\(\int \frac {1}{x (a+b \log (c x^n))^3} \, dx\) [85]

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2339, 30} \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[In]

Int[1/(x*(a + b*Log[c*x^n])^3),x]

[Out]

-1/2*1/(b*n*(a + b*Log[c*x^n])^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[In]

Integrate[1/(x*(a + b*Log[c*x^n])^3),x]

[Out]

-1/2*1/(b*n*(a + b*Log[c*x^n])^2)

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(-\frac {1}{2 b n {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}\)	\(21\)
default	\(-\frac {1}{2 b n {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}\)	\(21\)
parallelrisch	\(-\frac {1}{2 b n {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}\)	\(21\)
risch	\(-\frac {2}{b n {\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b +2 a \right )}^{2}}\)	\(109\)

[In]

int(1/x/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (20) = 40\).

Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 \, {\left (b^{3} n^{3} \log \left (x\right )^{2} + b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n + 2 \, {\left (b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )\right )}} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).

Time = 1.87 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{a^{3}} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac {\log {\left (x \right )}}{\left (a + b \log {\left (c \right )}\right )^{3}} & \text {for}\: n = 0 \\- \frac {1}{2 a^{2} b n + 4 a b^{2} n \log {\left (c x^{n} \right )} + 2 b^{3} n \log {\left (c x^{n} \right )}^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x/(a+b*ln(c*x**n))**3,x)

[Out]

Piecewise((log(x)/a**3, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)/(a + b*log(c))**3, Eq(n, 0)), (-1/(2*a**2*b
*n + 4*a*b**2*n*log(c*x**n) + 2*b**3*n*log(c*x**n)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 \, {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} b n} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

-1/2/((b*log(c*x^n) + a)^2*b*n)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2 \, {\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{2} b n} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^3} \, dx=-\frac {1}{2\,n\,a^2\,b+4\,n\,a\,b^2\,\ln \left (c\,x^n\right )+2\,n\,b^3\,{\ln \left (c\,x^n\right )}^2} \]

[In]

int(1/(x*(a + b*log(c*x^n))^3),x)

[Out]

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