\(\int \frac {1}{2} (-2-x \log (2)) \, dx\) [10046]

Optimal result

Integrand size = 11, antiderivative size = 24 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=1-x-\frac {1}{4} \log (2) \log \left (3 e^{x^2} \left (1+e^6\right )\right ) \]

[Out]

1-x-1/4*ln(3*(1+exp(6))*exp(x^2))*ln(2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {9} \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-\frac {(x \log (2)+2)^2}{4 \log (2)} \]

[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{align*} \text {integral}& = -\frac {(2+x \log (2))^2}{4 \log (2)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-x-\frac {1}{4} x^2 \log (2) \]

[In]

Integrate[(-2 - x*Log[2])/2,x]

[Out]

-x - (x^2*Log[2])/4

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-\frac {1}{4} \, x^{2} \log \left (2\right ) - x \]

[In]

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

[Out]

-1/4*x^2*log(2) - x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.42 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=- \frac {x^{2} \log {\left (2 \right )}}{4} - x \]

[In]

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

[Out]

-x**2*log(2)/4 - x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-\frac {1}{4} \, x^{2} \log \left (2\right ) - x \]

[In]

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

[Out]

-1/4*x^2*log(2) - x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-\frac {1}{4} \, x^{2} \log \left (2\right ) - x \]

[In]

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

[Out]

-1/4*x^2*log(2) - x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2} (-2-x \log (2)) \, dx=-\frac {\ln \left (2\right )\,x^2}{4}-x \]

[In]

int(- (x*log(2))/2 - 1,x)

[Out]

- x - (x^2*log(2))/4