3.101.46 \(\int \frac {1}{2} (-2-x \log (2)) \, dx\)

Optimal. Leaf size=24 \[ 1-x-\frac {1}{4} \log (2) \log \left (3 e^{x^2} \left (1+e^6\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {9} \begin {gather*} -\frac {(x \log (2)+2)^2}{4 \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - x*Log[2])/2,x]

[Out]

-1/4*(2 + x*Log[2])^2/Log[2]

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {(2+x \log (2))^2}{4 \log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.54 \begin {gather*} -x-\frac {1}{4} x^2 \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - x*Log[2])/2,x]

[Out]

-x - (x^2*Log[2])/4

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fricas [A]  time = 0.66, size = 11, normalized size = 0.46 \begin {gather*} -\frac {1}{4} \, x^{2} \log \relax (2) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2*log(2) - x

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giac [A]  time = 0.18, size = 11, normalized size = 0.46 \begin {gather*} -\frac {1}{4} \, x^{2} \log \relax (2) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2*log(2) - x

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maple [A]  time = 0.02, size = 10, normalized size = 0.42

method	result	size
gosper	\(-\frac {x \left (x \ln \relax (2)+4\right )}{4}\)	\(10\)
default	\(-\frac {x^{2} \ln \relax (2)}{4}-x\)	\(12\)
norman	\(-\frac {x^{2} \ln \relax (2)}{4}-x\)	\(12\)
risch	\(-\frac {x^{2} \ln \relax (2)}{4}-x\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x*(x*ln(2)+4)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.46 \begin {gather*} -\frac {1}{4} \, x^{2} \log \relax (2) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x^2*log(2) - x

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mupad [B]  time = 0.05, size = 11, normalized size = 0.46 \begin {gather*} -\frac {\ln \relax (2)\,x^2}{4}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (x*log(2))/2 - 1,x)

[Out]

- x - (x^2*log(2))/4

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sympy [A]  time = 0.05, size = 10, normalized size = 0.42 \begin {gather*} - \frac {x^{2} \log {\relax (2 )}}{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2*log(2)/4 - x

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