3.23.31 \(\int -\frac {1}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -1+\frac {2}{9} \left (2+\frac {e^2}{3}\right )+\frac {1}{x} \]

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Rubi [A]  time = 0.00, antiderivative size = 3, normalized size of antiderivative = 0.17, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {30} \begin {gather*} \frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-x^(-2),x]

[Out]

x^(-1)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 3, normalized size = 0.17 \begin {gather*} \frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-x^(-2),x]

[Out]

x^(-1)

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fricas [A]  time = 0.58, size = 3, normalized size = 0.17 \begin {gather*} \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x

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giac [A]  time = 0.60, size = 3, normalized size = 0.17 \begin {gather*} \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x^2,x, algorithm="giac")

[Out]

1/x

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maple [A]  time = 0.01, size = 4, normalized size = 0.22

method	result	size
gosper	\(\frac {1}{x}\)	\(4\)
default	\(\frac {1}{x}\)	\(4\)
norman	\(\frac {1}{x}\)	\(4\)
risch	\(\frac {1}{x}\)	\(4\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/x^2,x,method=_RETURNVERBOSE)

[Out]

1/x

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maxima [A]  time = 0.41, size = 3, normalized size = 0.17 \begin {gather*} \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x

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mupad [B]  time = 0.01, size = 3, normalized size = 0.17 \begin {gather*} \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/x^2,x)

[Out]

1/x

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sympy [A]  time = 0.05, size = 2, normalized size = 0.11 \begin {gather*} \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/x**2,x)

[Out]

1/x

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