\(\int \frac {\log (x^{-n} (a+b x^n))}{x} \, dx\) [395]

Optimal result

Integrand size = 18, antiderivative size = 47 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=-\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (b+a x^{-n}\right )}{n}-\frac {\operatorname {PolyLog}\left (2,1+\frac {a x^{-n}}{b}\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2511, 2504, 2441, 2352} \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,\frac {a x^{-n}}{b}+1\right )}{n}-\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )}{n} \]

[In]

Int[Log[(a + b*x^n)/x^n]/x,x]

[Out]

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

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])

Rule 2511

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=-\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (b+a x^{-n}\right )+\operatorname {PolyLog}\left (2,\frac {b+a x^{-n}}{b}\right )}{n} \]

[In]

Integrate[Log[(a + b*x^n)/x^n]/x,x]

[Out]

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

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {-\operatorname {dilog}\left (-\frac {a \,x^{-n}}{b}\right )-\ln \left (b +a \,x^{-n}\right ) \ln \left (-\frac {a \,x^{-n}}{b}\right )}{n}\)	\(44\)
default	\(\frac {-\operatorname {dilog}\left (-\frac {a \,x^{-n}}{b}\right )-\ln \left (b +a \,x^{-n}\right ) \ln \left (-\frac {a \,x^{-n}}{b}\right )}{n}\)	\(44\)
risch	\(-\ln \left (x \right ) \ln \left (x^{n}\right )+\frac {n \ln \left (x \right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{-n}\right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{2}}{2}+\frac {i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (a +b \,x^{n}\right )\right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{2}}{2}-\frac {i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{3}}{2}-\frac {i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{-n}\right ) \operatorname {csgn}\left (i \left (a +b \,x^{n}\right )\right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )}{2}+\frac {\ln \left (a +b \,x^{n}\right ) \ln \left (-\frac {x^{n} b}{a}\right )}{n}+\frac {\operatorname {dilog}\left (-\frac {x^{n} b}{a}\right )}{n}\)	\(187\)

[In]

int(ln((a+b*x^n)/(x^n))/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\frac {n^{2} \log \left (x\right )^{2} - 2 \, n \log \left (x\right ) \log \left (\frac {b x^{n} + a}{a}\right ) + 2 \, n \log \left (x\right ) \log \left (\frac {b x^{n} + a}{x^{n}}\right ) - 2 \, {\rm Li}_2\left (-\frac {b x^{n} + a}{a} + 1\right )}{2 \, n} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int \frac {\log {\left (a x^{- n} + b \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(log(a/x**n + b)/x, x)

Maxima [F]

\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(log((b*x^n + a)/x^n)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int \frac {\ln \left (\frac {a+b\,x^n}{x^n}\right )}{x} \,d x \]

[In]

int(log((a + b*x^n)/x^n)/x,x)

[Out]

int(log((a + b*x^n)/x^n)/x, x)