3.36.22 \(\int \frac {1}{4} (1+e^6 (4-4 x)) \, dx\)

Optimal. Leaf size=25 \[ -\frac {3}{5}+\frac {x}{4}-e^6 \left (-x+\frac {x^2}{2}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12} \begin {gather*} \frac {x}{4}-\frac {1}{2} e^6 (1-x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^6*(4 - 4*x))/4,x]

[Out]

-1/2*(E^6*(1 - x)^2) + x/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 {1}{4} \int \left (1+e^6 (4-4 x)\right ) \, dx\\ &=-\frac {1}{2} e^6 (1-x)^2+\frac {x}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 0.84 \begin {gather*} \frac {x}{4}+e^6 x-\frac {e^6 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^6*(4 - 4*x))/4,x]

[Out]

x/4 + E^6*x - (E^6*x^2)/2

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fricas [A]  time = 0.59, size = 15, normalized size = 0.60 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 2 \, x\right )} e^{6} + \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x+4)*exp(3)^2+1/4,x, algorithm="fricas")

[Out]

-1/2*(x^2 - 2*x)*e^6 + 1/4*x

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giac [A]  time = 0.14, size = 15, normalized size = 0.60 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 2 \, x\right )} e^{6} + \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x+4)*exp(3)^2+1/4,x, algorithm="giac")

[Out]

-1/2*(x^2 - 2*x)*e^6 + 1/4*x

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maple [A]  time = 0.02, size = 16, normalized size = 0.64

method	result	size
risch	\(-\frac {x^{2} {\mathrm e}^{6}}{2}+x \,{\mathrm e}^{6}+\frac {x}{4}\)	\(16\)
gosper	\(-\frac {x \left (2 x \,{\mathrm e}^{6}-4 \,{\mathrm e}^{6}-1\right )}{4}\)	\(19\)
norman	\(\left ({\mathrm e}^{6}+\frac {1}{4}\right ) x -\frac {x^{2} {\mathrm e}^{6}}{2}\)	\(19\)
default	\(\frac {{\mathrm e}^{6} \left (-2 x^{2}+4 x \right )}{4}+\frac {x}{4}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*x+4)*exp(3)^2+1/4,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.47, size = 15, normalized size = 0.60 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} - 2 \, x\right )} e^{6} + \frac {1}{4} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x+4)*exp(3)^2+1/4,x, algorithm="maxima")

[Out]

-1/2*(x^2 - 2*x)*e^6 + 1/4*x

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mupad [B]  time = 0.07, size = 17, normalized size = 0.68 \begin {gather*} -\frac {\left (4\,x-4\right )\,\left ({\mathrm {e}}^6\,\left (4\,x-4\right )-2\right )}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4 - (exp(6)*(4*x - 4))/4,x)

[Out]

-((4*x - 4)*(exp(6)*(4*x - 4) - 2))/32

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sympy [A]  time = 0.05, size = 15, normalized size = 0.60 \begin {gather*} - \frac {x^{2} e^{6}}{2} + x \left (\frac {1}{4} + e^{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x+4)*exp(3)**2+1/4,x)

[Out]

-x**2*exp(6)/2 + x*(1/4 + exp(6))

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