\(\int (d x)^{-1+n} \log ^3(c x^n) \, dx\) [156]

Optimal result

Integrand size = 16, antiderivative size = 74 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=-\frac {6 (d x)^n}{d n}+\frac {6 (d x)^n \log \left (c x^n\right )}{d n}-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {(d x)^n \log ^3\left (c x^n\right )}{d n} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2342, 2341} \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\frac {(d x)^n \log ^3\left (c x^n\right )}{d n}-\frac {3 (d x)^n \log ^2\left (c x^n\right )}{d n}+\frac {6 (d x)^n \log \left (c x^n\right )}{d n}-\frac {6 (d x)^n}{d n} \]

[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 2341

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 2342

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.54 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\frac {(d x)^n \left (-6+6 \log \left (c x^n\right )-3 \log ^2\left (c x^n\right )+\log ^3\left (c x^n\right )\right )}{d n} \]

[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)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93

method	result	size
parallelrisch	\(-\frac {-\left (d x \right )^{n -1} \ln \left (c \,x^{n}\right )^{3} x +3 \left (d x \right )^{n -1} \ln \left (c \,x^{n}\right )^{2} x -6 \left (d x \right )^{n -1} x \ln \left (c \,x^{n}\right )+6 \left (d x \right )^{n -1} x}{n}\)	\(69\)
risch	\(\text {Expression too large to display}\)	\(2008\)

[In]

int((d*x)^(n-1)*ln(c*x^n)^3,x,method=_RETURNVERBOSE)

[Out]

-(-(d*x)^(n-1)*ln(c*x^n)^3*x+3*(d*x)^(n-1)*ln(c*x^n)^2*x-6*(d*x)^(n-1)*x*ln(c*x^n)+6*(d*x)^(n-1)*x)/n

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\frac {{\left (n^{3} \log \left (x\right )^{3} + \log \left (c\right )^{3} + 3 \, {\left (n^{2} \log \left (c\right ) - n^{2}\right )} \log \left (x\right )^{2} - 3 \, \log \left (c\right )^{2} + 3 \, {\left (n \log \left (c\right )^{2} - 2 \, n \log \left (c\right ) + 2 \, n\right )} \log \left (x\right ) + 6 \, \log \left (c\right ) - 6\right )} d^{n - 1} x^{n}}{n} \]

[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

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\begin {cases} \frac {x \left (d x\right )^{n - 1} \log {\left (c x^{n} \right )}^{3}}{n} - \frac {3 x \left (d x\right )^{n - 1} \log {\left (c x^{n} \right )}^{2}}{n} + \frac {6 x \left (d x\right )^{n - 1} \log {\left (c x^{n} \right )}}{n} - \frac {6 x \left (d x\right )^{n - 1}}{n} & \text {for}\: n \neq 0 \\\frac {\log {\left (c \right )}^{3} \log {\left (x \right )}}{d} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x)**(-1+n)*ln(c*x**n)**3,x)

[Out]

Piecewise((x*(d*x)**(n - 1)*log(c*x**n)**3/n - 3*x*(d*x)**(n - 1)*log(c*x**n)**2/n + 6*x*(d*x)**(n - 1)*log(c*
x**n)/n - 6*x*(d*x)**(n - 1)/n, Ne(n, 0)), (log(c)**3*log(x)/d, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=-\frac {3 \, d^{n - 1} x^{n} \log \left (c x^{n}\right )^{2}}{n} + \frac {\left (d x\right )^{n} \log \left (c x^{n}\right )^{3}}{d n} + \frac {6 \, {\left (\frac {d^{n} x^{n} \log \left (c x^{n}\right )}{n} - \frac {d^{n} x^{n}}{n}\right )}}{d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (74) = 148\).

Time = 0.42 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.19 \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\frac {d^{n} n^{2} x^{n} \log \left (x\right )^{3}}{d} + \frac {3 \, d^{n} n x^{n} \log \left (c\right ) \log \left (x\right )^{2}}{d} + \frac {3 \, d^{n} x^{n} \log \left (c\right )^{2} \log \left (x\right )}{d} - \frac {3 \, d^{n} n x^{n} \log \left (x\right )^{2}}{d} + \frac {d^{n} x^{n} \log \left (c\right )^{3}}{d n} - \frac {6 \, d^{n} x^{n} \log \left (c\right ) \log \left (x\right )}{d} - \frac {3 \, d^{n} x^{n} \log \left (c\right )^{2}}{d n} + \frac {6 \, d^{n} x^{n} \log \left (x\right )}{d} + \frac {6 \, d^{n} x^{n} \log \left (c\right )}{d n} - \frac {6 \, d^{n} x^{n}}{d n} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (d x)^{-1+n} \log ^3\left (c x^n\right ) \, dx=\int {\ln \left (c\,x^n\right )}^3\,{\left (d\,x\right )}^{n-1} \,d x \]

[In]

int(log(c*x^n)^3*(d*x)^(n - 1),x)

[Out]

int(log(c*x^n)^3*(d*x)^(n - 1), x)