Integrand size = 18, antiderivative size = 47 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=-\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (b+a x^{-n}\right )}{n}-\frac {\operatorname {PolyLog}\left (2,1+\frac {a x^{-n}}{b}\right )}{n} \] Output:
-ln(-a/b/(x^n))*ln(b+a/(x^n))/n-polylog(2,1+a/b/(x^n))/n
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=-\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (b+a x^{-n}\right )+\operatorname {PolyLog}\left (2,\frac {b+a x^{-n}}{b}\right )}{n} \] Input:
Integrate[Log[(a + b*x^n)/x^n]/x,x]
Output:
-((Log[-(a/(b*x^n))]*Log[b + a/x^n] + PolyLog[2, (b + a/x^n)/b])/n)
Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2911, 2904, 2841, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 2911 |
\(\displaystyle \int \frac {\log \left (a x^{-n}+b\right )}{x}dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -\frac {\int x^n \log \left (a x^{-n}+b\right )dx^{-n}}{n}\) |
\(\Big \downarrow \) 2841 |
\(\displaystyle -\frac {\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )-a \int \frac {\log \left (-\frac {a x^{-n}}{b}\right )}{a x^{-n}+b}dx^{-n}}{n}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle -\frac {\operatorname {PolyLog}\left (2,\frac {a x^{-n}}{b}+1\right )+\log \left (-\frac {a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )}{n}\) |
Input:
Int[Log[(a + b*x^n)/x^n]/x,x]
Output:
-((Log[-(a/(b*x^n))]*Log[b + a/x^n] + PolyLog[2, 1 + a/(b*x^n)])/n)
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
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] - Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[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])
Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*((f_.)*(x_))^(m_.), x_Symbo l] :> Int[(f*x)^m*(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] && !BinomialMatchQ[v, x]
Time = 6.91 (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 | \(-\ln \left (x \right ) \ln \left (x^{n}\right )+\frac {n \ln \left (x \right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left (a +b \,x^{n}\right )\right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{2} \operatorname {csgn}\left (i x^{-n}\right )}{2}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right )^{3}}{2}-\frac {i \pi \ln \left (x \right ) \operatorname {csgn}\left (i \left (a +b \,x^{n}\right )\right ) \operatorname {csgn}\left (i x^{-n} \left (a +b \,x^{n}\right )\right ) \operatorname {csgn}\left (i x^{-n}\right )}{2}+\frac {\ln \left (a +b \,x^{n}\right ) \ln \left (-\frac {x^{n} b}{a}\right )}{n}+\frac {\operatorname {dilog}\left (-\frac {x^{n} b}{a}\right )}{n}\) | \(187\) |
Input:
int(ln((a+b*x^n)/(x^n))/x,x,method=_RETURNVERBOSE)
Output:
1/n*(-dilog(-a/b/(x^n))-ln(b+a/(x^n))*ln(-a/b/(x^n)))
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\frac {n^{2} \log \left (x\right )^{2} - 2 \, n \log \left (x\right ) \log \left (\frac {b x^{n} + a}{a}\right ) + 2 \, n \log \left (x\right ) \log \left (\frac {b x^{n} + a}{x^{n}}\right ) - 2 \, {\rm Li}_2\left (-\frac {b x^{n} + a}{a} + 1\right )}{2 \, n} \] Input:
integrate(log((a+b*x^n)/(x^n))/x,x, algorithm="fricas")
Output:
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
\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int \frac {\log {\left (a x^{- n} + b \right )}}{x}\, dx \] Input:
integrate(ln((a+b*x**n)/(x**n))/x,x)
Output:
Integral(log(a/x**n + b)/x, x)
\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{x} \,d x } \] Input:
integrate(log((a+b*x^n)/(x^n))/x,x, algorithm="maxima")
Output:
a*n*integrate(log(x)/(b*x*x^n + a*x), x) + log(b*x^n + a)*log(x) - log(x)* log(x^n)
\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{x} \,d x } \] Input:
integrate(log((a+b*x^n)/(x^n))/x,x, algorithm="giac")
Output:
integrate(log((b*x^n + a)/x^n)/x, x)
Timed out. \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int \frac {\ln \left (\frac {a+b\,x^n}{x^n}\right )}{x} \,d x \] Input:
int(log((a + b*x^n)/x^n)/x,x)
Output:
int(log((a + b*x^n)/x^n)/x, x)
\[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{x} \, dx=\int \frac {\mathrm {log}\left (\frac {x^{n} b +a}{x^{n}}\right )}{x}d x \] Input:
int(log((a+b*x^n)/(x^n))/x,x)
Output:
int(log((x**n*b + a)/x**n)/x,x)