3.39.92 \(\int \frac {-8-x^2}{x^3} \, dx\)

Optimal. Leaf size=11 \[ 1+\frac {4}{x^2}-\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} \frac {4}{x^2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 - x^2)/x^3,x]

[Out]

4/x^2 - Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {8}{x^3}-\frac {1}{x}\right ) \, dx\\ &=\frac {4}{x^2}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.91 \begin {gather*} \frac {4}{x^2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 - x^2)/x^3,x]

[Out]

4/x^2 - Log[x]

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fricas [A]  time = 0.53, size = 13, normalized size = 1.18 \begin {gather*} -\frac {x^{2} \log \relax (x) - 4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-8)/x^3,x, algorithm="fricas")

[Out]

-(x^2*log(x) - 4)/x^2

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giac [A]  time = 0.11, size = 17, normalized size = 1.55 \begin {gather*} \frac {x^{2} + 8}{2 \, x^{2}} - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-8)/x^3,x, algorithm="giac")

[Out]

1/2*(x^2 + 8)/x^2 - 1/2*log(x^2)

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maple [A]  time = 0.02, size = 11, normalized size = 1.00

method	result	size
default	\(\frac {4}{x^{2}}-\ln \relax (x )\)	\(11\)
norman	\(\frac {4}{x^{2}}-\ln \relax (x )\)	\(11\)
risch	\(\frac {4}{x^{2}}-\ln \relax (x )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2-8)/x^3,x,method=_RETURNVERBOSE)

[Out]

4/x^2-ln(x)

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maxima [A]  time = 0.35, size = 12, normalized size = 1.09 \begin {gather*} \frac {4}{x^{2}} - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-8)/x^3,x, algorithm="maxima")

[Out]

4/x^2 - 1/2*log(x^2)

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mupad [B]  time = 0.02, size = 10, normalized size = 0.91 \begin {gather*} \frac {4}{x^2}-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2 + 8)/x^3,x)

[Out]

4/x^2 - log(x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.64 \begin {gather*} - \log {\relax (x )} + \frac {4}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2-8)/x**3,x)

[Out]

-log(x) + 4/x**2

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