3.1850 \(\int \frac{1}{(a+\frac{b}{x^2}) x} \, dx\)

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

[Out]

Log[b + a*x^2]/(2*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0048052, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 260} \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x),x]

[Out]

Log[b + a*x^2]/(2*a)

Rule 263

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]

Rule 260

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]

Rubi steps

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

Mathematica [A]  time = 0.0021292, size = 15, normalized size = 1. \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x),x]

[Out]

Log[b + a*x^2]/(2*a)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01804, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x^2 + b)/a

________________________________________________________________________________________

Fricas [A]  time = 1.55414, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x^2 + b)/a

________________________________________________________________________________________

Sympy [A]  time = 0.103085, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a x^{2} + b \right )}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x**2 + b)/(2*a)

________________________________________________________________________________________

Giac [A]  time = 1.16807, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(a*x^2 + b))/a