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

Optimal. Leaf size=20 \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

[Out]

-(1/(b*n*(a + b*Log[c*x^n])))

________________________________________________________________________________________

Rubi [A]  time = 0.024049, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2302, 30} \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*Log[c*x^n])^2),x]

[Out]

-(1/(b*n*(a + b*Log[c*x^n])))

Rule 2302

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]

Rule 30

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=\frac{\operatorname{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]  time = 0.0045306, size = 20, normalized size = 1. \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*n*(a + b*Log[c*x^n])))

________________________________________________________________________________________

Maple [A]  time = 0.034, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{bn \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.13194, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*log(c*x^n) + a)*b*n)

________________________________________________________________________________________

Fricas [A]  time = 0.804778, size = 59, normalized size = 2.95 \begin{align*} -\frac{1}{b^{2} n^{2} \log \left (x\right ) + b^{2} n \log \left (c\right ) + a b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 49.4998, size = 70, normalized size = 3.5 \begin{align*} \begin{cases} \frac{\tilde{\infty } \log{\left (x \right )}}{\log{\left (c \right )}^{2}} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\tilde{\infty } n \log{\left (x \right )} & \text{for}\: a = - b \left (n \log{\left (x \right )} + \log{\left (c \right )}\right ) \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\\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^{2} \log{\left (x \right )} + b^{2} n \log{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.18794, size = 28, normalized size = 1.4 \begin{align*} -\frac{1}{{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)