3.1.31 \(\int \frac {\log (c (a+\frac {b}{x})^p)}{x} \, dx\) [31]

Optimal. Leaf size=40 \[ -\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )-p \text {Li}_2\left (1+\frac {b}{a x}\right ) \]

[Out]

-ln(c*(a+b/x)^p)*ln(-b/a/x)-p*polylog(2,1+b/a/x)

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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2441, 2352} \begin {gather*} \log \left (-\frac {b}{a x}\right ) \left (-\log \left (c \left (a+\frac {b}{x}\right )^p\right )\right )-p \text {PolyLog}\left (2,\frac {b}{a x}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p]/x,x]

[Out]

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

Rule 2352

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

Rule 2441

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 2504

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])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 41, normalized size = 1.02 \begin {gather*} -\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )-p \text {Li}_2\left (\frac {a+\frac {b}{x}}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p]/x,x]

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (39) = 78\).
time = 0.26, size = 83, normalized size = 2.08 \begin {gather*} \frac {1}{2} \, b p {\left (\frac {2 \, \log \left (a + \frac {b}{x}\right ) \log \left (x\right )}{b} + \frac {\log \left (x\right )^{2}}{b} - \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b}\right )} - p \log \left (a + \frac {b}{x}\right ) \log \left (x\right ) + \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(c*((a*x + b)/x)^p)/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(c*(a + b/x)**p)/x, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (39) = 78\).
time = 4.63, size = 152, normalized size = 3.80 \begin {gather*} -\frac {\frac {b^{3} p \log \left (\frac {a x + b}{x}\right )}{a^{2} - \frac {2 \, {\left (a x + b\right )} a}{x} + \frac {{\left (a x + b\right )}^{2}}{x^{2}}} + \frac {b^{3} p \log \left (-a + \frac {a x + b}{x}\right )}{a^{2}} - \frac {b^{3} p \log \left (\frac {a x + b}{x}\right )}{a^{2}} - \frac {a b^{3} p - a b^{3} \log \left (c\right ) - \frac {{\left (a x + b\right )} b^{3} p}{x}}{a^{3} - \frac {2 \, {\left (a x + b\right )} a^{2}}{x} + \frac {{\left (a x + b\right )}^{2} a}{x^{2}}}}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*p*log((a*x + b)/x)/(a^2 - 2*(a*x + b)*a/x + (a*x + b)^2/x^2) + b^3*p*log(-a + (a*x + b)/x)/a^2 - b^3
*p*log((a*x + b)/x)/a^2 - (a*b^3*p - a*b^3*log(c) - (a*x + b)*b^3*p/x)/(a^3 - 2*(a*x + b)*a^2/x + (a*x + b)^2*
a/x^2))/b^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x)^p)/x,x)

[Out]

int(log(c*(a + b/x)^p)/x, x)

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