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

Optimal. Leaf size=27 \[ \frac{\log (x)}{a}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]

[Out]

Log[x]/a - (b*Log[b + a*Log[c*x^n]])/(a^2*n)

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Rubi [A]  time = 0.023225, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{\log (x)}{a}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + (b*x)/Log[c*x^n])^(-1),x]

[Out]

Log[x]/a - (b*Log[b + a*Log[c*x^n]])/(a^2*n)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0150973, size = 34, normalized size = 1.26 \[ \frac{\log \left (c x^n\right )}{a n}-\frac{b \log \left (a \log \left (c x^n\right )+b\right )}{a^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[c*x^n]/(a*n) - (b*Log[b + a*Log[c*x^n]])/(a^2*n)

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Maple [A]  time = 0.006, size = 35, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{b\ln \left ( b+a\ln \left ( c{x}^{n} \right ) \right ) }{{a}^{2}n}} \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]

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

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Maxima [A]  time = 1.04503, size = 45, normalized size = 1.67 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{b \log \left (\frac{a \log \left (c\right ) + a \log \left (x^{n}\right ) + b}{a}\right )}{a^{2} 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(x)/a - b*log((a*log(c) + a*log(x^n) + b)/a)/(a^2*n)

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Fricas [A]  time = 1.8948, size = 77, normalized size = 2.85 \begin{align*} \frac{a n \log \left (x\right ) - b \log \left (a n \log \left (x\right ) + a \log \left (c\right ) + b\right )}{a^{2} 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]

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

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

Exception raised: TypeError

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Giac [A]  time = 1.30061, size = 72, normalized size = 2.67 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{b \log \left (\frac{1}{4} \,{\left (\pi a n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) + b\right )}^{2}\right )}{2 \, a^{2} 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]

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