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*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)
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Rubi [A] time = 0.0531129, antiderivative size = 74, 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} \[ \frac{(d x)^n \log ^3\left (c x^n\right )}{d n}-\frac{3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac{6 (d x)^n \log \left (c x^n\right )}{d n}-\frac{6 (d x)^n}{d n} \]
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 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 (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*}
Mathematica [A] time = 0.0076953, 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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Maple [C] time = 0.257, size = 2008, normalized size = 27.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^(-1+n)*ln(c*x^n)^3,x)
[Out]
1/n*x*exp(1/2*(-1+n)*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d
*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)^3+3/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+2*ln(c)-2)/n*x*exp(1/2*(-1+n)*(-I*
csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*P
i+2*ln(x)+2*ln(d)))*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*cs
gn(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)*c
sgn(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)^2*csgn(I*c)+4*I*ln(c)*Pi*csgn(I*c*x^n)^2
*csgn(I*c)-4*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+4*I*Pi*csgn(I*c*x^n)^3+4*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-
4*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*ln(c)*Pi*csgn(I*c*x^n)^3+4*I*Pi*csgn(I*x^n)*csgn(I*c*x^n)
*csgn(I*c)+4*ln(c)^2-8*ln(c)+8)/n*x*exp(1/2*(-1+n)*(-I*csgn(I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*
d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))*ln(x^n)+1/8*(-48+I*Pi^3*csgn(I*x^n)
^3*csgn(I*c*x^n)^3*csgn(I*c)^3+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)^8-3*I
*Pi^3*csgn(I*c*x^n)^8*csgn(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)^3-12*I*l
n(c)^2*Pi*csgn(I*c*x^n)^3+6*Pi^2*csgn(I*c*x^n)^4*csgn(I*c)^2-24*ln(c)^2-12*Pi^2*csgn(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)*csgn(I*c*x^n)^4*csgn(I*c)-12*Pi
^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-6*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*csgn(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*csg
n(I*x^n)*csgn(I*c*x^n)^2+24*I*Pi*csgn(I*c*x^n)^2*csgn(I*c)+24*I*ln(c)*Pi*csgn(I*c*x^n)^3-I*Pi^3*csgn(I*x^n)^3*
csgn(I*c*x^n)^6+6*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4-12*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^5+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+48*ln(c)-12*Pi^2*csgn(
I*c*x^n)^5*csgn(I*c)-24*I*Pi*csgn(I*c*x^n)^3+I*Pi^3*csgn(I*c*x^n)^9-6*ln(c)*Pi^2*csgn(I*c*x^n)^6+8*ln(c)^3+9*I
*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^5*csgn(I*c)^2-3*I*Pi^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4*csgn(I*c)^3+9*I*Pi^3*cs
gn(I*x^n)*csgn(I*c*x^n)^7*csgn(I*c)+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)-24*I*ln(c)*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*csg
n(I*c)-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)-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)-24*I*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+6*Pi^2*csgn(I*c*x^n)^6-9*I*Pi^3
*csgn(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+12*I*ln(c)^2*Pi*csgn(I*c*x^n)^2*csgn(I*c))/n*x*exp(1/2*(-1+n)*(-I*csgn(I*d*x)^3*Pi+I*c
sgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*ln(x)+2*ln(d)))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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]
Exception raised: ValueError
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Fricas [A] time = 0.995372, size = 197, normalized size = 2.66 \begin{align*} \frac{{\left (n^{3} \log \left (x\right )^{3} + \log \left (c\right )^{3} + 3 \,{\left (n^{2} \log \left (c\right ) - n^{2}\right )} \log \left (x\right )^{2} - 3 \, \log \left (c\right )^{2} + 3 \,{\left (n \log \left (c\right )^{2} - 2 \, n \log \left (c\right ) + 2 \, n\right )} \log \left (x\right ) + 6 \, \log \left (c\right ) - 6\right )} d^{n - 1} x^{n}}{n} \end{align*}
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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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**(-1+n)*ln(c*x**n)**3,x)
[Out]
Timed out
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Giac [B] time = 1.26781, size = 219, normalized size = 2.96 \begin{align*} \frac{d^{n} n^{2} x^{n} \log \left (x\right )^{3}}{d} + \frac{3 \, d^{n} n x^{n} \log \left (c\right ) \log \left (x\right )^{2}}{d} + \frac{3 \, d^{n} x^{n} \log \left (c\right )^{2} \log \left (x\right )}{d} - \frac{3 \, d^{n} n x^{n} \log \left (x\right )^{2}}{d} + \frac{d^{n} x^{n} \log \left (c\right )^{3}}{d n} - \frac{6 \, d^{n} x^{n} \log \left (c\right ) \log \left (x\right )}{d} - \frac{3 \, d^{n} x^{n} \log \left (c\right )^{2}}{d n} + \frac{6 \, d^{n} x^{n} \log \left (x\right )}{d} + \frac{6 \, d^{n} x^{n} \log \left (c\right )}{d n} - \frac{6 \, d^{n} x^{n}}{d n} \end{align*}
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 + 3*d^n*x^n*log(c)^2*log(x)/d - 3*d^n*n*x^n*log(x)^2/d
+ d^n*x^n*log(c)^3/(d*n) - 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*log(x)/d + 6*d^n*x
^n*log(c)/(d*n) - 6*d^n*x^n/(d*n)