3.393 \(\int \frac{\log (\frac{a+b x}{x})}{x} \, dx\)

Optimal. Leaf size=35 \[ -\text{PolyLog}\left (2,\frac{a}{b x}+1\right )-\log \left (\frac{a}{x}+b\right ) \log \left (-\frac{a}{b x}\right ) \]

[Out]

-(Log[b + a/x]*Log[-(a/(b*x))]) - PolyLog[2, 1 + a/(b*x)]

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Rubi [A]  time = 0.055865, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2461, 2454, 2394, 2315} \[ -\text{PolyLog}\left (2,\frac{a}{b x}+1\right )-\log \left (\frac{a}{x}+b\right ) \log \left (-\frac{a}{b x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[(a + b*x)/x]/x,x]

[Out]

-(Log[b + a/x]*Log[-(a/(b*x))]) - PolyLog[2, 1 + a/(b*x)]

Rule 2461

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*((f_.)*(x_))^(m_.), x_Symbol] :> Int[(f*x)^m*(a + b*Log[c*Expa
ndToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{\log \left (\frac{a+b x}{x}\right )}{x} \, dx &=\int \frac{\log \left (b+\frac{a}{x}\right )}{x} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{\log (b+a x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\log \left (b+\frac{a}{x}\right ) \log \left (-\frac{a}{b x}\right )+a \operatorname{Subst}\left (\int \frac{\log \left (-\frac{a x}{b}\right )}{b+a x} \, dx,x,\frac{1}{x}\right )\\ &=-\log \left (b+\frac{a}{x}\right ) \log \left (-\frac{a}{b x}\right )-\text{Li}_2\left (1+\frac{a}{b x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0037919, size = 36, normalized size = 1.03 \[ -\text{PolyLog}\left (2,\frac{\frac{a}{x}+b}{b}\right )-\log \left (\frac{a}{x}+b\right ) \log \left (-\frac{a}{b x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(a + b*x)/x]/x,x]

[Out]

-(Log[b + a/x]*Log[-(a/(b*x))]) - PolyLog[2, (b + a/x)/b]

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Maple [A]  time = 0.084, size = 34, normalized size = 1. \begin{align*} -{\it dilog} \left ( -{\frac{a}{bx}} \right ) -\ln \left ( b+{\frac{a}{x}} \right ) \ln \left ( -{\frac{a}{bx}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((b*x+a)/x)/x,x)

[Out]

-dilog(-a/b/x)-ln(b+a/x)*ln(-a/b/x)

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Maxima [A]  time = 1.05408, size = 90, normalized size = 2.57 \begin{align*} -{\left (\log \left (b x + a\right ) - \log \left (x\right )\right )} \log \left (x\right ) + \log \left (b x + a\right ) \log \left (x\right ) - \log \left (\frac{b x}{a} + 1\right ) \log \left (x\right ) - \frac{1}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\frac{b x + a}{x}\right ) -{\rm Li}_2\left (-\frac{b x}{a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(b*x + a) - log(x))*log(x) + log(b*x + a)*log(x) - log(b*x/a + 1)*log(x) - 1/2*log(x)^2 + log(x)*log((b*x
+ a)/x) - dilog(-b*x/a)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{b x + a}{x}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x + a)/x)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\frac{a}{x} + b \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a/x + b)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{b x + a}{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x + a)/x)/x, x)