∫ 64+44x−52x2−87x3+33x4+35x5−10x6−4x7+x8+(−64+20x+32x2−9x3−4x4+x5) log(−16+5x+4x2−x3 −4+x2 )+log(x)(−128+60x+64x2−25x3−8x4+3x5+(128−40x−64x2+18x3+8x4−2x5) log(−16+5x+4x2−x3 −4+x2 )) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3.61.91 \(\int \frac {64+44 x-52 x^2-87 x^3+33 x^4+35 x^5-10 x^6-4 x^7+x^8+(-64+20 x+32 x^2-9 x^3-4 x^4+x^5) \log (\frac {-16+5 x+4 x^2-x^3}{-4+x^2})+\log (x) (-128+60 x+64 x^2-25 x^3-8 x^4+3 x^5+(128-40 x-64 x^2+18 x^3+8 x^4-2 x^5) \log (\frac {-16+5 x+4 x^2-x^3}{-4+x^2}))}{-64 x^3+20 x^4+32 x^5-9 x^6-4 x^7+x^8} \, dx\)

Optimal. Leaf size=32 \[ 2+\frac {1}{x}+x-\frac {\log (x) \left (1-\log \left (4-x+\frac {x}{-4+x^2}\right )\right )}{x^2} \]

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Rubi [F] time = 54.52, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {64+44 x-52 x^2-87 x^3+33 x^4+35 x^5-10 x^6-4 x^7+x^8+\left (-64+20 x+32 x^2-9 x^3-4 x^4+x^5\right ) \log \left (\frac {-16+5 x+4 x^2-x^3}{-4+x^2}\right )+\log (x) \left (-128+60 x+64 x^2-25 x^3-8 x^4+3 x^5+\left (128-40 x-64 x^2+18 x^3+8 x^4-2 x^5\right ) \log \left (\frac {-16+5 x+4 x^2-x^3}{-4+x^2}\right )\right )}{-64 x^3+20 x^4+32 x^5-9 x^6-4 x^7+x^8} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(64 + 44*x - 52*x^2 - 87*x^3 + 33*x^4 + 35*x^5 - 10*x^6 - 4*x^7 + x^8 + (-64 + 20*x + 32*x^2 - 9*x^3 - 4*x

^4 + x^5)*Log[(-16 + 5*x + 4*x^2 - x^3)/(-4 + x^2)] + Log[x]*(-128 + 60*x + 64*x^2 - 25*x^3 - 8*x^4 + 3*x^5 +

(128 - 40*x - 64*x^2 + 18*x^3 + 8*x^4 - 2*x^5)*Log[(-16 + 5*x + 4*x^2 - x^3)/(-4 + x^2)]))/(-64*x^3 + 20*x^4 +

32*x^5 - 9*x^6 - 4*x^7 + x^8),x]

[Out]

x^(-1) + x - (90*31^(1/3)*((2 - (3*I)*Sqrt[3])^(2/3)*(28 - (11*I)*Sqrt[3]) + 31^(1/3)*(8 + (19*I)*Sqrt[3]))*Ar

cTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt

[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*

(2*I + 3*Sqrt[3]))]])/(Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2

- (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 -

(3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) - (39*31^(1/3)*((2 - (3*I)*Sqrt[3])^(2/3)*(215 - (28*I)*Sqrt[3]) + 3

1^(1/3)*(127 + (104*I)*Sqrt[3]))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (

3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3])

+ (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(8*Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2

/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 -

(3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) + (31^(1/3)*(2 - (3*I)*Sqrt[3])

^(2/3)*(696 - Sqrt[3]*(951*I - (6*I - 107*Sqrt[3])*(62 - (93*I)*Sqrt[3])^(1/3)))*ArcTan[(31^(1/3)*(2 - (3*I)*S

qrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*

Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(Sqrt

[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*

I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I

+ 3*Sqrt[3]))) + (15*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3)*(3123 - Sqrt[3]*(236*I - (287*I - 90*Sqrt[3])*(62 - (9

3*I)*Sqrt[3])^(1/3)))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3

])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2

- (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(128*Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 +

(12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sq

rt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) + (4*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3)*

(129 - Sqrt[3]*(54*I - (9*I - 8*Sqrt[3])*(62 - (93*I)*Sqrt[3])^(1/3)))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (

62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(

2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(Sqrt[31*31^(1/

3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt

[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3

]))) + (99*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3)*(24 - Sqrt[3]*(5*I - (2*I - Sqrt[3])*(62 - (93*I)*Sqrt[3])^(1/3)

))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))

/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(

1/3)*(2*I + 3*Sqrt[3]))]])/(Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*

I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)

*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) - (87*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3)*(15 + Sqrt[3]*(24*I +

(62 - (93*I)*Sqrt[3])^(1/3)*(3*I + 2*Sqrt[3])))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1

/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (

12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(2*Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3]

)^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3

) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) + (627*31^(1/3)*

(2 - (3*I)*Sqrt[3])^(2/3)*(33 + Sqrt[3]*(28*I + (62 - (93*I)*Sqrt[3])^(1/3)*(5*I + 2*Sqrt[3])))*ArcTan[(31^(1/

3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 - (3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1

/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqr

t[3]))]])/(128*Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)

*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*S

qrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) + (35*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3)*(3 + Sqrt[3]*(42*I + (62 - (93*I)*S

qrt[3])^(1/3)*(3*I + 4*Sqrt[3])))*ArcTan[(31^(1/3)*(2 - (3*I)*Sqrt[3] + (62 - (93*I)*Sqrt[3])^(1/3)) + 2*(2 -

(3*I)*Sqrt[3])^(2/3)*(4 - 3*x))/Sqrt[3*(31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/3)*(23 + (12*I)*Sqrt[3])

+ (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))]])/(Sqrt[31*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3) - 31^(2/

3)*(23 + (12*I)*Sqrt[3]) + (62*I)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3])]*((31*I)*31^(1/3) + (31*I)*(2 -

(3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + 3*Sqrt[3]))) + 20*ArcTanh[x/2] - (33*(2 - (3*

I)*Sqrt[3])^(1/3)*(31^(2/3) + 8*(2 - (3*I)*Sqrt[3])^(1/3) + 31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(

31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(31*31^(1/3) + 31*(2 - (3*I)*Sqr

t[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3)) + (209*(2 - (3*I)*Sqrt[3])^(1/3)*(2*31^(2/3) - 11*(2 - (3*I)

*Sqrt[3])^(1/3) + 2*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) -

(2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(128*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*S

qrt[3])^(4/3))) - (29*(2 - (3*I)*Sqrt[3])^(1/3)*(2*31^(2/3) - 5*(2 - (3*I)*Sqrt[3])^(1/3) + 2*31^(1/3)*(2 - (3

*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)]

)/(2*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (35*(2 - (3*I)*Sqrt[

3])^(1/3)*(4*31^(2/3) - (2 - (3*I)*Sqrt[3])^(1/3) + 4*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/

3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(3*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3

])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (10*(2 - (3*I)*Sqrt[3])^(1/3)*(5*31^(2/3) + 28*(2 - (3*I)*Sq

rt[3])^(1/3) + 5*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2

- (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^

(4/3)) - (4*(2 - (3*I)*Sqrt[3])^(1/3)*(8*31^(2/3) + 43*(2 - (3*I)*Sqrt[3])^(1/3) + 8*31^(1/3)*(2 - (3*I)*Sqrt[

3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(3*(31*

31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (13*(2 - (3*I)*Sqrt[3])^(1/3)

*(22*31^(2/3) + 215*(2 - (3*I)*Sqrt[3])^(1/3) + 22*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3)

+ (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(24*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])

^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) - (15*(2 - (3*I)*Sqrt[3])^(1/3)*(30*31^(2/3) + 347*(2 - (3*I)*Sq

rt[3])^(1/3) + 30*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2

- (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)])/(128*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqr

t[3])^(4/3))) - ((2 - (3*I)*Sqrt[3])^(1/3)*(107*31^(2/3) + 232*(2 - (3*I)*Sqrt[3])^(1/3) + 107*31^(1/3)*(2 - (

3*I)*Sqrt[3])^(2/3))*Log[31^(1/3)*(31^(1/3) + (2 - (3*I)*Sqrt[3])^(2/3)) - (2 - (3*I)*Sqrt[3])^(1/3)*(4 - 3*x)

])/(3*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + 10*Log[2 - x] - Log

[x]/x^2 - 10*Log[2 + x] + (33*(2 - (3*I)*Sqrt[3])^(1/3)*(31^(2/3) + 8*(2 - (3*I)*Sqrt[3])^(1/3) + 31^(1/3)*(2

- (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[

3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(

1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(2*(31*31^(1/3) + 31*(2 - (

3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) - (209*(2 - (3*I)*Sqrt[3])^(1/3)*(2*31^(2/3) - 11*(

2 - (3*I)*Sqrt[3])^(1/3) + 2*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/

3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(

1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3]

)^(2/3)*x^2])/(256*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (29*(2

- (3*I)*Sqrt[3])^(1/3)*(2*31^(2/3) - 5*(2 - (3*I)*Sqrt[3])^(1/3) + 2*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[

(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqr

t[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sq

rt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(4*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/

3)*(2 - (3*I)*Sqrt[3])^(4/3))) - (35*(2 - (3*I)*Sqrt[3])^(1/3)*(4*31^(2/3) - (2 - (3*I)*Sqrt[3])^(1/3) + 4*31^

(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*

I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*S

qrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(6*(31*31^(1/3) +

31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) - (5*(2 - (3*I)*Sqrt[3])^(1/3)*(5*31^(2/3)

+ 28*(2 - (3*I)*Sqrt[3])^(1/3) + 5*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) +

31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[

3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*

Sqrt[3])^(2/3)*x^2])/(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3)) + (2*(2

- (3*I)*Sqrt[3])^(1/3)*(8*31^(2/3) + 43*(2 - (3*I)*Sqrt[3])^(1/3) + 8*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log

[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sq

rt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*S

qrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(3*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2

/3)*(2 - (3*I)*Sqrt[3])^(4/3))) - (13*(2 - (3*I)*Sqrt[3])^(1/3)*(22*31^(2/3) + 215*(2 - (3*I)*Sqrt[3])^(1/3) +

22*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1

/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 -

(3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(48*(31*31^

(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (15*(2 - (3*I)*Sqrt[3])^(1/3)*(3

0*31^(2/3) + 347*(2 - (3*I)*Sqrt[3])^(1/3) + 30*31^(1/3)*(2 - (3*I)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt

[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) + 31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)

*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x - (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)

*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(256*(31*31^(1/3) + 31*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3

])^(4/3))) + ((2 - (3*I)*Sqrt[3])^(1/3)*(107*31^(2/3) + 232*(2 - (3*I)*Sqrt[3])^(1/3) + 107*31^(1/3)*(2 - (3*I

)*Sqrt[3])^(2/3))*Log[(-5*I)*(2 - (3*I)*Sqrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*(2*I + Sqrt[3]) +

31^(1/3)*(13*I + 4*Sqrt[3]) - (2*I)*31^(1/3)*x - 3*Sqrt[3]*31^(1/3)*x - I*31^(2/3)*(2 - (3*I)*Sqrt[3])^(1/3)*x

- (8*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x + (3*I)*(2 - (3*I)*Sqrt[3])^(2/3)*x^2])/(6*(31*31^(1/3) + 31*(2 - (3*I)*S

qrt[3])^(2/3) + 31^(2/3)*(2 - (3*I)*Sqrt[3])^(4/3))) + (Log[x]*Log[(16 - 5*x - 4*x^2 + x^3)/(4 - x^2)])/x^2 -

(25*Defer[Int][Log[x]/(-16 + 5*x + 4*x^2 - x^3), x])/4 - (25*Defer[Int][Log[x]/(16 - 5*x - 4*x^2 + x^3), x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-64-44 x+52 x^2+87 x^3-33 x^4-35 x^5+10 x^6+4 x^7-x^8-\left (-64+20 x+32 x^2-9 x^3-4 x^4+x^5\right ) \log \left (\frac {-16+5 x+4 x^2-x^3}{-4+x^2}\right )-\log (x) \left (-128+60 x+64 x^2-25 x^3-8 x^4+3 x^5+\left (128-40 x-64 x^2+18 x^3+8 x^4-2 x^5\right ) \log \left (\frac {-16+5 x+4 x^2-x^3}{-4+x^2}\right )\right )}{x^3 \left (64-20 x-32 x^2+9 x^3+4 x^4-x^5\right )} \, dx\\ &=\int \left (-\frac {87}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {64}{(-2+x) x^3 (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {44}{(-2+x) x^2 (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {52}{(-2+x) x (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {33 x}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {35 x^2}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {10 x^3}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {4 x^4}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {x^5}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {25 \log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {128 \log (x)}{(-2+x) x^3 (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {60 \log (x)}{(-2+x) x^2 (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {64 \log (x)}{(-2+x) x (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {8 x \log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}+\frac {3 x^2 \log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )}-\frac {(-1+2 \log (x)) \log \left (-\frac {16-5 x-4 x^2+x^3}{-4+x^2}\right )}{x^3}\right ) \, dx\\ &=3 \int \frac {x^2 \log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-4 \int \frac {x^4}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-8 \int \frac {x \log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-10 \int \frac {x^3}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-25 \int \frac {\log (x)}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+33 \int \frac {x}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+35 \int \frac {x^2}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+44 \int \frac {1}{(-2+x) x^2 (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-52 \int \frac {1}{(-2+x) x (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+60 \int \frac {\log (x)}{(-2+x) x^2 (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+64 \int \frac {1}{(-2+x) x^3 (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+64 \int \frac {\log (x)}{(-2+x) x (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-87 \int \frac {1}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-128 \int \frac {\log (x)}{(-2+x) x^3 (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx+\int \frac {x^5}{(-2+x) (2+x) \left (16-5 x-4 x^2+x^3\right )} \, dx-\int \frac {(-1+2 \log (x)) \log \left (-\frac {16-5 x-4 x^2+x^3}{-4+x^2}\right )}{x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.10, size = 41, normalized size = 1.28 \begin {gather*} \frac {1}{x}+x-\frac {\log (x)}{x^2}+\frac {\log (x) \log \left (-\frac {16-5 x-4 x^2+x^3}{-4+x^2}\right )}{x^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(64 + 44*x - 52*x^2 - 87*x^3 + 33*x^4 + 35*x^5 - 10*x^6 - 4*x^7 + x^8 + (-64 + 20*x + 32*x^2 - 9*x^3

- 4*x^4 + x^5)*Log[(-16 + 5*x + 4*x^2 - x^3)/(-4 + x^2)] + Log[x]*(-128 + 60*x + 64*x^2 - 25*x^3 - 8*x^4 + 3*

x^5 + (128 - 40*x - 64*x^2 + 18*x^3 + 8*x^4 - 2*x^5)*Log[(-16 + 5*x + 4*x^2 - x^3)/(-4 + x^2)]))/(-64*x^3 + 20

*x^4 + 32*x^5 - 9*x^6 - 4*x^7 + x^8),x]

[Out]

x^(-1) + x - Log[x]/x^2 + (Log[x]*Log[-((16 - 5*x - 4*x^2 + x^3)/(-4 + x^2))])/x^2

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fricas [A] time = 0.61, size = 37, normalized size = 1.16 \begin {gather*} \frac {x^{3} + {\left (\log \left (-\frac {x^{3} - 4 \, x^{2} - 5 \, x + 16}{x^{2} - 4}\right ) - 1\right )} \log \relax (x) + x}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5+8*x^4+18*x^3-64*x^2-40*x+128)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+3*x^5-8*x^4-25*x^3+64*x^2+6

0*x-128)*log(x)+(x^5-4*x^4-9*x^3+32*x^2+20*x-64)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+x^8-4*x^7-10*x^6+35*x^5+33*x

^4-87*x^3-52*x^2+44*x+64)/(x^8-4*x^7-9*x^6+32*x^5+20*x^4-64*x^3),x, algorithm="fricas")

[Out]

(x^3 + (log(-(x^3 - 4*x^2 - 5*x + 16)/(x^2 - 4)) - 1)*log(x) + x)/x^2

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giac [A] time = 0.35, size = 47, normalized size = 1.47 \begin {gather*} x + \frac {\log \left (-x^{3} + 4 \, x^{2} + 5 \, x - 16\right ) \log \relax (x)}{x^{2}} - \frac {\log \left (x^{2} - 4\right ) \log \relax (x)}{x^{2}} + \frac {1}{x} - \frac {\log \relax (x)}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5+8*x^4+18*x^3-64*x^2-40*x+128)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+3*x^5-8*x^4-25*x^3+64*x^2+6

0*x-128)*log(x)+(x^5-4*x^4-9*x^3+32*x^2+20*x-64)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+x^8-4*x^7-10*x^6+35*x^5+33*x

^4-87*x^3-52*x^2+44*x+64)/(x^8-4*x^7-9*x^6+32*x^5+20*x^4-64*x^3),x, algorithm="giac")

[Out]

x + log(-x^3 + 4*x^2 + 5*x - 16)*log(x)/x^2 - log(x^2 - 4)*log(x)/x^2 + 1/x - log(x)/x^2

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maple [C] time = 0.24, size = 270, normalized size = 8.44


method	result	size

risch	\(\frac {\ln \relax (x ) \ln \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}}+\frac {-i \pi \,\mathrm {csgn}\left (\frac {i}{x^{2}-4}\right ) \mathrm {csgn}\left (i \left (x^{3}-4 x^{2}-5 x +16\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}-4}\right ) \ln \relax (x )+i \pi \,\mathrm {csgn}\left (\frac {i}{x^{2}-4}\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}-4}\right )^{2} \ln \relax (x )+i \pi \,\mathrm {csgn}\left (i \left (x^{3}-4 x^{2}-5 x +16\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}-4}\right )^{2} \ln \relax (x )+i \pi \mathrm {csgn}\left (\frac {i \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}-4}\right )^{3} \ln \relax (x )-2 i \pi \mathrm {csgn}\left (\frac {i \left (x^{3}-4 x^{2}-5 x +16\right )}{x^{2}-4}\right )^{2} \ln \relax (x )+2 i \pi \ln \relax (x )+2 x^{3}-2 \ln \relax (x ) \ln \left (x^{2}-4\right )+2 x -2 \ln \relax (x )}{2 x^{2}}\)	\(270\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^5+8*x^4+18*x^3-64*x^2-40*x+128)*ln((-x^3+4*x^2+5*x-16)/(x^2-4))+3*x^5-8*x^4-25*x^3+64*x^2+60*x-128

)*ln(x)+(x^5-4*x^4-9*x^3+32*x^2+20*x-64)*ln((-x^3+4*x^2+5*x-16)/(x^2-4))+x^8-4*x^7-10*x^6+35*x^5+33*x^4-87*x^3

-52*x^2+44*x+64)/(x^8-4*x^7-9*x^6+32*x^5+20*x^4-64*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x^2*ln(x)*ln(x^3-4*x^2-5*x+16)+1/2*(-I*Pi*csgn(I/(x^2-4))*csgn(I*(x^3-4*x^2-5*x+16))*csgn(I/(x^2-4)*(x^3-4*x

^2-5*x+16))*ln(x)+I*Pi*csgn(I/(x^2-4))*csgn(I/(x^2-4)*(x^3-4*x^2-5*x+16))^2*ln(x)+I*Pi*csgn(I*(x^3-4*x^2-5*x+1

6))*csgn(I/(x^2-4)*(x^3-4*x^2-5*x+16))^2*ln(x)+I*Pi*csgn(I/(x^2-4)*(x^3-4*x^2-5*x+16))^3*ln(x)-2*I*Pi*csgn(I/(

x^2-4)*(x^3-4*x^2-5*x+16))^2*ln(x)+2*I*Pi*ln(x)+2*x^3-2*ln(x)*ln(x^2-4)+2*x-2*ln(x))/x^2

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maxima [A] time = 0.35, size = 48, normalized size = 1.50 \begin {gather*} \frac {x^{3} + \log \left (-x^{3} + 4 \, x^{2} + 5 \, x - 16\right ) \log \relax (x) - \log \left (x + 2\right ) \log \relax (x) - \log \left (x - 2\right ) \log \relax (x) + x - \log \relax (x)}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5+8*x^4+18*x^3-64*x^2-40*x+128)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+3*x^5-8*x^4-25*x^3+64*x^2+6

0*x-128)*log(x)+(x^5-4*x^4-9*x^3+32*x^2+20*x-64)*log((-x^3+4*x^2+5*x-16)/(x^2-4))+x^8-4*x^7-10*x^6+35*x^5+33*x

^4-87*x^3-52*x^2+44*x+64)/(x^8-4*x^7-9*x^6+32*x^5+20*x^4-64*x^3),x, algorithm="maxima")

[Out]

(x^3 + log(-x^3 + 4*x^2 + 5*x - 16)*log(x) - log(x + 2)*log(x) - log(x - 2)*log(x) + x - log(x))/x^2

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mupad [B] time = 4.72, size = 42, normalized size = 1.31 \begin {gather*} x-\frac {\ln \relax (x)}{x^2}+\frac {1}{x}+\frac {\ln \left (\frac {-x^3+4\,x^2+5\,x-16}{x^2-4}\right )\,\ln \relax (x)}{x^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(44*x + log((5*x + 4*x^2 - x^3 - 16)/(x^2 - 4))*(20*x + 32*x^2 - 9*x^3 - 4*x^4 + x^5 - 64) - log(x)*(log(

(5*x + 4*x^2 - x^3 - 16)/(x^2 - 4))*(40*x + 64*x^2 - 18*x^3 - 8*x^4 + 2*x^5 - 128) - 60*x - 64*x^2 + 25*x^3 +

8*x^4 - 3*x^5 + 128) - 52*x^2 - 87*x^3 + 33*x^4 + 35*x^5 - 10*x^6 - 4*x^7 + x^8 + 64)/(64*x^3 - 20*x^4 - 32*x^

5 + 9*x^6 + 4*x^7 - x^8),x)

[Out]

x - log(x)/x^2 + 1/x + (log((5*x + 4*x^2 - x^3 - 16)/(x^2 - 4))*log(x))/x^2

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sympy [A] time = 0.76, size = 37, normalized size = 1.16 \begin {gather*} x + \frac {1}{x} + \frac {\log {\relax (x )} \log {\left (\frac {- x^{3} + 4 x^{2} + 5 x - 16}{x^{2} - 4} \right )}}{x^{2}} - \frac {\log {\relax (x )}}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**5+8*x**4+18*x**3-64*x**2-40*x+128)*ln((-x**3+4*x**2+5*x-16)/(x**2-4))+3*x**5-8*x**4-25*x**3

+64*x**2+60*x-128)*ln(x)+(x**5-4*x**4-9*x**3+32*x**2+20*x-64)*ln((-x**3+4*x**2+5*x-16)/(x**2-4))+x**8-4*x**7-1

0*x**6+35*x**5+33*x**4-87*x**3-52*x**2+44*x+64)/(x**8-4*x**7-9*x**6+32*x**5+20*x**4-64*x**3),x)

[Out]

x + 1/x + log(x)*log((-x**3 + 4*x**2 + 5*x - 16)/(x**2 - 4))/x**2 - log(x)/x**2

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