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

Optimal. Leaf size=18 \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

Log[a + b*Log[c*x^n]]/(b*n)

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Rubi [A]  time = 0.0107713, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {31} \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x*Log[c*x^n])^(-1),x]

[Out]

Log[a + b*Log[c*x^n]]/(b*n)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0170556, size = 18, normalized size = 1. \[ \frac{\log \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*Log[c*x^n]]/(b*n)

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Maple [A]  time = 0.004, size = 19, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0808, size = 32, normalized size = 1.78 \begin{align*} \frac{\log \left (\frac{b \log \left (c\right ) + b \log \left (x^{n}\right ) + a}{b}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91867, size = 51, normalized size = 2.83 \begin{align*} \frac{\log \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/(a*x+b*x*log(c*x^n)),x, algorithm="fricas")

[Out]

log(b*n*log(x) + b*log(c) + a)/(b*n)

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Sympy [A]  time = 1.09954, size = 34, normalized size = 1.89 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b \log{\left (c \right )}} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21578, size = 61, normalized size = 3.39 \begin{align*} \frac{\log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + a\right )}^{2}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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