3.20.55 \(\int \frac {-4+8 x+(-2+e (-1+2 x)) \log (4)}{4+e \log (4)} \, dx\)

Optimal. Leaf size=20 \[ -x+x \left (x-\frac {2}{e+\frac {4}{\log (4)}}\right ) \]

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Rubi [B]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 2.65, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12} \begin {gather*} \frac {4 x^2}{4+e \log (4)}+\frac {(e (1-2 x)+2)^2 \log (4)}{4 e (4+e \log (4))}-\frac {4 x}{4+e \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 8*x + (-2 + E*(-1 + 2*x))*Log[4])/(4 + E*Log[4]),x]

[Out]

(-4*x)/(4 + E*Log[4]) + (4*x^2)/(4 + E*Log[4]) + ((2 + E*(1 - 2*x))^2*Log[4])/(4*E*(4 + E*Log[4]))

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int (-4+8 x+(-2+e (-1+2 x)) \log (4)) \, dx}{4+e \log (4)}\\ &=-\frac {4 x}{4+e \log (4)}+\frac {4 x^2}{4+e \log (4)}+\frac {(2+e (1-2 x))^2 \log (4)}{4 e (4+e \log (4))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.70 \begin {gather*} \frac {-4 x+4 x^2+(-2-e) x \log (4)+e x^2 \log (4)}{4+e \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 8*x + (-2 + E*(-1 + 2*x))*Log[4])/(4 + E*Log[4]),x]

[Out]

(-4*x + 4*x^2 + (-2 - E)*x*Log[4] + E*x^2*Log[4])/(4 + E*Log[4])

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fricas [A]  time = 1.05, size = 36, normalized size = 1.80 \begin {gather*} \frac {2 \, x^{2} + {\left ({\left (x^{2} - x\right )} e - 2 \, x\right )} \log \relax (2) - 2 \, x}{e \log \relax (2) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*((2*x-1)*exp(1)-2)*log(2)+8*x-4)/(2*exp(1)*log(2)+4),x, algorithm="fricas")

[Out]

(2*x^2 + ((x^2 - x)*e - 2*x)*log(2) - 2*x)/(e*log(2) + 2)

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giac [A]  time = 0.30, size = 36, normalized size = 1.80 \begin {gather*} \frac {2 \, x^{2} + {\left ({\left (x^{2} - x\right )} e - 2 \, x\right )} \log \relax (2) - 2 \, x}{e \log \relax (2) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*((2*x-1)*exp(1)-2)*log(2)+8*x-4)/(2*exp(1)*log(2)+4),x, algorithm="giac")

[Out]

(2*x^2 + ((x^2 - x)*e - 2*x)*log(2) - 2*x)/(e*log(2) + 2)

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maple [A]  time = 0.03, size = 28, normalized size = 1.40

method	result	size
norman	\(x^{2}-\frac {\left ({\mathrm e} \ln \relax (2)+2 \ln \relax (2)+2\right ) x}{{\mathrm e} \ln \relax (2)+2}\)	\(28\)
gosper	\(\frac {x \left (x \,{\mathrm e} \ln \relax (2)-{\mathrm e} \ln \relax (2)-2 \ln \relax (2)+2 x -2\right )}{{\mathrm e} \ln \relax (2)+2}\)	\(33\)
default	\(\frac {2 \ln \relax (2) \left ({\mathrm e} \left (x^{2}-x \right )-2 x \right )+4 x^{2}-4 x}{2 \,{\mathrm e} \ln \relax (2)+4}\)	\(39\)
risch	\(\frac {2 \ln \relax (2) {\mathrm e} x^{2}}{2 \,{\mathrm e} \ln \relax (2)+4}-\frac {2 x \,{\mathrm e} \ln \relax (2)}{2 \,{\mathrm e} \ln \relax (2)+4}-\frac {4 x \ln \relax (2)}{2 \,{\mathrm e} \ln \relax (2)+4}+\frac {4 x^{2}}{2 \,{\mathrm e} \ln \relax (2)+4}-\frac {4 x}{2 \,{\mathrm e} \ln \relax (2)+4}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*((2*x-1)*exp(1)-2)*ln(2)+8*x-4)/(2*exp(1)*ln(2)+4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.41, size = 36, normalized size = 1.80 \begin {gather*} \frac {2 \, x^{2} + {\left ({\left (x^{2} - x\right )} e - 2 \, x\right )} \log \relax (2) - 2 \, x}{e \log \relax (2) + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*((2*x-1)*exp(1)-2)*log(2)+8*x-4)/(2*exp(1)*log(2)+4),x, algorithm="maxima")

[Out]

(2*x^2 + ((x^2 - x)*e - 2*x)*log(2) - 2*x)/(e*log(2) + 2)

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mupad [B]  time = 0.12, size = 25, normalized size = 1.25 \begin {gather*} \frac {x^2\,\left (2\,\mathrm {e}\,\ln \relax (2)+4\right )}{2\,\left (\mathrm {e}\,\ln \relax (2)+2\right )}-\frac {x\,\left (\ln \relax (2)\,\left (\mathrm {e}+2\right )+2\right )}{\mathrm {e}\,\ln \relax (2)+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + 2*log(2)*(exp(1)*(2*x - 1) - 2) - 4)/(2*exp(1)*log(2) + 4),x)

[Out]

(x^2*(2*exp(1)*log(2) + 4))/(2*(exp(1)*log(2) + 2)) - (x*(log(2)*(exp(1) + 2) + 2))/(exp(1)*log(2) + 2)

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sympy [A]  time = 0.07, size = 27, normalized size = 1.35 \begin {gather*} x^{2} + \frac {x \left (-2 - e \log {\relax (2 )} - 2 \log {\relax (2 )}\right )}{e \log {\relax (2 )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*((2*x-1)*exp(1)-2)*ln(2)+8*x-4)/(2*exp(1)*ln(2)+4),x)

[Out]

x**2 + x*(-2 - E*log(2) - 2*log(2))/(E*log(2) + 2)

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