3.2613 \(\int \frac{1}{x \left (2+b x^n\right )} \, dx\)

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

[Out]

Log[x]/2 - Log[2 + b*x^n]/(2*n)

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Rubi [A]  time = 0.0290522, antiderivative size = 22, 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 \[ \frac{\log (x)}{2}-\frac{\log \left (b x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 + b*x^n)),x]

[Out]

Log[x]/2 - Log[2 + b*x^n]/(2*n)

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Rubi in Sympy [A]  time = 5.01186, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{n} \right )}}{2 n} - \frac{\log{\left (b x^{n} + 2 \right )}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(2+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0123414, size = 22, normalized size = 1. \[ \frac{n \log (x)-\log \left (b x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(2 + b*x^n)),x]

[Out]

(n*Log[x] - Log[2 + b*x^n])/(2*n)

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Maple [A]  time = 0.004, size = 24, normalized size = 1.1 \[{\frac{\ln \left ({x}^{n} \right ) }{2\,n}}-{\frac{\ln \left ( 2+b{x}^{n} \right ) }{2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(2+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.42195, size = 31, normalized size = 1.41 \[ -\frac{\log \left (b x^{n} + 2\right )}{2 \, n} + \frac{\log \left (x^{n}\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.223746, size = 27, normalized size = 1.23 \[ \frac{n \log \left (x\right ) - \log \left (b x^{n} + 2\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(n*log(x) - log(b*x^n + 2))/n

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Sympy [A]  time = 2.17578, size = 31, normalized size = 1.41 \[ \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x^{n} + \frac{2}{b} \right )}}{2 n} & \text{otherwise} \end{cases} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + 2)*x), x)