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

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 = 0.00734425, 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. \[ -\frac{1}{2 x^2}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[x^(-3) + x^(-2) + x^(-1),x]

[Out]

-1/(2*x^2) - x^(-1) + Log[x]

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Rubi in Sympy [A]  time = 0.929624, size = 12, normalized size = 0.8 \[ \log{\left (x \right )} - \frac{1}{x} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3+1/x**2+1/x,x)

[Out]

log(x) - 1/x - 1/(2*x**2)

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Mathematica [A]  time = 0.00290225, 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 = 0.002, size = 14, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}}}-{x}^{-1}+\ln \left ( x \right ) \]

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 = 1.32805, size = 18, normalized size = 1.2 \[ -\frac{1}{x} - \frac{1}{2 \, x^{2}} + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/x - 1/2/x^2 + log(x)

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Fricas [A]  time = 0.202237, size = 23, normalized size = 1.53 \[ \frac{2 \, x^{2} \log \left (x\right ) - 2 \, x - 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x^2*log(x) - 2*x - 1)/x^2

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Sympy [A]  time = 0.080193, size = 12, normalized size = 0.8 \[ \log{\left (x \right )} - \frac{2 x + 1}{2 x^{2}} \]

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/XCAS [A]  time = 0.246294, size = 19, normalized size = 1.27 \[ -\frac{1}{x} - \frac{1}{2 \, x^{2}} +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/x - 1/2/x^2 + ln(abs(x))