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

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

[Out]

Log[x] - Log[1 + b*x^2]/2

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Rubi [A]  time = 0.0090262, antiderivative size = 15, normalized size of antiderivative = 1., 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} \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(1 + b*x^2)),x]

[Out]

Log[x] - Log[1 + b*x^2]/2

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]]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0041693, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 + b*x^2)),x]

[Out]

Log[x] - Log[1 + b*x^2]/2

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ( b{x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.05713, size = 23, normalized size = 1.53 \begin{align*} -\frac{1}{2} \, \log \left (b x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(b*x^2 + 1) + 1/2*log(x^2)

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Fricas [A]  time = 1.24878, size = 41, normalized size = 2.73 \begin{align*} -\frac{1}{2} \, \log \left (b x^{2} + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(b*x^2 + 1) + log(x)

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Sympy [A]  time = 0.120866, size = 12, normalized size = 0.8 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + \frac{1}{b} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**2 + 1/b)/2

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Giac [A]  time = 2.17215, size = 24, normalized size = 1.6 \begin{align*} \frac{1}{2} \, \log \left (x^{2}\right ) - \frac{1}{2} \, \log \left ({\left | b x^{2} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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