3.98.56 \(\int \frac {12+x^2}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -3+x-\frac {4+x+2 (4+x)}{x}-\frac {e^3}{\log (2)} \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.28, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14} \begin {gather*} x-\frac {12}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 + x^2)/x^2,x]

[Out]

-12/x + 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 (1+\frac {12}{x^2}\right ) \, dx\\ &=-\frac {12}{x}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.28 \begin {gather*} -\frac {12}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + x^2)/x^2,x]

[Out]

-12/x + x

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fricas [A]  time = 0.78, size = 9, normalized size = 0.36 \begin {gather*} \frac {x^{2} - 12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12)/x^2,x, algorithm="fricas")

[Out]

(x^2 - 12)/x

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giac [A]  time = 0.21, size = 7, normalized size = 0.28 \begin {gather*} x - \frac {12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12)/x^2,x, algorithm="giac")

[Out]

x - 12/x

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maple [A]  time = 0.01, size = 8, normalized size = 0.32

method	result	size
default	\(x -\frac {12}{x}\)	\(8\)
risch	\(x -\frac {12}{x}\)	\(8\)
gosper	\(\frac {x^{2}-12}{x}\)	\(10\)
norman	\(\frac {x^{2}-12}{x}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+12)/x^2,x,method=_RETURNVERBOSE)

[Out]

x-12/x

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maxima [A]  time = 0.34, size = 7, normalized size = 0.28 \begin {gather*} x - \frac {12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12)/x^2,x, algorithm="maxima")

[Out]

x - 12/x

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mupad [B]  time = 0.02, size = 7, normalized size = 0.28 \begin {gather*} x-\frac {12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 12)/x^2,x)

[Out]

x - 12/x

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sympy [A]  time = 0.08, size = 3, normalized size = 0.12 \begin {gather*} x - \frac {12}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+12)/x**2,x)

[Out]

x - 12/x

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