3.1852 \(\int \frac{1}{(a+\frac{b}{x^2}) x^3} \, dx\)

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

[Out]

-Log[a + b/x^2]/(2*b)

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Rubi [A]  time = 0.0036942, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {260} \[ -\frac{\log \left (a+\frac{b}{x^2}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)*x^3),x]

[Out]

-Log[a + b/x^2]/(2*b)

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^3} \, dx &=-\frac{\log \left (a+\frac{b}{x^2}\right )}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.004896, size = 22, normalized size = 1.47 \[ \frac{\log (x)}{b}-\frac{\log \left (a x^2+b\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)*x^3),x]

[Out]

Log[x]/b - Log[b + a*x^2]/(2*b)

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Maple [A]  time = 0.004, size = 21, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,b}}+{\frac{\ln \left ( x \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.02235, size = 18, normalized size = 1.2 \begin{align*} -\frac{\log \left (a + \frac{b}{x^{2}}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)/x^3,x, algorithm="maxima")

[Out]

-1/2*log(a + b/x^2)/b

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Fricas [A]  time = 1.66191, size = 49, normalized size = 3.27 \begin{align*} -\frac{\log \left (a x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)/x^3,x, algorithm="fricas")

[Out]

-1/2*(log(a*x^2 + b) - 2*log(x))/b

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Sympy [A]  time = 0.202757, size = 15, normalized size = 1. \begin{align*} \frac{\log{\left (x \right )}}{b} - \frac{\log{\left (x^{2} + \frac{b}{a} \right )}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)/x**3,x)

[Out]

log(x)/b - log(x**2 + b/a)/(2*b)

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Giac [A]  time = 1.14458, size = 32, normalized size = 2.13 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, b} - \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)/x^3,x, algorithm="giac")

[Out]

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