3.86.56 \(\int \frac {1}{2} (4+\log (x^2)) \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 2295} \begin {gather*} \frac {1}{2} x \log \left (x^2\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + Log[x^2])/2,x]

[Out]

x + (x*Log[x^2])/2

Rule 12

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} x+\frac {1}{2} x \log \left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + Log[x^2])/2,x]

[Out]

x + (x*Log[x^2])/2

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fricas [A]  time = 0.61, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x \log \left (x^{2}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x^2) + x

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giac [A]  time = 0.15, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x \log \left (x^{2}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2+1/2*log(x^2),x, algorithm="giac")

[Out]

1/2*x*log(x^2) + x

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maple [A]  time = 0.02, size = 10, normalized size = 0.91

method	result	size
default	\(x +\frac {x \ln \left (x^{2}\right )}{2}\)	\(10\)
norman	\(x +\frac {x \ln \left (x^{2}\right )}{2}\)	\(10\)
risch	\(x +\frac {x \ln \left (x^{2}\right )}{2}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/2*x*ln(x^2)

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maxima [A]  time = 0.36, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x \log \left (x^{2}\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*log(x^2) + x

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mupad [B]  time = 5.16, size = 9, normalized size = 0.82 \begin {gather*} \frac {x\,\left (\ln \left (x^2\right )+2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x^2)/2 + 2,x)

[Out]

(x*(log(x^2) + 2))/2

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sympy [A]  time = 0.08, size = 8, normalized size = 0.73 \begin {gather*} \frac {x \log {\left (x^{2} \right )}}{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2+1/2*ln(x**2),x)

[Out]

x*log(x**2)/2 + x

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