3.1904 \(\int (\frac{1}{x^3}+\frac{1}{x^2}+\frac{1}{x}) \, dx\)

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

[Out]

-1/(2*x^2) - x^(-1) + Log[x]

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Rubi [A]  time = 0.0020757, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ -\frac{1}{2 x^2}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-3) + x^(-2) + x^(-1),x]

[Out]

-1/(2*x^2) - x^(-1) + Log[x]

Rubi steps

\begin{align*} \int \left (\frac{1}{x^3}+\frac{1}{x^2}+\frac{1}{x}\right ) \, dx &=-\frac{1}{2 x^2}-\frac{1}{x}+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0022421, size = 15, normalized size = 1. \[ -\frac{1}{2 x^2}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-3) + x^(-2) + x^(-1),x]

[Out]

-1/(2*x^2) - x^(-1) + Log[x]

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{x}^{-1}+\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3+1/x^2+1/x,x)

[Out]

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

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Maxima [A]  time = 0.95597, size = 18, normalized size = 1.2 \begin{align*} -\frac{1}{x} - \frac{1}{2 \, x^{2}} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x - 1/2/x^2 + log(x)

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Fricas [A]  time = 1.94769, size = 46, normalized size = 3.07 \begin{align*} \frac{2 \, x^{2} \log \left (x\right ) - 2 \, x - 1}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x^2*log(x) - 2*x - 1)/x^2

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Sympy [A]  time = 0.080251, size = 12, normalized size = 0.8 \begin{align*} \log{\left (x \right )} - \frac{2 x + 1}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - (2*x + 1)/(2*x**2)

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Giac [A]  time = 1.05113, size = 19, normalized size = 1.27 \begin{align*} -\frac{1}{x} - \frac{1}{2 \, x^{2}} + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x - 1/2/x^2 + log(abs(x))