\(\int \frac {1}{x (a+b x^{-n})} \, dx\) [2737]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\log \left (b+a x^n\right )}{a n} \]

[Out]

ln(b+a*x^n)/a/n

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {269, 266} \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\log \left (a x^n+b\right )}{a n} \]

[In]

Int[1/(x*(a + b/x^n)),x]

[Out]

Log[b + a*x^n]/(a*n)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\log \left (b+a x^n\right )}{a n} \]

[In]

Integrate[1/(x*(a + b/x^n)),x]

[Out]

Log[b + a*x^n]/(a*n)

Maple [A] (verified)

Time = 3.67 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {\ln \left (b +a \,x^{n}\right )}{a n}\)	\(16\)
default	\(\frac {\ln \left (b +a \,x^{n}\right )}{a n}\)	\(16\)
parallelrisch	\(\frac {\ln \left (b +a \,x^{n}\right )}{a n}\)	\(16\)
norman	\(\frac {\ln \left (a \,{\mathrm e}^{n \ln \left (x \right )}+b \right )}{a n}\)	\(18\)
risch	\(\frac {\ln \left (x^{n}+\frac {b}{a}\right )}{a n}\)	\(18\)

[In]

int(1/x/(a+b/(x^n)),x,method=_RETURNVERBOSE)

[Out]

ln(b+a*x^n)/a/n

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\log \left (a x^{n} + b\right )}{a n} \]

[In]

integrate(1/x/(a+b/(x^n)),x, algorithm="fricas")

[Out]

log(a*x^n + b)/(a*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (10) = 20\).

Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {x^{n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} + \frac {\log {\left (\frac {a}{b} + x^{- n} \right )}}{a n} & \text {otherwise} \end {cases} \]

[In]

integrate(1/x/(a+b/(x**n)),x)

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (x**n/(b*n), Eq(a, 0)), (log(x)/a, Eq(b, 0)), (log(x)/
(a + b), Eq(n, 0)), (log(x)/a + log(a/b + x**(-n))/(a*n), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\log \left (a + \frac {b}{x^{n}}\right )}{a n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{a n} \]

[In]

integrate(1/x/(a+b/(x^n)),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{n}}\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b/(x^n)),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^n)*x), x)

Mupad [B] (verification not implemented)

Time = 5.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x \left (a+b x^{-n}\right )} \, dx=\frac {\ln \left (a+\frac {b}{x^n}\right )+n\,\ln \left (x\right )}{a\,n} \]

[In]

int(1/(x*(a + b/x^n)),x)

[Out]

(log(a + b/x^n) + n*log(x))/(a*n)