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

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{x}+\log (x) \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{2 x^2}-\frac {1}{x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-3) + x^(-2) + x^(-1),x]

[Out]

-1/2*1/x^2 - x^(-1) + Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {1}{x^3}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^(-3) + x^(-2) + x^(-1),x]

[Out]

IntegrateAlgebraic[x^(-3) + x^(-2) + x^(-1), x]

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fricas [A]  time = 1.23, size = 17, normalized size = 1.13 \begin {gather*} \frac {2 \, x^{2} \log \relax (x) - 2 \, x - 1}{2 \, x^{2}} \end {gather*}

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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giac [A]  time = 1.08, size = 14, normalized size = 0.93 \begin {gather*} -\frac {1}{x} - \frac {1}{2 \, x^{2}} + \log \left ({\left | x \right |}\right ) \end {gather*}

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))

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} \ln \relax (x )-\frac {1}{x}-\frac {1}{2 x^{2}} \end {gather*}

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 = 1.04, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{x} - \frac {1}{2 \, x^{2}} + \log \relax (x) \end {gather*}

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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mupad [B]  time = 0.03, size = 11, normalized size = 0.73 \begin {gather*} \ln \relax (x)-\frac {x+\frac {1}{2}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x + 1/x^2 + 1/x^3,x)

[Out]

log(x) - (x + 1/2)/x^2

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sympy [A]  time = 0.09, size = 14, normalized size = 0.93 \begin {gather*} \log {\relax (x )} + \frac {- 2 x - 1}{2 x^{2}} \end {gather*}

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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