3.156 \(\int (d x)^{-1+n} \log ^3(c x^n) \, dx\)
Optimal. Leaf size=74 \[ \frac {(d x)^n \log ^3\left (c x^n\right )}{d n}-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {6 (d x)^n \log \left (c x^n\right )}{d n}-\frac {6 (d x)^n}{d n} \]
[Out]
-6*(d*x)^n/d/n+6*(d*x)^n*ln(c*x^n)/d/n-3*(d*x)^n*ln(c*x^n)^2/d/n+(d*x)^n*ln(c*x^n)^3/d/n
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Rubi [A] time = 0.05, antiderivative size = 74, 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} \[ \frac {(d x)^n \log ^3\left (c x^n\right )}{d n}-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {6 (d x)^n \log \left (c x^n\right )}{d n}-\frac {6 (d x)^n}{d n} \]
Antiderivative was successfully verified.
[In]
Int[(d*x)^(-1 + n)*Log[c*x^n]^3,x]
[Out]
(-6*(d*x)^n)/(d*n) + (6*(d*x)^n*Log[c*x^n])/(d*n) - (3*(d*x)^n*Log[c*x^n]^2)/(d*n) + ((d*x)^n*Log[c*x^n]^3)/(d
*n)
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 (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx &=\frac {(d x)^n \log ^3\left (c x^n\right )}{d n}-3 \int (d x)^{-1+n} \log ^2\left (c x^n\right ) \, dx\\ &=-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {(d x)^n \log ^3\left (c x^n\right )}{d n}+6 \int (d x)^{-1+n} \log \left (c x^n\right ) \, dx\\ &=-\frac {6 (d x)^n}{d n}+\frac {6 (d x)^n \log \left (c x^n\right )}{d n}-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {(d x)^n \log ^3\left (c x^n\right )}{d n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.54 \[ \frac {(d x)^n \left (\log ^3\left (c x^n\right )-3 \log ^2\left (c x^n\right )+6 \log \left (c x^n\right )-6\right )}{d n} \]
Antiderivative was successfully verified.
[In]
Integrate[(d*x)^(-1 + n)*Log[c*x^n]^3,x]
[Out]
((d*x)^n*(-6 + 6*Log[c*x^n] - 3*Log[c*x^n]^2 + Log[c*x^n]^3))/(d*n)
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fricas [A] time = 0.44, size = 73, normalized size = 0.99 \[ \frac {{\left (n^{3} \log \relax (x)^{3} + \log \relax (c)^{3} + 3 \, {\left (n^{2} \log \relax (c) - n^{2}\right )} \log \relax (x)^{2} - 3 \, \log \relax (c)^{2} + 3 \, {\left (n \log \relax (c)^{2} - 2 \, n \log \relax (c) + 2 \, n\right )} \log \relax (x) + 6 \, \log \relax (c) - 6\right )} d^{n - 1} x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^(-1+n)*log(c*x^n)^3,x, algorithm="fricas")
[Out]
(n^3*log(x)^3 + log(c)^3 + 3*(n^2*log(c) - n^2)*log(x)^2 - 3*log(c)^2 + 3*(n*log(c)^2 - 2*n*log(c) + 2*n)*log(
x) + 6*log(c) - 6)*d^(n - 1)*x^n/n
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giac [B] time = 0.53, size = 170, normalized size = 2.30 \[ \frac {d^{n} n^{2} x^{n} \log \relax (x)^{3}}{d} + \frac {3 \, d^{n} n x^{n} \log \relax (c) \log \relax (x)^{2}}{d} + \frac {\frac {1}{d}^{n} x^{n} {\left | d \right |}^{2 \, n} \log \relax (c)^{3}}{d n} + \frac {3 \, d^{n} x^{n} \log \relax (c)^{2} \log \relax (x)}{d} - \frac {3 \, d^{n} n x^{n} \log \relax (x)^{2}}{d} - \frac {6 \, d^{n} x^{n} \log \relax (c) \log \relax (x)}{d} - \frac {3 \, d^{n} x^{n} \log \relax (c)^{2}}{d n} + \frac {6 \, d^{n} x^{n} \log \relax (x)}{d} + \frac {6 \, d^{n} x^{n} \log \relax (c)}{d n} - \frac {6 \, d^{n} x^{n}}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^(-1+n)*log(c*x^n)^3,x, algorithm="giac")
[Out]
d^n*n^2*x^n*log(x)^3/d + 3*d^n*n*x^n*log(c)*log(x)^2/d + (1/d)^n*x^n*abs(d)^(2*n)*log(c)^3/(d*n) + 3*d^n*x^n*l
og(c)^2*log(x)/d - 3*d^n*n*x^n*log(x)^2/d - 6*d^n*x^n*log(c)*log(x)/d - 3*d^n*x^n*log(c)^2/(d*n) + 6*d^n*x^n*l
og(x)/d + 6*d^n*x^n*log(c)/(d*n) - 6*d^n*x^n/(d*n)
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maple [C] time = 0.30, size = 2008, normalized size = 27.14 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^(n-1)*ln(c*x^n)^3,x)
[Out]
1/n*x*exp(1/2*(n-1)*(-I*Pi*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d*x)^2+I*Pi*csgn(I*x)*csgn(I*
d*x)^2-I*Pi*csgn(I*d*x)^3+2*ln(d)+2*ln(x)))*ln(x^n)^3+3/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*c)*csg
n(I*x^n)*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*ln(c)-2)/n*x*exp(1/2*(n-1)*(-I*Pi
*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d*x)^2+I*Pi*csgn(I*x)*csgn(I*d*x)^2-I*Pi*csgn(I*d*x)^3+
2*ln(d)+2*ln(x)))*ln(x^n)^2+3/4*(-Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn
(I*c)-Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+2*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^5-4*Pi^2*csgn(I*x^n)*csg
n(I*c*x^n)^4*csgn(I*c)+2*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-Pi^2*csgn(I*c*x^n)^6+2*Pi^2*csgn(I*c*x^n
)^5*csgn(I*c)-Pi^2*csgn(I*c*x^n)^4*csgn(I*c)^2+4*I*Pi*csgn(I*c*x^n)^3-4*I*ln(c)*Pi*csgn(I*c*x^n)^3+4*I*ln(c)*P
i*csgn(I*c*x^n)^2*csgn(I*c)-4*I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+4*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2+4*ln(c)^2-8*ln(c)+8)/n*x*exp(1/2*(n-1)*(-I*Pi*csgn(I*d)*csgn(I*x)*csgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d
*x)^2+I*Pi*csgn(I*x)*csgn(I*d*x)^2-I*Pi*csgn(I*d*x)^3+2*ln(d)+2*ln(x)))*ln(x^n)+1/8*(-48-12*Pi^2*csgn(I*c*x^n)
^5*csgn(I*c)+6*Pi^2*csgn(I*c*x^n)^4*csgn(I*c)^2-12*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^5+6*Pi^2*csgn(I*x^n)^2*csgn(
I*c*x^n)^4+6*Pi^2*csgn(I*c*x^n)^6+I*Pi^3*csgn(I*c*x^n)^9-24*I*Pi*csgn(I*c*x^n)^3-6*ln(c)*Pi^2*csgn(I*c*x^n)^6+
48*ln(c)-24*ln(c)^2+24*I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+24*I*ln(c)*Pi*csgn(I*c*x^n)^3-12*I*ln(c)^2*Pi*csgn(I*c*x
^n)^3-I*Pi^3*csgn(I*x^n)^3*csgn(I*c*x^n)^6+3*I*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^7-3*I*Pi^3*csgn(I*x^n)*csgn(I*
c*x^n)^8-6*ln(c)*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+12*ln(c)*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^5+12*ln(c)*Pi^2*cs
gn(I*c*x^n)^5*csgn(I*c)-6*ln(c)*Pi^2*csgn(I*c*x^n)^4*csgn(I*c)^2+24*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-12*Pi^2*c
sgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)+6*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+24*Pi^2*csgn(I*x^n)*cs
gn(I*c*x^n)^4*csgn(I*c)-12*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+8*ln(c)^3-3*I*Pi^3*csgn(I*c*x^n)^8*csg
n(I*c)+3*I*Pi^3*csgn(I*c*x^n)^7*csgn(I*c)^2-I*Pi^3*csgn(I*c*x^n)^6*csgn(I*c)^3-3*I*Pi^3*csgn(I*x^n)^3*csgn(I*c
*x^n)^4*csgn(I*c)^2-9*I*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^6*csgn(I*c)+9*I*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*cs
gn(I*c)^2-3*I*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)^3+24*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
-12*I*ln(c)^2*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+9*I*Pi^3*csgn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)-9*I*Pi^3*c
sgn(I*x^n)*csgn(I*c*x^n)^6*csgn(I*c)^2+3*I*Pi^3*csgn(I*x^n)*csgn(I*c*x^n)^5*csgn(I*c)^3+12*I*ln(c)^2*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2-24*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*ln(c)*Pi*csgn(I*c*x^n)^2*csgn(I*c)+12*I*l
n(c)^2*Pi*csgn(I*c*x^n)^2*csgn(I*c)+3*I*Pi^3*csgn(I*x^n)^3*csgn(I*c*x^n)^5*csgn(I*c)+I*Pi^3*csgn(I*x^n)^3*csgn
(I*c*x^n)^3*csgn(I*c)^3+12*ln(c)*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-6*ln(c)*Pi^2*csgn(I*x^n)^2*csgn(
I*c*x^n)^2*csgn(I*c)^2-24*ln(c)*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+12*ln(c)*Pi^2*csgn(I*x^n)*csgn(I*c*
x^n)^3*csgn(I*c)^2-24*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c))/n*x*exp(1/2*(n-1)*(-I*Pi*csgn(I*d)*csgn(I*x)*c
sgn(I*d*x)+I*Pi*csgn(I*d)*csgn(I*d*x)^2+I*Pi*csgn(I*x)*csgn(I*d*x)^2-I*Pi*csgn(I*d*x)^3+2*ln(d)+2*ln(x)))
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maxima [A] time = 0.59, size = 75, normalized size = 1.01 \[ -\frac {3 \, d^{n - 1} x^{n} \log \left (c x^{n}\right )^{2}}{n} + \frac {\left (d x\right )^{n} \log \left (c x^{n}\right )^{3}}{d n} + \frac {6 \, {\left (\frac {d^{n} x^{n} \log \left (c x^{n}\right )}{n} - \frac {d^{n} x^{n}}{n}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)^(-1+n)*log(c*x^n)^3,x, algorithm="maxima")
[Out]
-3*d^(n - 1)*x^n*log(c*x^n)^2/n + (d*x)^n*log(c*x^n)^3/(d*n) + 6*(d^n*x^n*log(c*x^n)/n - d^n*x^n/n)/d
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left (c\,x^n\right )}^3\,{\left (d\,x\right )}^{n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(log(c*x^n)^3*(d*x)^(n - 1),x)
[Out]
int(log(c*x^n)^3*(d*x)^(n - 1), x)
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sympy [A] time = 123.18, size = 292, normalized size = 3.95 \[ \begin {cases} \tilde {\infty } x \log {\relax (c )}^{3} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\log {\relax (c )}^{3} \log {\relax (x )}}{d} & \text {for}\: n = 0 \\0^{n - 1} \left (n^{3} x \log {\relax (x )}^{3} - 3 n^{3} x \log {\relax (x )}^{2} + 6 n^{3} x \log {\relax (x )} - 6 n^{3} x + 3 n^{2} x \log {\relax (c )} \log {\relax (x )}^{2} - 6 n^{2} x \log {\relax (c )} \log {\relax (x )} + 6 n^{2} x \log {\relax (c )} + 3 n x \log {\relax (c )}^{2} \log {\relax (x )} - 3 n x \log {\relax (c )}^{2} + x \log {\relax (c )}^{3}\right ) & \text {for}\: d = 0 \\\frac {d^{n} n^{2} x^{n} \log {\relax (x )}^{3}}{d} + \frac {3 d^{n} n x^{n} \log {\relax (c )} \log {\relax (x )}^{2}}{d} - \frac {3 d^{n} n x^{n} \log {\relax (x )}^{2}}{d} + \frac {3 d^{n} x^{n} \log {\relax (c )}^{2} \log {\relax (x )}}{d} - \frac {6 d^{n} x^{n} \log {\relax (c )} \log {\relax (x )}}{d} + \frac {6 d^{n} x^{n} \log {\relax (x )}}{d} + \frac {d^{n} x^{n} \log {\relax (c )}^{3}}{d n} - \frac {3 d^{n} x^{n} \log {\relax (c )}^{2}}{d n} + \frac {6 d^{n} x^{n} \log {\relax (c )}}{d n} - \frac {6 d^{n} x^{n}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**(-1+n)*ln(c*x**n)**3,x)
[Out]
Piecewise((zoo*x*log(c)**3, Eq(d, 0) & Eq(n, 0)), (log(c)**3*log(x)/d, Eq(n, 0)), (0**(n - 1)*(n**3*x*log(x)**
3 - 3*n**3*x*log(x)**2 + 6*n**3*x*log(x) - 6*n**3*x + 3*n**2*x*log(c)*log(x)**2 - 6*n**2*x*log(c)*log(x) + 6*n
**2*x*log(c) + 3*n*x*log(c)**2*log(x) - 3*n*x*log(c)**2 + x*log(c)**3), Eq(d, 0)), (d**n*n**2*x**n*log(x)**3/d
+ 3*d**n*n*x**n*log(c)*log(x)**2/d - 3*d**n*n*x**n*log(x)**2/d + 3*d**n*x**n*log(c)**2*log(x)/d - 6*d**n*x**n
*log(c)*log(x)/d + 6*d**n*x**n*log(x)/d + d**n*x**n*log(c)**3/(d*n) - 3*d**n*x**n*log(c)**2/(d*n) + 6*d**n*x**
n*log(c)/(d*n) - 6*d**n*x**n/(d*n), True))
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