3.64.100 \(\int \frac {(160+40 x+(64 x^3+16 x^4) \log ^2(4)) \log (\frac {5-x^3 \log ^2(4)}{x})+(-80-40 x+(16 x^3+8 x^4) \log ^2(4)) \log ^2(\frac {5-x^3 \log ^2(4)}{x})}{-5+x^3 \log ^2(4)} \, dx\)
Optimal. Leaf size=24 \[ 4 x (4+x) \log ^2\left (\frac {5}{x}-x^2 \log ^2(4)\right ) \]
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Rubi [C] time = 10.85, antiderivative size = 3127, normalized size of antiderivative =
130.29, number of steps used = 145, number of rules used = 31, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.337,
Rules used = {6688, 12, 1887, 1861, 31, 634, 617, 204, 628, 2525, 1834, 1871, 260, 6742, 2528,
2523, 388, 200, 459, 292, 2524, 2418, 2392, 2391, 2416, 2390, 2301, 2394, 2393, 2357,
2337}
result too large to display
Warning: Unable to verify antiderivative.
[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 200
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 260
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 292
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 388
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 459
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 617
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 628
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 634
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 1834
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 1861
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 3]], s = Denominato
r[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] && Ne
gQ[a/b]
Rule 1871
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 1887
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 2301
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 2337
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 2357
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 2390
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 2391
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 2392
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 2393
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 2394
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 2416
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 2418
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 2523
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 2524
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 2525
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 2528
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 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
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 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\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 2.11, size = 49, normalized size = 2.04 \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
[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)
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fricas [A] time = 0.70, size = 28, normalized size = 1.17 \begin {gather*} 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-\frac {4 \, x^{3} \log \relax (2)^{2} - 5}{x}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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giac [B] time = 0.72, size = 60, normalized size = 2.50 \begin {gather*} 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \relax (2)^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \relax (2)^{2} + 5\right ) \log \relax (x) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \relax (x)^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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maple [A] time = 0.45, size = 29, normalized size = 1.21
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| method |
result |
size |
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| risch |
\(\left (4 x^{2}+16 x \right ) \ln \left (\frac {-4 x^{3} \ln \relax (2)^{2}+5}{x}\right )^{2}\) |
\(29\) |
| norman |
\(16 x \ln \left (\frac {-4 x^{3} \ln \relax (2)^{2}+5}{x}\right )^{2}+4 x^{2} \ln \left (\frac {-4 x^{3} \ln \relax (2)^{2}+5}{x}\right )^{2}\) |
\(46\) |
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Verification of antiderivative is not currently implemented for this CAS.
[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
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maxima [B] time = 1.06, size = 60, normalized size = 2.50 \begin {gather*} 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \relax (2)^{2} + 5\right )^{2} - 8 \, {\left (x^{2} + 4 \, x\right )} \log \left (-4 \, x^{3} \log \relax (2)^{2} + 5\right ) \log \relax (x) + 4 \, {\left (x^{2} + 4 \, x\right )} \log \relax (x)^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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mupad [B] time = 4.39, size = 25, normalized size = 1.04 \begin {gather*} 4\,x\,{\ln \left (-\frac {4\,x^3\,{\ln \relax (2)}^2-5}{x}\right )}^2\,\left (x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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sympy [A] time = 0.21, size = 24, normalized size = 1.00 \begin {gather*} \left (4 x^{2} + 16 x\right ) \log {\left (\frac {- 4 x^{3} \log {\relax (2 )}^{2} + 5}{x} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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
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