3.2620 \(\int \frac {1}{x (2+b x^n)} \, dx\)

Optimal. Leaf size=22 \[ \frac {\log (x)}{2}-\frac {\log \left (b x^n+2\right )}{2 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac {\log (x)}{2}-\frac {\log \left (b x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 + b*x^n)),x]

[Out]

Log[x]/2 - Log[2 + b*x^n]/(2*n)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {1}{x \left (2+b x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{2 n}-\frac {b \operatorname {Subst}\left (\int \frac {1}{2+b x} \, dx,x,x^n\right )}{2 n}\\ &=\frac {\log (x)}{2}-\frac {\log \left (2+b x^n\right )}{2 n}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ \frac {n \log (x)-\log \left (b x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(2 + b*x^n)),x]

[Out]

(n*Log[x] - Log[2 + b*x^n])/(2*n)

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 20, normalized size = 0.91 \[ \frac {n \log \relax (x) - \log \left (b x^{n} + 2\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^n),x, algorithm="fricas")

[Out]

1/2*(n*log(x) - log(b*x^n + 2))/n

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + 2\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^n),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + 2)*x), x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 24, normalized size = 1.09 \[ \frac {\ln \left (x^{n}\right )}{2 n}-\frac {\ln \left (b \,x^{n}+2\right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(2+b*x^n),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.69, size = 23, normalized size = 1.05 \[ -\frac {\log \left (b x^{n} + 2\right )}{2 \, n} + \frac {\log \left (x^{n}\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x^n),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 18, normalized size = 0.82 \[ \frac {\ln \relax (x)}{2}-\frac {\ln \left (b\,x^n+2\right )}{2\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x^n + 2)),x)

[Out]

log(x)/2 - log(b*x^n + 2)/(2*n)

________________________________________________________________________________________

sympy [A]  time = 0.59, size = 29, normalized size = 1.32 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{2} - \frac {\log {\left (x^{n} + \frac {2}{b} \right )}}{2 n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(2+b*x**n),x)

[Out]

Piecewise((log(x)/2, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)/(b + 2), Eq(n, 0)), (log(x)/2 - log(x**n + 2/b
)/(2*n), True))

________________________________________________________________________________________