3.57 \(\int x^3 (a+b \log (c x^n))^3 \, dx\)
Optimal. Leaf size=77 \[ \frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{3}{128} b^3 n^3 x^4 \]
[Out]
(-3*b^3*n^3*x^4)/128 + (3*b^2*n^2*x^4*(a + b*Log[c*x^n]))/32 - (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 + (x^4*(a +
b*Log[c*x^n])^3)/4
________________________________________________________________________________________
Rubi [A] time = 0.0626123, antiderivative size = 77, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used =
{2305, 2304} \[ \frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac{3}{128} b^3 n^3 x^4 \]
Antiderivative was successfully verified.
[In]
Int[x^3*(a + b*Log[c*x^n])^3,x]
[Out]
(-3*b^3*n^3*x^4)/128 + (3*b^2*n^2*x^4*(a + b*Log[c*x^n]))/32 - (3*b*n*x^4*(a + b*Log[c*x^n])^2)/16 + (x^4*(a +
b*Log[c*x^n])^3)/4
Rule 2305
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
Rule 2304
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]
Rubi steps
\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac{1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{3}{128} b^3 n^3 x^4+\frac{3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3\\ \end{align*}
Mathematica [A] time = 0.0292536, size = 66, normalized size = 0.86 \[ \frac{1}{4} \left (x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac{3}{32} b n x^4 \left (-4 b n \left (a+b \log \left (c x^n\right )\right )+8 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(a + b*Log[c*x^n])^3,x]
[Out]
(x^4*(a + b*Log[c*x^n])^3 - (3*b*n*x^4*(b^2*n^2 - 4*b*n*(a + b*Log[c*x^n]) + 8*(a + b*Log[c*x^n])^2))/32)/4
________________________________________________________________________________________
Maple [C] time = 0.318, size = 2649, normalized size = 34.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(a+b*ln(c*x^n))^3,x)
[Out]
1/4*b^3*x^4*ln(x^n)^3+3/16*b^2*x^4*(2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)-2*I*b*Pi*csgn(I*c*x^n)^3+2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+4*b*ln(c)-b*n+4*a)*ln(x^n)^2+3/32*b*x^4*(8
*ln(c)^2*b^2-2*Pi^2*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-4*a*b*n+b^2*n^2+8*a^2-2*I*Pi*b^2*n*csgn(I*c*x^n)^2*csgn(I*
c)+8*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2+8*I*Pi*a*b*csgn(I*c*x^n)^2*csgn(I*c)-2*I*Pi*b^2*n*csgn(I*x^n)*
csgn(I*c*x^n)^2+8*I*ln(c)*Pi*b^2*csgn(I*c*x^n)^2*csgn(I*c)+8*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)^2+4*Pi^2*b^2*c
sgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+4*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-2*Pi^2*b^2*csgn(I*x^
n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-8*Pi^2*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)-8*I*ln(c)*Pi*b^2*csgn(I*c*x^
n)^3-8*I*Pi*a*b*csgn(I*c*x^n)^3+2*I*Pi*b^2*n*csgn(I*c*x^n)^3-8*I*ln(c)*Pi*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)-8*I*Pi*a*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-2*Pi^2*b^2*csgn(I*c*x^n)^6+16*ln(c)*a*b-4*ln(c)*b^2*n+2*I*P
i*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+4*Pi^2*b^2*csgn(I*c*x^n)^5*csgn(I*c)+4*Pi^2*b^2*csgn(I*x^n)*csgn(I
*c*x^n)^5-2*Pi^2*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4)*ln(x^n)+1/128*x^4*(32*a^3+12*a*b^2*n^2-24*a^2*b*n-6*I*Pi*b
^3*n^2*csgn(I*c*x^n)^3-4*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6+12*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^7-
24*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+24*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+32*ln(c)^3
*b^3-96*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+24*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^3+6*I*Pi*b^3*n^2*cs
gn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b^3*n^2*csgn(I*c*x^n)^2*csgn(I*c)+24*I*Pi*a*b^2*n*csgn(I*c*x^n)^3-12*Pi^2*b^3
*n*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+6*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+24*Pi^2*b^3*
n*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+12*I*Pi^3*b^3*csgn(I*c*x^n)^7*csgn(I*c)^2-4*I*Pi^3*b^3*csgn(I*c*x^n)^6
*csgn(I*c)^3-48*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^3-48*I*Pi*a^2*b*csgn(I*c*x^n)^3-24*ln(c)*Pi^2*b^3*csgn(I*x^n)^2
*csgn(I*c*x^n)^4+48*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+48*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^5*csgn(I*c)-12*
I*Pi^3*b^3*csgn(I*c*x^n)^8*csgn(I*c)-12*Pi^2*b^3*n*csgn(I*c*x^n)^5*csgn(I*c)+6*Pi^2*b^3*n*csgn(I*c*x^n)^4*csgn
(I*c)^2-3*b^3*n^3+24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^
6*csgn(I*c)^2+12*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^3+48*I*ln(c)^2*Pi*b^3*csgn(I*x^n)*csgn(I*c*x
^n)^2+48*I*ln(c)^2*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c)-12*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8+48*Pi^2*a*b^2*cs
gn(I*x^n)*csgn(I*c*x^n)^5-24*ln(c)^2*b^3*n+12*ln(c)*b^3*n^2+96*ln(c)*a^2*b+96*ln(c)^2*a*b^2+48*Pi^2*a*b^2*csgn
(I*c*x^n)^5*csgn(I*c)-24*Pi^2*a*b^2*csgn(I*c*x^n)^4*csgn(I*c)^2-24*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^4*csgn(I*c)^2+
6*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4-12*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5+6*Pi^2*b^3*n*csgn(I*c*x^n
)^6+4*I*Pi^3*b^3*csgn(I*c*x^n)^9-24*ln(c)*Pi^2*b^3*csgn(I*c*x^n)^6-24*Pi^2*a*b^2*csgn(I*c*x^n)^6-96*I*ln(c)*Pi
*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-36*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^6*csgn(I*c)+36*I*Pi^3*b^3
*csgn(I*x^n)^2*csgn(I*c*x^n)^5*csgn(I*c)^2-12*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)^3+36*I*Pi^3*b
^3*csgn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)-96*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^3+48*ln(c)*Pi^2*b^3*csgn(I*x^n)*csg
n(I*c*x^n)^3*csgn(I*c)^2+48*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2+48*I*Pi*a^2*b*csgn(I*c*x^n)^2*csgn(I*c)+48*
Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+12*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^5*csgn(I*c)-12*I*
Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^4*csgn(I*c)^2+4*I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^3*csgn(I*c)^3-24*l
n(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-96*ln(c)*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c
)-12*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+48*ln(c)*Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I
*c)+48*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-24*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c
)^2-48*ln(c)*a*b^2*n-24*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^2*csgn(I*c)+96*I*ln(c)*Pi*a*b^2*csgn(I*c*x^n)^2*csgn(I*
c)-48*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*Pi*a*b^2*n*csgn(I*c*x^n)^2*csgn(I*c)-48*I*ln(c)^2*Pi
*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+96*I*ln(c)*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2-6*I*Pi*b^3*n^2*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn
(I*c*x^n)^2)
________________________________________________________________________________________
Maxima [A] time = 1.13463, size = 182, normalized size = 2.36 \begin{align*} \frac{1}{4} \, b^{3} x^{4} \log \left (c x^{n}\right )^{3} + \frac{3}{4} \, a b^{2} x^{4} \log \left (c x^{n}\right )^{2} - \frac{3}{16} \, a^{2} b n x^{4} + \frac{3}{4} \, a^{2} b x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a^{3} x^{4} + \frac{3}{32} \,{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} a b^{2} - \frac{3}{128} \,{\left (8 \, n x^{4} \log \left (c x^{n}\right )^{2} +{\left (n^{2} x^{4} - 4 \, n x^{4} \log \left (c x^{n}\right )\right )} n\right )} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*log(c*x^n))^3,x, algorithm="maxima")
[Out]
1/4*b^3*x^4*log(c*x^n)^3 + 3/4*a*b^2*x^4*log(c*x^n)^2 - 3/16*a^2*b*n*x^4 + 3/4*a^2*b*x^4*log(c*x^n) + 1/4*a^3*
x^4 + 3/32*(n^2*x^4 - 4*n*x^4*log(c*x^n))*a*b^2 - 3/128*(8*n*x^4*log(c*x^n)^2 + (n^2*x^4 - 4*n*x^4*log(c*x^n))
*n)*b^3
________________________________________________________________________________________
Fricas [B] time = 0.866165, size = 517, normalized size = 6.71 \begin{align*} \frac{1}{4} \, b^{3} n^{3} x^{4} \log \left (x\right )^{3} + \frac{1}{4} \, b^{3} x^{4} \log \left (c\right )^{3} - \frac{3}{16} \,{\left (b^{3} n - 4 \, a b^{2}\right )} x^{4} \log \left (c\right )^{2} + \frac{3}{32} \,{\left (b^{3} n^{2} - 4 \, a b^{2} n + 8 \, a^{2} b\right )} x^{4} \log \left (c\right ) - \frac{1}{128} \,{\left (3 \, b^{3} n^{3} - 12 \, a b^{2} n^{2} + 24 \, a^{2} b n - 32 \, a^{3}\right )} x^{4} + \frac{3}{16} \,{\left (4 \, b^{3} n^{2} x^{4} \log \left (c\right ) -{\left (b^{3} n^{3} - 4 \, a b^{2} n^{2}\right )} x^{4}\right )} \log \left (x\right )^{2} + \frac{3}{32} \,{\left (8 \, b^{3} n x^{4} \log \left (c\right )^{2} - 4 \,{\left (b^{3} n^{2} - 4 \, a b^{2} n\right )} x^{4} \log \left (c\right ) +{\left (b^{3} n^{3} - 4 \, a b^{2} n^{2} + 8 \, a^{2} b n\right )} x^{4}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*log(c*x^n))^3,x, algorithm="fricas")
[Out]
1/4*b^3*n^3*x^4*log(x)^3 + 1/4*b^3*x^4*log(c)^3 - 3/16*(b^3*n - 4*a*b^2)*x^4*log(c)^2 + 3/32*(b^3*n^2 - 4*a*b^
2*n + 8*a^2*b)*x^4*log(c) - 1/128*(3*b^3*n^3 - 12*a*b^2*n^2 + 24*a^2*b*n - 32*a^3)*x^4 + 3/16*(4*b^3*n^2*x^4*l
og(c) - (b^3*n^3 - 4*a*b^2*n^2)*x^4)*log(x)^2 + 3/32*(8*b^3*n*x^4*log(c)^2 - 4*(b^3*n^2 - 4*a*b^2*n)*x^4*log(c
) + (b^3*n^3 - 4*a*b^2*n^2 + 8*a^2*b*n)*x^4)*log(x)
________________________________________________________________________________________
Sympy [B] time = 7.59318, size = 338, normalized size = 4.39 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b n x^{4} \log{\left (x \right )}}{4} - \frac{3 a^{2} b n x^{4}}{16} + \frac{3 a^{2} b x^{4} \log{\left (c \right )}}{4} + \frac{3 a b^{2} n^{2} x^{4} \log{\left (x \right )}^{2}}{4} - \frac{3 a b^{2} n^{2} x^{4} \log{\left (x \right )}}{8} + \frac{3 a b^{2} n^{2} x^{4}}{32} + \frac{3 a b^{2} n x^{4} \log{\left (c \right )} \log{\left (x \right )}}{2} - \frac{3 a b^{2} n x^{4} \log{\left (c \right )}}{8} + \frac{3 a b^{2} x^{4} \log{\left (c \right )}^{2}}{4} + \frac{b^{3} n^{3} x^{4} \log{\left (x \right )}^{3}}{4} - \frac{3 b^{3} n^{3} x^{4} \log{\left (x \right )}^{2}}{16} + \frac{3 b^{3} n^{3} x^{4} \log{\left (x \right )}}{32} - \frac{3 b^{3} n^{3} x^{4}}{128} + \frac{3 b^{3} n^{2} x^{4} \log{\left (c \right )} \log{\left (x \right )}^{2}}{4} - \frac{3 b^{3} n^{2} x^{4} \log{\left (c \right )} \log{\left (x \right )}}{8} + \frac{3 b^{3} n^{2} x^{4} \log{\left (c \right )}}{32} + \frac{3 b^{3} n x^{4} \log{\left (c \right )}^{2} \log{\left (x \right )}}{4} - \frac{3 b^{3} n x^{4} \log{\left (c \right )}^{2}}{16} + \frac{b^{3} x^{4} \log{\left (c \right )}^{3}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(a+b*ln(c*x**n))**3,x)
[Out]
a**3*x**4/4 + 3*a**2*b*n*x**4*log(x)/4 - 3*a**2*b*n*x**4/16 + 3*a**2*b*x**4*log(c)/4 + 3*a*b**2*n**2*x**4*log(
x)**2/4 - 3*a*b**2*n**2*x**4*log(x)/8 + 3*a*b**2*n**2*x**4/32 + 3*a*b**2*n*x**4*log(c)*log(x)/2 - 3*a*b**2*n*x
**4*log(c)/8 + 3*a*b**2*x**4*log(c)**2/4 + b**3*n**3*x**4*log(x)**3/4 - 3*b**3*n**3*x**4*log(x)**2/16 + 3*b**3
*n**3*x**4*log(x)/32 - 3*b**3*n**3*x**4/128 + 3*b**3*n**2*x**4*log(c)*log(x)**2/4 - 3*b**3*n**2*x**4*log(c)*lo
g(x)/8 + 3*b**3*n**2*x**4*log(c)/32 + 3*b**3*n*x**4*log(c)**2*log(x)/4 - 3*b**3*n*x**4*log(c)**2/16 + b**3*x**
4*log(c)**3/4
________________________________________________________________________________________
Giac [B] time = 1.26319, size = 354, normalized size = 4.6 \begin{align*} \frac{1}{4} \, b^{3} n^{3} x^{4} \log \left (x\right )^{3} - \frac{3}{16} \, b^{3} n^{3} x^{4} \log \left (x\right )^{2} + \frac{3}{4} \, b^{3} n^{2} x^{4} \log \left (c\right ) \log \left (x\right )^{2} + \frac{3}{32} \, b^{3} n^{3} x^{4} \log \left (x\right ) - \frac{3}{8} \, b^{3} n^{2} x^{4} \log \left (c\right ) \log \left (x\right ) + \frac{3}{4} \, b^{3} n x^{4} \log \left (c\right )^{2} \log \left (x\right ) + \frac{3}{4} \, a b^{2} n^{2} x^{4} \log \left (x\right )^{2} - \frac{3}{128} \, b^{3} n^{3} x^{4} + \frac{3}{32} \, b^{3} n^{2} x^{4} \log \left (c\right ) - \frac{3}{16} \, b^{3} n x^{4} \log \left (c\right )^{2} + \frac{1}{4} \, b^{3} x^{4} \log \left (c\right )^{3} - \frac{3}{8} \, a b^{2} n^{2} x^{4} \log \left (x\right ) + \frac{3}{2} \, a b^{2} n x^{4} \log \left (c\right ) \log \left (x\right ) + \frac{3}{32} \, a b^{2} n^{2} x^{4} - \frac{3}{8} \, a b^{2} n x^{4} \log \left (c\right ) + \frac{3}{4} \, a b^{2} x^{4} \log \left (c\right )^{2} + \frac{3}{4} \, a^{2} b n x^{4} \log \left (x\right ) - \frac{3}{16} \, a^{2} b n x^{4} + \frac{3}{4} \, a^{2} b x^{4} \log \left (c\right ) + \frac{1}{4} \, a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*log(c*x^n))^3,x, algorithm="giac")
[Out]
1/4*b^3*n^3*x^4*log(x)^3 - 3/16*b^3*n^3*x^4*log(x)^2 + 3/4*b^3*n^2*x^4*log(c)*log(x)^2 + 3/32*b^3*n^3*x^4*log(
x) - 3/8*b^3*n^2*x^4*log(c)*log(x) + 3/4*b^3*n*x^4*log(c)^2*log(x) + 3/4*a*b^2*n^2*x^4*log(x)^2 - 3/128*b^3*n^
3*x^4 + 3/32*b^3*n^2*x^4*log(c) - 3/16*b^3*n*x^4*log(c)^2 + 1/4*b^3*x^4*log(c)^3 - 3/8*a*b^2*n^2*x^4*log(x) +
3/2*a*b^2*n*x^4*log(c)*log(x) + 3/32*a*b^2*n^2*x^4 - 3/8*a*b^2*n*x^4*log(c) + 3/4*a*b^2*x^4*log(c)^2 + 3/4*a^2
*b*n*x^4*log(x) - 3/16*a^2*b*n*x^4 + 3/4*a^2*b*x^4*log(c) + 1/4*a^3*x^4