\(\int \frac {1}{x (a+b \log (c x^n))^2} \, dx\) [77]

Optimal result

Integrand size = 16, antiderivative size = 20 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, 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} \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[1/(x*(a + b*Log[c*x^n])^2),x]

[Out]

-(1/(b*n*(a + b*Log[c*x^n])))

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[1/(x*(a + b*Log[c*x^n])^2),x]

[Out]

-(1/(b*n*(a + b*Log[c*x^n])))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05

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

[In]

int(1/x/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{b^{2} n^{2} \log \left (x\right ) + b^{2} n \log \left (c\right ) + a b n} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).

Time = 0.87 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{a^{2}} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac {\log {\left (x \right )}}{\left (a + b \log {\left (c \right )}\right )^{2}} & \text {for}\: n = 0 \\- \frac {1}{a b n + b^{2} n \log {\left (c x^{n} \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x/(a+b*ln(c*x**n))**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^2,x, algorithm="maxima")

[Out]

-1/((b*log(c*x^n) + a)*b*n)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} b n} \]

[In]

integrate(1/x/(a+b*log(c*x^n))^2,x, algorithm="giac")

[Out]

-1/((b*n*log(x) + b*log(c) + a)*b*n)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx=-\frac {1}{n\,\ln \left (c\,x^n\right )\,b^2+a\,n\,b} \]

[In]

int(1/(x*(a + b*log(c*x^n))^2),x)

[Out]

-1/(b^2*n*log(c*x^n) + a*b*n)