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

Optimal. Leaf size=22 \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

[Out]

-1/(2*b*n*(a + b*Log[c*x^n])^2)

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Rubi [A]  time = 0.0235673, antiderivative size = 22, 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} \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*Log[c*x^n])^3),x]

[Out]

-1/(2*b*n*(a + b*Log[c*x^n])^2)

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 )^3} \, dx &=\frac{\operatorname{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]  time = 0.0038617, size = 22, normalized size = 1. \[ -\frac{1}{2 b n \left (a+b \log \left (c x^n\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*b*n*(a + b*Log[c*x^n])^2)

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Maple [A]  time = 0.035, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,bn \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09555, size = 27, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 0.792293, size = 150, normalized size = 6.82 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

[Out]

Timed out

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Giac [A]  time = 1.28818, size = 28, normalized size = 1.27 \begin{align*} -\frac{1}{2 \,{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}^{2} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)