3.63 \(\int \frac {(a+b \log (c x^n))^3}{x^3} \, dx\)

Optimal. Leaf size=77 \[ -\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3}{8 x^2} \]

[Out]

-3/8*b^3*n^3/x^2-3/4*b^2*n^2*(a+b*ln(c*x^n))/x^2-3/4*b*n*(a+b*ln(c*x^n))^2/x^2-1/2*(a+b*ln(c*x^n))^3/x^2

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Rubi [A]  time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3/x^3,x]

[Out]

(-3*b^3*n^3)/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n]))/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2)/(4*x^2) - (a + b*
Log[c*x^n])^3/(2*x^2)

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
]

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {1}{2} (3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\\ &=-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {1}{2} \left (3 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac {3 b^3 n^3}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.78 \[ -\frac {4 \left (a+b \log \left (c x^n\right )\right )^3+3 b n \left (2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+2 b \log \left (c x^n\right )+b n\right )\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3/x^3,x]

[Out]

-1/8*(4*(a + b*Log[c*x^n])^3 + 3*b*n*(2*(a + b*Log[c*x^n])^2 + b*n*(2*a + b*n + 2*b*Log[c*x^n])))/x^2

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fricas [B]  time = 0.42, size = 189, normalized size = 2.45 \[ -\frac {4 \, b^{3} n^{3} \log \relax (x)^{3} + 3 \, b^{3} n^{3} + 4 \, b^{3} \log \relax (c)^{3} + 6 \, a b^{2} n^{2} + 6 \, a^{2} b n + 4 \, a^{3} + 6 \, {\left (b^{3} n + 2 \, a b^{2}\right )} \log \relax (c)^{2} + 6 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \relax (c) + 2 \, a b^{2} n^{2}\right )} \log \relax (x)^{2} + 6 \, {\left (b^{3} n^{2} + 2 \, a b^{2} n + 2 \, a^{2} b\right )} \log \relax (c) + 6 \, {\left (b^{3} n^{3} + 2 \, b^{3} n \log \relax (c)^{2} + 2 \, a b^{2} n^{2} + 2 \, a^{2} b n + 2 \, {\left (b^{3} n^{2} + 2 \, a b^{2} n\right )} \log \relax (c)\right )} \log \relax (x)}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^3,x, algorithm="fricas")

[Out]

-1/8*(4*b^3*n^3*log(x)^3 + 3*b^3*n^3 + 4*b^3*log(c)^3 + 6*a*b^2*n^2 + 6*a^2*b*n + 4*a^3 + 6*(b^3*n + 2*a*b^2)*
log(c)^2 + 6*(b^3*n^3 + 2*b^3*n^2*log(c) + 2*a*b^2*n^2)*log(x)^2 + 6*(b^3*n^2 + 2*a*b^2*n + 2*a^2*b)*log(c) +
6*(b^3*n^3 + 2*b^3*n*log(c)^2 + 2*a*b^2*n^2 + 2*a^2*b*n + 2*(b^3*n^2 + 2*a*b^2*n)*log(c))*log(x))/x^2

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giac [B]  time = 0.35, size = 203, normalized size = 2.64 \[ -\frac {b^{3} n^{3} \log \relax (x)^{3}}{2 \, x^{2}} - \frac {3 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \relax (c) + 2 \, a b^{2} n^{2}\right )} \log \relax (x)^{2}}{4 \, x^{2}} - \frac {3 \, {\left (b^{3} n^{3} + 2 \, b^{3} n^{2} \log \relax (c) + 2 \, b^{3} n \log \relax (c)^{2} + 2 \, a b^{2} n^{2} + 4 \, a b^{2} n \log \relax (c) + 2 \, a^{2} b n\right )} \log \relax (x)}{4 \, x^{2}} - \frac {3 \, b^{3} n^{3} + 6 \, b^{3} n^{2} \log \relax (c) + 6 \, b^{3} n \log \relax (c)^{2} + 4 \, b^{3} \log \relax (c)^{3} + 6 \, a b^{2} n^{2} + 12 \, a b^{2} n \log \relax (c) + 12 \, a b^{2} \log \relax (c)^{2} + 6 \, a^{2} b n + 12 \, a^{2} b \log \relax (c) + 4 \, a^{3}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^3,x, algorithm="giac")

[Out]

-1/2*b^3*n^3*log(x)^3/x^2 - 3/4*(b^3*n^3 + 2*b^3*n^2*log(c) + 2*a*b^2*n^2)*log(x)^2/x^2 - 3/4*(b^3*n^3 + 2*b^3
*n^2*log(c) + 2*b^3*n*log(c)^2 + 2*a*b^2*n^2 + 4*a*b^2*n*log(c) + 2*a^2*b*n)*log(x)/x^2 - 1/8*(3*b^3*n^3 + 6*b
^3*n^2*log(c) + 6*b^3*n*log(c)^2 + 4*b^3*log(c)^3 + 6*a*b^2*n^2 + 12*a*b^2*n*log(c) + 12*a*b^2*log(c)^2 + 6*a^
2*b*n + 12*a^2*b*log(c) + 4*a^3)/x^2

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maple [C]  time = 0.29, size = 2673, normalized size = 34.71 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^3/x^3,x)

[Out]

-1/2*b^3/x^2*ln(x^n)^3-3/4*(I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-
I*Pi*b^3*csgn(I*c*x^n)^3+I*Pi*b^3*csgn(I*c)*csgn(I*c*x^n)^2+2*b^3*ln(c)+b^3*n+2*a*b^2)/x^2*ln(x^n)^2-3/8*(-4*I
*Pi*a*b^2*csgn(I*c*x^n)^3+2*Pi^2*b^3*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-Pi^2*b^3*csgn(I*c)^2*csgn(I*x^n)^
2*csgn(I*c*x^n)^2+4*a^2*b+4*b^3*ln(c)^2-Pi^2*b^3*csgn(I*c)^2*csgn(I*c*x^n)^4-Pi^2*b^3*csgn(I*x^n)^2*csgn(I*c*x
^n)^4+2*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5+2*Pi^2*b^3*csgn(I*c)*csgn(I*c*x^n)^5+4*a*b^2*n+8*a*b^2*ln(c)+4*b^
3*n*ln(c)-Pi^2*b^3*csgn(I*c*x^n)^6+2*b^3*n^2-2*I*n*Pi*b^3*csgn(I*c*x^n)^3-4*Pi^2*b^3*csgn(I*c)*csgn(I*x^n)*csg
n(I*c*x^n)^4+2*Pi^2*b^3*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-4*I*Pi*b^3*csgn(I*c*x^n)^3*ln(c)+4*I*Pi*b^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2*ln(c)+4*I*Pi*b^3*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)+4*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x
^n)^2+4*I*Pi*a*b^2*csgn(I*c)*csgn(I*c*x^n)^2-2*I*n*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*a*b^2*csg
n(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-4*I*Pi*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(c)+2*I*n*Pi*b^3*csgn(I*c*x^
n)^2*csgn(I*c)+2*I*n*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2)/x^2*ln(x^n)-1/16*(8*a^3-24*I*Pi*a*b^2*csgn(I*c)*csgn(
I*x^n)*csgn(I*c*x^n)*ln(c)-6*Pi^2*a*b^2*csgn(I*c*x^n)^6-6*Pi^2*b^3*csgn(I*c*x^n)^6*ln(c)+24*a*b^2*n*ln(c)+8*b^
3*ln(c)^3+24*a*b^2*ln(c)^2+24*a^2*b*ln(c)+12*b^3*n*ln(c)^2+12*b^3*n^2*ln(c)+I*Pi^3*b^3*csgn(I*c*x^n)^9+12*a*b^
2*n^2+12*a^2*b*n-3*Pi^2*b^3*n*csgn(I*c*x^n)^6+6*b^3*n^3-12*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-12
*I*n*ln(c)*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+24*I*Pi*a*b^2*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)-12*I*Pi*a^
2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-24*I*Pi*a*b^2*csgn(I*c*x^n)^3*ln(c)+12*I*Pi*a^2*b*csgn(I*x^n)*csgn(I*c
*x^n)^2-3*Pi^2*b^3*n*csgn(I*x^n)^2*csgn(I*c*x^n)^4+6*Pi^2*b^3*n*csgn(I*c)*csgn(I*c*x^n)^5-3*Pi^2*b^3*n*csgn(I*
c)^2*csgn(I*c*x^n)^4+6*Pi^2*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^5+9*I*Pi^3*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)
^7-9*I*Pi^3*b^3*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^6+3*I*Pi^3*b^3*csgn(I*c)^3*csgn(I*x^n)*csgn(I*c*x^n)^5+1
2*I*Pi*b^3*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(c)^2+12*I*Pi*b^3*csgn(I*c)*csgn(I*c*x^n)^2*ln(c)^2-6*Pi^2*b^3*csgn(I
*c)^2*csgn(I*c*x^n)^4*ln(c)-6*Pi^2*a*b^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+12*Pi^2*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)
^5+12*Pi^2*a*b^2*csgn(I*c)*csgn(I*c*x^n)^5-6*Pi^2*a*b^2*csgn(I*c)^2*csgn(I*c*x^n)^4-6*Pi^2*b^3*csgn(I*x^n)^2*c
sgn(I*c*x^n)^4*ln(c)+12*Pi^2*b^3*csgn(I*x^n)*csgn(I*c*x^n)^5*ln(c)+12*Pi^2*b^3*csgn(I*c)*csgn(I*c*x^n)^5*ln(c)
-12*I*Pi*a^2*b*csgn(I*c*x^n)^3-I*Pi^3*b^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6+3*I*Pi^3*b^3*csgn(I*x^n)^2*csgn(I*c*x^
n)^7-3*I*Pi^3*b^3*csgn(I*x^n)*csgn(I*c*x^n)^8-3*I*Pi^3*b^3*csgn(I*c)*csgn(I*c*x^n)^8+3*I*Pi^3*b^3*csgn(I*c)^2*
csgn(I*c*x^n)^7-I*Pi^3*b^3*csgn(I*c)^3*csgn(I*c*x^n)^6-12*I*Pi*b^3*csgn(I*c*x^n)^3*ln(c)^2-6*I*Pi*b^3*n^2*csgn
(I*c*x^n)^3-12*I*ln(c)*Pi*b^3*n*csgn(I*c*x^n)^3+12*I*Pi*a*b^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2+12*I*Pi*a*b^2*n*cs
gn(I*c*x^n)^2*csgn(I*c)+12*I*ln(c)*Pi*b^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-12*I*Pi*b^3*csgn(I*c)*csgn(I*x^n)*csgn
(I*c*x^n)*ln(c)^2+24*I*Pi*a*b^2*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(c)+12*I*n*ln(c)*Pi*b^3*csgn(I*c*x^n)^2*csgn(I*c
)+12*Pi^2*a*b^2*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3+6*Pi^2*b^3*n*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-3
*Pi^2*b^3*n*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2-12*Pi^2*b^3*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+6*Pi
^2*b^3*n*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3-6*Pi^2*b^3*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*ln(c)-24
*Pi^2*b^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4*ln(c)+12*Pi^2*b^3*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3*ln(c
)+12*Pi^2*a*b^2*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3-6*Pi^2*a*b^2*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2
-24*Pi^2*a*b^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)^4+12*Pi^2*b^3*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3*ln(c)
+12*I*Pi*a^2*b*csgn(I*c)*csgn(I*c*x^n)^2+I*Pi^3*b^3*csgn(I*c)^3*csgn(I*x^n)^3*csgn(I*c*x^n)^3+3*I*Pi^3*b^3*csg
n(I*c)*csgn(I*x^n)^3*csgn(I*c*x^n)^5-3*I*Pi^3*b^3*csgn(I*c)^2*csgn(I*x^n)^3*csgn(I*c*x^n)^4-9*I*Pi^3*b^3*csgn(
I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^6+9*I*Pi^3*b^3*csgn(I*c)^2*csgn(I*x^n)^2*csgn(I*c*x^n)^5-3*I*Pi^3*b^3*csgn(I*
c)^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+6*I*Pi*b^3*n^2*csgn(I*x^n)*csgn(I*c*x^n)^2+6*I*Pi*b^3*n^2*csgn(I*c)*csgn(I*
c*x^n)^2-6*I*Pi*b^3*n^2*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-12*I*Pi*a*b^2*n*csgn(I*c*x^n)^3)/x^2

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maxima [A]  time = 0.60, size = 135, normalized size = 1.75 \[ -\frac {3}{8} \, {\left (n {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} + \frac {2 \, n \log \left (c x^{n}\right )^{2}}{x^{2}}\right )} b^{3} - \frac {3}{4} \, a b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac {b^{3} \log \left (c x^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b n}{4 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^3/x^3,x, algorithm="maxima")

[Out]

-3/8*(n*(n^2/x^2 + 2*n*log(c*x^n)/x^2) + 2*n*log(c*x^n)^2/x^2)*b^3 - 3/4*a*b^2*(n^2/x^2 + 2*n*log(c*x^n)/x^2)
- 1/2*b^3*log(c*x^n)^3/x^2 - 3/2*a*b^2*log(c*x^n)^2/x^2 - 3/4*a^2*b*n/x^2 - 3/2*a^2*b*log(c*x^n)/x^2 - 1/2*a^3
/x^2

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mupad [B]  time = 3.68, size = 111, normalized size = 1.44 \[ -\frac {\frac {a^3}{2}+\frac {3\,a^2\,b\,n}{4}+\frac {3\,a\,b^2\,n^2}{4}+\frac {3\,b^3\,n^3}{8}}{x^2}-\frac {\ln \left (c\,x^n\right )\,\left (3\,a^2\,b+3\,a\,b^2\,n+\frac {3\,b^3\,n^2}{2}\right )}{2\,x^2}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {3\,n\,b^3}{2}+3\,a\,b^2\right )}{2\,x^2}-\frac {b^3\,{\ln \left (c\,x^n\right )}^3}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))^3/x^3,x)

[Out]

- (a^3/2 + (3*b^3*n^3)/8 + (3*a*b^2*n^2)/4 + (3*a^2*b*n)/4)/x^2 - (log(c*x^n)*(3*a^2*b + (3*b^3*n^2)/2 + 3*a*b
^2*n))/(2*x^2) - (log(c*x^n)^2*(3*a*b^2 + (3*b^3*n)/2))/(2*x^2) - (b^3*log(c*x^n)^3)/(2*x^2)

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sympy [B]  time = 1.16, size = 338, normalized size = 4.39 \[ - \frac {a^{3}}{2 x^{2}} - \frac {3 a^{2} b n \log {\relax (x )}}{2 x^{2}} - \frac {3 a^{2} b n}{4 x^{2}} - \frac {3 a^{2} b \log {\relax (c )}}{2 x^{2}} - \frac {3 a b^{2} n^{2} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {3 a b^{2} n^{2} \log {\relax (x )}}{2 x^{2}} - \frac {3 a b^{2} n^{2}}{4 x^{2}} - \frac {3 a b^{2} n \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {3 a b^{2} n \log {\relax (c )}}{2 x^{2}} - \frac {3 a b^{2} \log {\relax (c )}^{2}}{2 x^{2}} - \frac {b^{3} n^{3} \log {\relax (x )}^{3}}{2 x^{2}} - \frac {3 b^{3} n^{3} \log {\relax (x )}^{2}}{4 x^{2}} - \frac {3 b^{3} n^{3} \log {\relax (x )}}{4 x^{2}} - \frac {3 b^{3} n^{3}}{8 x^{2}} - \frac {3 b^{3} n^{2} \log {\relax (c )} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {3 b^{3} n^{2} \log {\relax (c )} \log {\relax (x )}}{2 x^{2}} - \frac {3 b^{3} n^{2} \log {\relax (c )}}{4 x^{2}} - \frac {3 b^{3} n \log {\relax (c )}^{2} \log {\relax (x )}}{2 x^{2}} - \frac {3 b^{3} n \log {\relax (c )}^{2}}{4 x^{2}} - \frac {b^{3} \log {\relax (c )}^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**3/x**3,x)

[Out]

-a**3/(2*x**2) - 3*a**2*b*n*log(x)/(2*x**2) - 3*a**2*b*n/(4*x**2) - 3*a**2*b*log(c)/(2*x**2) - 3*a*b**2*n**2*l
og(x)**2/(2*x**2) - 3*a*b**2*n**2*log(x)/(2*x**2) - 3*a*b**2*n**2/(4*x**2) - 3*a*b**2*n*log(c)*log(x)/x**2 - 3
*a*b**2*n*log(c)/(2*x**2) - 3*a*b**2*log(c)**2/(2*x**2) - b**3*n**3*log(x)**3/(2*x**2) - 3*b**3*n**3*log(x)**2
/(4*x**2) - 3*b**3*n**3*log(x)/(4*x**2) - 3*b**3*n**3/(8*x**2) - 3*b**3*n**2*log(c)*log(x)**2/(2*x**2) - 3*b**
3*n**2*log(c)*log(x)/(2*x**2) - 3*b**3*n**2*log(c)/(4*x**2) - 3*b**3*n*log(c)**2*log(x)/(2*x**2) - 3*b**3*n*lo
g(c)**2/(4*x**2) - b**3*log(c)**3/(2*x**2)

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