\(\int \frac {(160+40 x+(64 x^3+16 x^4) \log ^2(4)) \log (\frac {5-x^3 \log ^2(4)}{x})+(-80-40 x+(16 x^3+8 x^4) \log ^2(4)) \log ^2(\frac {5-x^3 \log ^2(4)}{x})}{-5+x^3 \log ^2(4)} \, dx\) [6400]

Optimal result

Integrand size = 92, antiderivative size = 24 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 x (4+x) \log ^2\left (\frac {5}{x}-x^2 \log ^2(4)\right ) \]

[Out]

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

Rubi [C] (warning: unable to verify)

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

Time = 5.07 (sec) , antiderivative size = 3127, normalized size of antiderivative = 130.29, number of steps used = 145, number of rules used = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.337, Rules used = {6820, 12, 1901, 1875, 31, 648, 631, 210, 642, 2605, 1848, 1885, 266, 6874, 2608, 2603, 396, 206, 470, 298, 2604, 2465, 2439, 2438, 2463, 2437, 2338, 2441, 2440, 2404, 2375} \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx =\text {Too large to display} \]

[In]

Int[((160 + 40*x + (64*x^3 + 16*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x] + (-80 - 40*x + (16*x^3 + 8*x^4)*Log[
4]^2)*Log[(5 - x^3*Log[4]^2)/x]^2)/(-5 + x^3*Log[4]^2),x]

[Out]

(8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5]*Log[x])/3 - (8*(4*5^(1/3) + (-5/Log[4])^(2/3) -
4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5]*Log[x])/3 - (8*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5]*Log
[x])/(3*Log[4]^(4/3)) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5]*Log[x])/(3*Log[4]^(2/3)) + (8*(4*5^(1/3)
+ (5/Log[4])^(2/3))*Log[5]*Log[x])/(3*Log[4]^(2/3)) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*L
og[5]*Log[x])/(3*Log[4]^(2/3)) + 16*Log[x]^2 + (4*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5^(1/3) - x*(-Log[4])
^(2/3)]^2)/Log[4]^(2/3) - (4*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/
3)]^2)/Log[4]^(2/3) + (4*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5^(1/3) - x*Log[4]^(2/3)]^2)/Log[4]^(4/3) - (4*(4*5^
(1/3) + (5/Log[4])^(2/3))*Log[5^(1/3) - x*Log[4]^(2/3)]^2)/Log[4]^(2/3) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3)
)*Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[((I + Sqrt[3])*(5^(1/3) - x*Log[4]^(2/3)))/(5^(1/3)*(3*I + Sqrt[3]))])/
Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[((I
+ Sqrt[3])*(5^(1/3) - x*Log[4]^(2/3)))/(5^(1/3)*(3*I + Sqrt[3]))])/Log[4]^(2/3) + (8*(4*(-5)^(1/3) - (-5/Log[
4])^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3)]*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/3) - x*Log[4]^(2/3)))/((-5)^(1/
3) + 5^(1/3)))])/Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5^(1/3) - x*(-Log[4
])^(2/3)]*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/3) - x*Log[4]^(2/3)))/((-5)^(1/3) + 5^(1/3)))])/Log[4]^(2/3) + (8
*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5^(1/3) - x*Log[4]^(2/3)]*Log[((-5)^(1/3) + x*Log[4]^(2/3))/((-5)^(1/3) + 5^
(1/3))])/Log[4]^(4/3) - (8*(4*5^(1/3) + (5/Log[4])^(2/3))*Log[5^(1/3) - x*Log[4]^(2/3)]*Log[((-5)^(1/3) + x*Lo
g[4]^(2/3))/((-5)^(1/3) + 5^(1/3))])/Log[4]^(2/3) - 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*Log
[((-1/5)^(1/3)*(5^(1/3) - x*Log[4]^(2/3)))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)] + 8*(4*5
^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*Log[((-1/5)^(1/3)*(5^(1/3) - x*Lo
g[4]^(2/3)))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)] - 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-
Log[4]^(-1))^(2/3)*Log[-(((-1/5)^(1/3)*((-5)^(1/3) + x*Log[4]^(2/3)))/(1 - (-1)^(2/3)))]*Log[5^(1/3) + (-1)^(1
/3)*x*Log[4]^(2/3)] + 8*(4*5^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*Log[-
(((-1/5)^(1/3)*((-5)^(1/3) + x*Log[4]^(2/3)))/(1 - (-1)^(2/3)))]*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)] - 4*
(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)]^2 + 4*(4*5^(1/3)
+ (-5/Log[4])^(2/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)
]^2 + (8*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5^(1/3) - x*Log[4]^(2/3)]*Log[(-2*(-1)^(2/3)*(5^(1/3) + (-1)^(1/3)*x
*Log[4]^(2/3)))/(5^(1/3)*(3 - I*Sqrt[3]))])/Log[4]^(4/3) - (8*(4*5^(1/3) + (5/Log[4])^(2/3))*Log[5^(1/3) - x*L
og[4]^(2/3)]*Log[(-2*(-1)^(2/3)*(5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)))/(5^(1/3)*(3 - I*Sqrt[3]))])/Log[4]^(2/3
) + 32*Log[x]*Log[(5 - x^3*Log[4]^2)/x] - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*Log[5^(1/3) - x*(-Log[4])^(2/3
)]*Log[(5 - x^3*Log[4]^2)/x])/Log[4]^(2/3) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*Log[5^(1/3
) - x*(-Log[4])^(2/3)]*Log[(5 - x^3*Log[4]^2)/x])/Log[4]^(2/3) - (8*(5^(1/3) + 2*Log[4]^(2/3))^2*Log[5^(1/3) -
x*Log[4]^(2/3)]*Log[(5 - x^3*Log[4]^2)/x])/Log[4]^(4/3) + (8*(4*5^(1/3) + (5/Log[4])^(2/3))*Log[5^(1/3) - x*L
og[4]^(2/3)]*Log[(5 - x^3*Log[4]^2)/x])/Log[4]^(2/3) + 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*
Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)]*Log[(5 - x^3*Log[4]^2)/x] - 8*(4*5^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)
^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*Log[5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3)]*Log[(5 - x^3*Log[4]^2)/x] +
4*(2 + x)^2*Log[(5 - x^3*Log[4]^2)/x]^2 - 32*Log[x]*Log[1 - (x^3*Log[4]^2)/5] + (8*(4*(-5)^(1/3) - (-5/Log[4]
)^(2/3))*PolyLog[2, (2*(5^(1/3) - x*(-Log[4])^(2/3)))/(5^(1/3)*(3 - I*Sqrt[3]))])/Log[4]^(2/3) - (8*(4*(-5)^(1
/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*PolyLog[2, (2*(5^(1/3) - x*(-Log[4])^(2/3)))/(5^(1/3)*(3 - I*Sqrt[3]
))])/Log[4]^(2/3) + (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3))*PolyLog[2, (5^(1/3) - x*(-Log[4])^(2/3))/((-5)^(1/3)
+ 5^(1/3))])/Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3) - 4*Log[4]^(2/3))*PolyLog[2, (5^(1/3) - x*(-
Log[4])^(2/3))/((-5)^(1/3) + 5^(1/3))])/Log[4]^(2/3) + (8*(5^(1/3) + 2*Log[4]^(2/3))^2*PolyLog[2, (5^(1/3) - x
*Log[4]^(2/3))/((-5)^(1/3) + 5^(1/3))])/Log[4]^(4/3) - (8*(4*5^(1/3) + (5/Log[4])^(2/3))*PolyLog[2, (5^(1/3) -
x*Log[4]^(2/3))/((-5)^(1/3) + 5^(1/3))])/Log[4]^(2/3) - 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3
)*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3))/(5^(1/3)*(1 - (-1)^(2/3)))] + 8*(4*5^(1/3) + (-5/Log[4])^(2
/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3))/(5^(1/3
)*(1 - (-1)^(2/3)))] - 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*(-Log[4]^(-1))^(2/3)*PolyLog[2, (5^(1/3) + (-1)^(1/3)
*x*Log[4]^(2/3))/((-5)^(1/3) + 5^(1/3))] + 8*(4*5^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)^(1/3)*Log[4]^(2/3))*(-Log
[4]^(-1))^(2/3)*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x*Log[4]^(2/3))/((-5)^(1/3) + 5^(1/3))] + (8*(4*(-5)^(1/3) -
(-5/Log[4])^(2/3))*PolyLog[2, (x*(-Log[4])^(2/3))/5^(1/3)])/Log[4]^(2/3) - (8*(4*(-5)^(1/3) - (-5/Log[4])^(2/3
) - 4*Log[4]^(2/3))*PolyLog[2, (x*(-Log[4])^(2/3))/5^(1/3)])/Log[4]^(2/3) - 8*(4*5^(1/3) + (-5/Log[4])^(2/3))*
(-Log[4]^(-1))^(2/3)*PolyLog[2, -((-1/5)^(1/3)*x*Log[4]^(2/3))] + 8*(4*5^(1/3) + (-5/Log[4])^(2/3) - 4*(-1)^(1
/3)*Log[4]^(2/3))*(-Log[4]^(-1))^(2/3)*PolyLog[2, -((-1/5)^(1/3)*x*Log[4]^(2/3))] + (8*(5^(1/3) + 2*Log[4]^(2/
3))^2*PolyLog[2, (x*Log[4]^(2/3))/5^(1/3)])/Log[4]^(4/3) - (8*(4*5^(1/3) + (5/Log[4])^(2/3))*PolyLog[2, (x*Log
[4]^(2/3))/5^(1/3)])/Log[4]^(2/3) - (32*PolyLog[2, (x^3*Log[4]^2)/5])/3 + (8*(5^(1/3) + 2*Log[4]^(2/3))^2*Poly
Log[2, (2*(5 - x*(5*Log[4])^(2/3)))/(5*(3 - I*Sqrt[3]))])/Log[4]^(4/3) - (8*(4*5^(1/3) + (5/Log[4])^(2/3))*Pol
yLog[2, (2*(5 - x*(5*Log[4])^(2/3)))/(5*(3 - I*Sqrt[3]))])/Log[4]^(2/3)

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1848

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(Pq/(a + b*x
^n)), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1875

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 3]], s = Denominator[
Rt[-a/b, 3]]}, Dist[r*((B*r + A*s)/(3*a*s)), Int[1/(r - s*x), x], x] - Dist[r/(3*a*s), Int[(r*(B*r - 2*A*s) -
s*(B*r + A*s)*x)/(r^2 + r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && NegQ[a
/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + e*(x/d)]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2604

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b
*Log[c*RFx^p])^n/e), x] - Dist[b*n*(p/e), Int[Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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 8 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \left (\frac {(4+x) \left (5+2 x^3 \log ^2(4)\right )}{-5+x^3 \log ^2(4)}+(2+x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )\right ) \, dx \\ & = 8 \int \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \left (\frac {(4+x) \left (5+2 x^3 \log ^2(4)\right )}{-5+x^3 \log ^2(4)}+(2+x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )\right ) \, dx \\ & = 8 \int \left (\frac {(4+x) \left (5+2 x^3 \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)}+(2+x) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )\right ) \, dx \\ & = 8 \int \frac {(4+x) \left (5+2 x^3 \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx+8 \int (2+x) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right ) \, dx \\ & = 4 (2+x)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )-8 \int \frac {(2+x)^2 \left (-5-2 x^3 \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{x \left (5-x^3 \log ^2(4)\right )} \, dx+8 \int \left (8 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+2 x \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {15 (4+x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)}\right ) \, dx \\ & = 4 (2+x)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )-8 \int \left (8 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )-\frac {4 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{x}+2 x \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {3 \left (20+5 x+4 x^2 \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)}\right ) \, dx+16 \int x \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \, dx+64 \int \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \, dx+120 \int \frac {(4+x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx \\ & = 64 x \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+8 x^2 \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+4 (2+x)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )-8 \int \frac {x \left (-5-2 x^3 \log ^2(4)\right )}{5-x^3 \log ^2(4)} \, dx-16 \int x \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \, dx-24 \int \frac {\left (20+5 x+4 x^2 \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx+32 \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{x} \, dx-64 \int \frac {-5-2 x^3 \log ^2(4)}{5-x^3 \log ^2(4)} \, dx-64 \int \log \left (\frac {5-x^3 \log ^2(4)}{x}\right ) \, dx+120 \int \left (-\frac {\left (4 \sqrt [3]{5}-\sqrt [3]{-1} \left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}-\frac {\left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}-\frac {\left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)\right )}\right ) \, dx \\ & = -128 x-8 x^2+32 \log (x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+4 (2+x)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )+8 \int \frac {x \left (-5-2 x^3 \log ^2(4)\right )}{5-x^3 \log ^2(4)} \, dx-24 \int \left (-\frac {\left (20 \sqrt [3]{5}-5 \sqrt [3]{-1} \left (\frac {5}{\log (4)}\right )^{2/3}+20 (-\log (4))^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}-\frac {\left (20 \sqrt [3]{5}+5 \left (\frac {5}{\log (4)}\right )^{2/3}+20 \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}-\frac {\left (20 \sqrt [3]{5}+5 \left (-\frac {5}{\log (4)}\right )^{2/3}-20 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{15 \left (\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)\right )}\right ) \, dx-32 \int \frac {x \left (-3 x \log ^2(4)-\frac {5-x^3 \log ^2(4)}{x^2}\right ) \log (x)}{5-x^3 \log ^2(4)} \, dx+64 \int \frac {-5-2 x^3 \log ^2(4)}{5-x^3 \log ^2(4)} \, dx+120 \int \frac {x}{5-x^3 \log ^2(4)} \, dx+960 \int \frac {1}{5-x^3 \log ^2(4)} \, dx-\left (8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right )\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)} \, dx-\left (8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right )\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)} \, dx-\left (8 \sqrt [3]{5} \left (4-\frac {\sqrt [3]{-5}}{\log ^{\frac {2}{3}}(4)}\right )\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}-x (-\log (4))^{2/3}} \, dx \\ & = 32 \log (x) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {8 \sqrt [3]{-5} \left (\sqrt [3]{-5}-4 \log ^{\frac {2}{3}}(4)\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {4}{3}}(4)}+\frac {8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right ) \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\log ^{\frac {2}{3}}(4)}+8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3} \log \left (\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+4 (2+x)^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )-32 \int \left (-\frac {\log (x)}{x}+\frac {3 x^2 \log ^2(4) \log (x)}{-5+x^3 \log ^2(4)}\right ) \, dx-120 \int \frac {x}{5-x^3 \log ^2(4)} \, dx-960 \int \frac {1}{5-x^3 \log ^2(4)} \, dx+\left (64 \sqrt [3]{5}\right ) \int \frac {1}{\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)} \, dx+\left (64 \sqrt [3]{5}\right ) \int \frac {2 \sqrt [3]{5}+x \log ^{\frac {2}{3}}(4)}{5^{2/3}+\sqrt [3]{5} x \log ^{\frac {2}{3}}(4)+x^2 \log ^{\frac {4}{3}}(4)} \, dx+\left (8 \left (4 \left (\sqrt [3]{5}+(-\log (4))^{2/3}\right )-\sqrt [3]{-1} \left (\frac {5}{\log (4)}\right )^{2/3}\right )\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}-x (-\log (4))^{2/3}} \, dx+\left (8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}-4 \sqrt [3]{-1} \log ^{\frac {2}{3}}(4)\right )\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)} \, dx-\left (8 \left (4 \sqrt [3]{5}+\left (-\frac {5}{\log (4)}\right )^{2/3}\right ) \left (-\frac {1}{\log (4)}\right )^{2/3}\right ) \int \frac {x \left (-3 x \log ^2(4)-\frac {5-x^3 \log ^2(4)}{x^2}\right ) \log \left (\sqrt [3]{5}+\sqrt [3]{-1} x \log ^{\frac {2}{3}}(4)\right )}{5-x^3 \log ^2(4)} \, dx+\left (8 \left (\frac {5}{\log (4)}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)} \, dx-\left (8 \left (\frac {5}{\log (4)}\right )^{2/3}\right ) \int \frac {\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)}{5^{2/3}+\sqrt [3]{5} x \log ^{\frac {2}{3}}(4)+x^2 \log ^{\frac {4}{3}}(4)} \, dx-\frac {\left (8 \sqrt [3]{-5} \left (\sqrt [3]{-5}-4 \log ^{\frac {2}{3}}(4)\right )\right ) \int \frac {x \left (-3 x \log ^2(4)-\frac {5-x^3 \log ^2(4)}{x^2}\right ) \log \left (\sqrt [3]{5}-x (-\log (4))^{2/3}\right )}{5-x^3 \log ^2(4)} \, dx}{\log ^{\frac {4}{3}}(4)}-\frac {\left (8 \left (4 \sqrt [3]{5}+\left (\frac {5}{\log (4)}\right )^{2/3}\right )\right ) \int \frac {x \left (-3 x \log ^2(4)-\frac {5-x^3 \log ^2(4)}{x^2}\right ) \log \left (\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)\right )}{5-x^3 \log ^2(4)} \, dx}{\log ^{\frac {2}{3}}(4)}+\frac {\left (8 \left (\sqrt [3]{5}+2 \log ^{\frac {2}{3}}(4)\right )^2\right ) \int \frac {\log \left (\frac {5-x^3 \log ^2(4)}{x}\right )}{\sqrt [3]{5}-x \log ^{\frac {2}{3}}(4)} \, dx}{\log ^{\frac {2}{3}}(4)} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).

Time = 1.93 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=8 \left (2 x \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )+\frac {1}{2} x^2 \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )\right ) \]

[In]

Integrate[((160 + 40*x + (64*x^3 + 16*x^4)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x] + (-80 - 40*x + (16*x^3 + 8*x^4
)*Log[4]^2)*Log[(5 - x^3*Log[4]^2)/x]^2)/(-5 + x^3*Log[4]^2),x]

[Out]

8*(2*x*Log[(5 - x^3*Log[4]^2)/x]^2 + (x^2*Log[(5 - x^3*Log[4]^2)/x]^2)/2)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21

method	result	size
risch	\(\left (4 x^{2}+16 x \right ) \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}\)	\(29\)
norman	\(16 x \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}+4 x^{2} \ln \left (\frac {-4 x^{3} \ln \left (2\right )^{2}+5}{x}\right )^{2}\)	\(46\)
parallelrisch	\(4 x^{2} \ln \left (-\frac {4 x^{3} \ln \left (2\right )^{2}-5}{x}\right )^{2}+16 \ln \left (-\frac {4 x^{3} \ln \left (2\right )^{2}-5}{x}\right )^{2} x\)	\(48\)

[In]

int(((4*(8*x^4+16*x^3)*ln(2)^2-40*x-80)*ln((-4*x^3*ln(2)^2+5)/x)^2+(4*(16*x^4+64*x^3)*ln(2)^2+40*x+160)*ln((-4
*x^3*ln(2)^2+5)/x))/(4*x^3*ln(2)^2-5),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-\frac {4 \, x^{3} \log \left (2\right )^{2} - 5}{x}\right )^{2} \]

[In]

integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+1
60)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3*log(2)^2-5),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=\left (4 x^{2} + 16 x\right ) \log {\left (\frac {- 4 x^{3} \log {\left (2 \right )}^{2} + 5}{x} \right )}^{2} \]

[In]

integrate(((4*(8*x**4+16*x**3)*ln(2)**2-40*x-80)*ln((-4*x**3*ln(2)**2+5)/x)**2+(4*(16*x**4+64*x**3)*ln(2)**2+4
0*x+160)*ln((-4*x**3*ln(2)**2+5)/x))/(4*x**3*ln(2)**2-5),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 0.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right ) \log \left (x\right ) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} \]

[In]

integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+1
60)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3*log(2)^2-5),x, algorithm="maxima")

[Out]

4*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)^2 - 8*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)*log(x) + 4*(x^2 + 4*x)*log(x
)^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).

Time = 0.49 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \left (2\right )^{2} + 5\right ) \log \left (x\right ) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} \]

[In]

integrate(((4*(8*x^4+16*x^3)*log(2)^2-40*x-80)*log((-4*x^3*log(2)^2+5)/x)^2+(4*(16*x^4+64*x^3)*log(2)^2+40*x+1
60)*log((-4*x^3*log(2)^2+5)/x))/(4*x^3*log(2)^2-5),x, algorithm="giac")

[Out]

4*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)^2 - 8*(x^2 + 4*x)*log(-4*x^3*log(2)^2 + 5)*log(x) + 4*(x^2 + 4*x)*log(x
)^2

Mupad [B] (verification not implemented)

Time = 13.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (160+40 x+\left (64 x^3+16 x^4\right ) \log ^2(4)\right ) \log \left (\frac {5-x^3 \log ^2(4)}{x}\right )+\left (-80-40 x+\left (16 x^3+8 x^4\right ) \log ^2(4)\right ) \log ^2\left (\frac {5-x^3 \log ^2(4)}{x}\right )}{-5+x^3 \log ^2(4)} \, dx=4\,x\,{\ln \left (-\frac {4\,x^3\,{\ln \left (2\right )}^2-5}{x}\right )}^2\,\left (x+4\right ) \]

[In]

int((log(-(4*x^3*log(2)^2 - 5)/x)*(40*x + 4*log(2)^2*(64*x^3 + 16*x^4) + 160) - log(-(4*x^3*log(2)^2 - 5)/x)^2
*(40*x - 4*log(2)^2*(16*x^3 + 8*x^4) + 80))/(4*x^3*log(2)^2 - 5),x)

[Out]

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