log \left (2\right )^{2} + 4 \, {\left (\frac {{\left (\log \left (2\right ) - 1\right )} \log \left (\frac {2 \, x - \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}{2 \, x + \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}\right )}{{\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (x {\left (\log \left (2\right ) - 1\right )} - 9\right )}}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171}\right )} \log \left (2\right ) + 8 \, {\left (\frac {2 \, x + \log \left (2\right ) - 1}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171} + \frac {\log \left (\frac {2 \, x - \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}{2 \, x + \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}\right )}{{\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )}^{\frac {3}{2}}}\right )} \log \left (2\right ) + \frac {2 \, {\left (x + \log \left (2\right ) - 1\right )} e^{x}}{2 \, x^{2} + 2 \, x {\left (\log \left (2\right ) - 1\right )} - 9} - \frac {4 \, {\left (\log \left (2\right ) - 1\right )} \log \left (\frac {2 \, x - \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}{2 \, x + \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}\right )}{{\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 10\right )} x - 9 \, \log \left (2\right ) + 9\right )}}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171} - \frac {8 \, {\left (x {\left (\log \left (2\right ) - 1\right )} - 9\right )}}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171} - \frac {22 \, {\left (2 \, x + \log \left (2\right ) - 1\right )}}{2 \, {\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )} x^{2} + 2 \, {\left (\log \left (2\right )^{3} - 3 \, \log \left (2\right )^{2} + 21 \, \log \left (2\right ) - 19\right )} x - 9 \, \log \left (2\right )^{2} + 18 \, \log \left (2\right ) - 171} - \frac {4 \, \log \left (\frac {2 \, x - \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}{2 \, x + \sqrt {\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19} + \log \left (2\right ) - 1}\right )}{{\left (\log \left (2\right )^{2} - 2 \, \log \left (2\right ) + 19\right )}^{\frac {3}{2}}} \end {gather*}

integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*exp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*

x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3-4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="maxima")

-4*((2*x + log(2) - 1)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*lo

g(2)^2 + 18*log(2) - 171) + log((2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*l

og(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2))*log(2)^2 + 4*((log(2) - 1)*log((2*x - sqrt(log(2)

^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) +

19)^(3/2) + 2*(x*(log(2) - 1) - 9)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - 3*log(2)^2 + 21*log(2) -

19)*x - 9*log(2)^2 + 18*log(2) - 171))*log(2) + 8*((2*x + log(2) - 1)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(l

og(2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) + log((2*x - sqrt(log(2)^2 - 2*log(2)

+ 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2))*lo

g(2) + 2*(x + log(2) - 1)*e^x/(2*x^2 + 2*x*(log(2) - 1) - 9) - 4*(log(2) - 1)*log((2*x - sqrt(log(2)^2 - 2*log

(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2)

- 2*(2*(log(2)^2 - 2*log(2) + 10)*x - 9*log(2) + 9)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - 3*log(2)

^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) - 8*(x*(log(2) - 1) - 9)/(2*(log(2)^2 - 2*log(2) + 19)*

x^2 + 2*(log(2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) - 22*(2*x + log(2) - 1)/(2*

(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171)

- 4*log((2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1

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Fricas [A]
time = 0.37, size = 36, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left ({\left (x + \log \left (2\right ) - 1\right )} e^{x} - 2 \, x - 2 \, \log \left (2\right ) + 2\right )}}{2 \, x^{2} + 2 \, x \log \left (2\right ) - 2 \, x - 9} \end {gather*}

integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*exp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*

x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3-4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="fricas")

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. $2 (17) = 34$.
time = 0.35, size = 54, normalized size = 2.35 \begin {gather*} \frac {- 4 x - 4 \log {\left (2 \right )} + 4}{2 x^{2} + x \left (-2 + 2 \log {\left (2 \right )}\right ) - 9} + \frac {\left (2 x - 2 + 2 \log {\left (2 \right )}\right ) e^{x}}{2 x^{2} - 2 x + 2 x \log {\left (2 \right )} - 9} \end {gather*}

integrate(((4*(-1+x)*ln(2)**2+2*(4*x**2-8*x-5)*ln(2)+4*x**3-12*x**2-6*x-4)*exp(x)+8*ln(2)**2+2*(8*x-8)*ln(2)+8

*x**2-16*x+44)/(4*x**2*ln(2)**2+2*(4*x**3-4*x**2-18*x)*ln(2)+4*x**4-8*x**3-32*x**2+36*x+81),x)

(-4*x - 4*log(2) + 4)/(2*x**2 + x*(-2 + 2*log(2)) - 9) + (2*x - 2 + 2*log(2))*exp(x)/(2*x**2 - 2*x + 2*x*log(2

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Giac [A]
time = 0.43, size = 41, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (x e^{x} + e^{x} \log \left (2\right ) - 2 \, x - e^{x} - 2 \, \log \left (2\right ) + 2\right )}}{2 \, x^{2} + 2 \, x \log \left (2\right ) - 2 \, x - 9} \end {gather*}

integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*exp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*

x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3-4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="giac")

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,\ln \left (2\right )\,\left (8\,x-8\right )-16\,x-{\mathrm {e}}^x\,\left (6\,x-4\,{\ln \left (2\right )}^2\,\left (x-1\right )+2\,\ln \left (2\right )\,\left (-4\,x^2+8\,x+5\right )+12\,x^2-4\,x^3+4\right )+8\,{\ln \left (2\right )}^2+8\,x^2+44}{36\,x+4\,x^2\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (-4\,x^3+4\,x^2+18\,x\right )-32\,x^2-8\,x^3+4\,x^4+81} \,d x \end {gather*}

int((2*log(2)*(8*x - 8) - 16*x - exp(x)*(6*x - 4*log(2)^2*(x - 1) + 2*log(2)*(8*x - 4*x^2 + 5) + 12*x^2 - 4*x^

3 + 4) + 8*log(2)^2 + 8*x^2 + 44)/(36*x + 4*x^2*log(2)^2 - 2*log(2)*(18*x + 4*x^2 - 4*x^3) - 32*x^2 - 8*x^3 +

int((2*log(2)*(8*x - 8) - 16*x - exp(x)*(6*x - 4*log(2)^2*(x - 1) + 2*log(2)*(8*x - 4*x^2 + 5) + 12*x^2 - 4*x^

3 + 4) + 8*log(2)^2 + 8*x^2 + 44)/(36*x + 4*x^2*log(2)^2 - 2*log(2)*(18*x + 4*x^2 - 4*x^3) - 32*x^2 - 8*x^3 +

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3.71.60 \(\int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x (-4-6 x-12 x^2+4 x^3+(-5-8 x+4 x^2) \log (4)+(-1+x) \log ^2(4))}{81+36 x-32 x^2-8 x^3+4 x^4+(-18 x-4 x^2+4 x^3) \log (4)+x^2 \log ^2(4)} \, dx\) [7060]