3.621 \(\int \frac{(a+b \log (x))^{-n}}{x} \, dx\)

Optimal. Leaf size=23 \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]

[Out]

(a + b*Log[x])^(1 - n)/(b*(1 - n))

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Rubi [A]  time = 0.0324734, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*Log[x])^n),x]

[Out]

(a + b*Log[x])^(1 - n)/(b*(1 - 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{(a+b \log (x))^{-n}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int x^{-n} \, dx,x,a+b \log (x)\right )}{b}\\ &=\frac{(a+b \log (x))^{1-n}}{b (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.011729, size = 23, normalized size = 1. \[ \frac{(a+b \log (x))^{1-n}}{b (1-n)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*Log[x])^n),x]

[Out]

(a + b*Log[x])^(1 - n)/(b*(1 - n))

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Maple [A]  time = 0., size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\ln \left ( x \right ) \right ) ^{1-n}}{b \left ( 1-n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.44858, size = 62, normalized size = 2.7 \begin{align*} -\frac{b \log \left (x\right ) + a}{{\left (b n - b\right )}{\left (b \log \left (x\right ) + a\right )}^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 37.2145, size = 71, normalized size = 3.09 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 1 \\a^{- n} \log{\left (x \right )} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + \log{\left (x \right )} \right )}}{b} & \text{for}\: n = 1 \\- \frac{a}{b n \left (a + b \log{\left (x \right )}\right )^{n} - b \left (a + b \log{\left (x \right )}\right )^{n}} - \frac{b \log{\left (x \right )}}{b n \left (a + b \log{\left (x \right )}\right )^{n} - b \left (a + b \log{\left (x \right )}\right )^{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.0971, size = 30, normalized size = 1.3 \begin{align*} -\frac{{\left (b \log \left (x\right ) + a\right )}^{-n + 1}}{b{\left (n - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*log(x) + a)^(-n + 1)/(b*(n - 1))