3.88.6 \(\int \frac {x}{2} \, dx\)

Optimal. Leaf size=16 \[ 1-e^4+\frac {1}{4} \left (4+x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {12, 30} \begin {gather*} \frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/2,x]

[Out]

x^2/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 {\int x \, dx}{2}\\ &=\frac {x^2}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.44 \begin {gather*} \frac {x^2}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/2,x]

[Out]

x^2/4

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fricas [A]  time = 0.58, size = 5, normalized size = 0.31 \begin {gather*} \frac {1}{4} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2

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giac [A]  time = 0.21, size = 5, normalized size = 0.31 \begin {gather*} \frac {1}{4} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x,x, algorithm="giac")

[Out]

1/4*x^2

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maple [A]  time = 0.02, size = 6, normalized size = 0.38

method	result	size
gosper	\(\frac {x^{2}}{4}\)	\(6\)
default	\(\frac {x^{2}}{4}\)	\(6\)
norman	\(\frac {x^{2}}{4}\)	\(6\)
risch	\(\frac {x^{2}}{4}\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*x,x,method=_RETURNVERBOSE)

[Out]

1/4*x^2

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maxima [A]  time = 0.35, size = 5, normalized size = 0.31 \begin {gather*} \frac {1}{4} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^2

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mupad [B]  time = 0.03, size = 5, normalized size = 0.31 \begin {gather*} \frac {x^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/2,x)

[Out]

x^2/4

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sympy [A]  time = 0.01, size = 3, normalized size = 0.19 \begin {gather*} \frac {x^{2}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x,x)

[Out]

x**2/4

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