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

Optimal. Leaf size=23 \[ \frac {-2+e^x}{x+\frac {9}{2-2 x-\log (4)}} \]

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Rubi [C]  time = 3.75, 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, 6741, 6688, 6742, 1660, 8, 2270, 2178, 2177}

result too large to display

Antiderivative was successfully verified.

[In]

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),x]

[Out]

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 1660

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
]

Rule 2177

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] &&  !$UseGamma ===
True

Rule 2178

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

Rule 2270

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 6688

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

Rule 6741

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

Rule 6742

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

Rubi steps

\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 = 0.06, 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*}

Antiderivative was successfully verified.

[In]

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*
Log[4]^2),x]

[Out]

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

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fricas [A]  time = 0.96, size = 36, normalized size = 1.57 \begin {gather*} \frac {2 \, {\left ({\left (x + \log \relax (2) - 1\right )} e^{x} - 2 \, x - 2 \, \log \relax (2) + 2\right )}}{2 \, x^{2} + 2 \, x \log \relax (2) - 2 \, x - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*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")

[Out]

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

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giac [A]  time = 0.16, size = 41, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (x e^{x} + e^{x} \log \relax (2) - 2 \, x - e^{x} - 2 \, \log \relax (2) + 2\right )}}{2 \, x^{2} + 2 \, x \log \relax (2) - 2 \, x - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*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")

[Out]

2*(x*e^x + e^x*log(2) - 2*x - e^x - 2*log(2) + 2)/(2*x^2 + 2*x*log(2) - 2*x - 9)

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maple [A]  time = 0.18, size = 42, normalized size = 1.83

method	result	size
norman	\(\frac {-4 x +\left (2 \ln \relax (2)-2\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} x +4-4 \ln \relax (2)}{2 x \ln \relax (2)+2 x^{2}-2 x -9}\)	\(42\)
risch	\(\frac {-2 x +2-2 \ln \relax (2)}{x \ln \relax (2)+x^{2}-x -\frac {9}{2}}+\frac {2 \left (x +\ln \relax (2)-1\right ) {\mathrm e}^{x}}{2 x \ln \relax (2)+2 x^{2}-2 x -9}\)	\(52\)
default	\(\frac {2 \ln \relax (2)^{3} {\mathrm e}^{x}}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {3 \,{\mathrm e}^{x} \ln \relax (2)^{2}}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {49 \,{\mathrm e}^{x} \ln \relax (2)}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {{\mathrm e}^{x} \left (2 x \ln \relax (2)^{3}-6 x \ln \relax (2)^{2}-9 \ln \relax (2)^{2}+33 x \ln \relax (2)+18 \ln \relax (2)-29 x -90\right )}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {3 \,{\mathrm e}^{x} \left (2 x \ln \relax (2)^{2}-4 x \ln \relax (2)-9 \ln \relax (2)+20 x +9\right )}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {3 \,{\mathrm e}^{x} \left (x \ln \relax (2)-x -9\right )}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {32 \ln \relax (2) x}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {2 \,{\mathrm e}^{x} \left (\ln \relax (2)+2 x -1\right )}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {-\frac {2 \left (\ln \relax (2)^{2}-2 \ln \relax (2)+10\right ) x}{\ln \relax (2)^{2}-2 \ln \relax (2)+19}+\frac {8 \left (\frac {9 \ln \relax (2)}{8}-\frac {9}{8}\right )}{\ln \relax (2)^{2}-2 \ln \relax (2)+19}}{x \ln \relax (2)+x^{2}-x -\frac {9}{2}}+\frac {16 \ln \relax (2)^{3}}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {2 \ln \relax (2)^{2} {\mathrm e}^{x} x}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {2 \ln \relax (2)^{3} {\mathrm e}^{x} x}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {22 \ln \relax (2) {\mathrm e}^{x} x}{\left (\ln \relax (2)^{2}-2 \ln \relax (2)+19\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {48 \ln \relax (2)^{2}}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {320 \ln \relax (2)}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}+\frac {176 x +88 \ln \relax (2)-88}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {144 \arctanh \left (\frac {4 x +2 \ln \relax (2)-2}{2 \sqrt {\ln \relax (2)^{2}-2 \ln \relax (2)+19}}\right )}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \sqrt {\ln \relax (2)^{2}-2 \ln \relax (2)+19}}-\frac {16 \left (\left (2-2 \ln \relax (2)\right ) x +18\right )}{\left (-4 \ln \relax (2)^{2}+8 \ln \relax (2)-76\right ) \left (2 x \ln \relax (2)+2 x^{2}-2 x -9\right )}-\frac {72 \arctanh \left (\frac {4 x +2 \ln \relax (2)-2}{2 \sqrt {\ln \relax (2)^{2}-2 \ln \relax (2)+19}}\right )}{\left (2 \ln \relax (2)^{2}-4 \ln \relax (2)+38\right ) \sqrt {\ln \relax (2)^{2}-2 \ln \relax (2)+19}}\)	\(819\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

(-4*x+(2*ln(2)-2)*exp(x)+2*exp(x)*x+4-4*ln(2))/(2*x*ln(2)+2*x^2-2*x-9)

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maxima [B]  time = 0.53, size = 692, normalized size = 30.09 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-1)*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")

[Out]

-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
))/(log(2)^2 - 2*log(2) + 19)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 +
4*x^4 + 81),x)

[Out]

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 +
4*x^4 + 81), x)

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sympy [B]  time = 0.62, size = 54, normalized size = 2.35 \begin {gather*} \frac {- 4 x - 4 \log {\relax (2 )} + 4}{2 x^{2} + x \left (-2 + 2 \log {\relax (2 )}\right ) - 9} + \frac {\left (2 x - 2 + 2 \log {\relax (2 )}\right ) e^{x}}{2 x^{2} - 2 x + 2 x \log {\relax (2 )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((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)

[Out]

(-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
) - 9)

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