3.14.51 \(\int \frac {e^4 (156-18 x)-8 x+e^2 (-104+24 x)}{e^4} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.28, 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}{e^4}+\frac {4 (13-3 x)^2}{3 e^2}-(26-3 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(156 - 18*x) - 8*x + E^2*(-104 + 24*x))/E^4,x]

[Out]

(4*(13 - 3*x)^2)/(3*E^2) - (26 - 3*x)^2 - (4*x^2)/E^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 \left (e^4 (156-18 x)-8 x+e^2 (-104+24 x)\right ) \, dx}{e^4}\\ &=\frac {4 (13-3 x)^2}{3 e^2}-(26-3 x)^2-\frac {4 x^2}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.36 \begin {gather*} -\frac {2 \left (-2+3 e^2\right ) \left (-26 e^2 x-x^2+\frac {3 e^2 x^2}{2}\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(156 - 18*x) - 8*x + E^2*(-104 + 24*x))/E^4,x]

[Out]

(-2*(-2 + 3*E^2)*(-26*E^2*x - x^2 + (3*E^2*x^2)/2))/E^4

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fricas [A]  time = 1.61, size = 36, normalized size = 1.44 \begin {gather*} -{\left (4 \, x^{2} + 3 \, {\left (3 \, x^{2} - 52 \, x\right )} e^{4} - 4 \, {\left (3 \, x^{2} - 26 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x+156)*exp(2)^2+(24*x-104)*exp(2)-8*x)/exp(2)^2,x, algorithm="fricas")

[Out]

-(4*x^2 + 3*(3*x^2 - 52*x)*e^4 - 4*(3*x^2 - 26*x)*e^2)*e^(-4)

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giac [A]  time = 0.22, size = 36, normalized size = 1.44 \begin {gather*} -{\left (4 \, x^{2} + 3 \, {\left (3 \, x^{2} - 52 \, x\right )} e^{4} - 4 \, {\left (3 \, x^{2} - 26 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x+156)*exp(2)^2+(24*x-104)*exp(2)-8*x)/exp(2)^2,x, algorithm="giac")

[Out]

-(4*x^2 + 3*(3*x^2 - 52*x)*e^4 - 4*(3*x^2 - 26*x)*e^2)*e^(-4)

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maple [A]  time = 0.02, size = 27, normalized size = 1.08

method	result	size
gosper	\(-\left (-2+3 \,{\mathrm e}^{2}\right ) x \left (3 \,{\mathrm e}^{2} x -52 \,{\mathrm e}^{2}-2 x \right ) {\mathrm e}^{-4}\)	\(27\)
risch	\(-9 x^{2}+156 x +12 x^{2} {\mathrm e}^{-2}-104 x \,{\mathrm e}^{-2}-4 \,{\mathrm e}^{-4} x^{2}\)	\(29\)
norman	\(\left (\left (156 \,{\mathrm e}^{2}-104\right ) x -\left (9 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{-2}\)	\(36\)
default	\({\mathrm e}^{-4} \left ({\mathrm e}^{4} \left (-9 x^{2}+156 x \right )+{\mathrm e}^{2} \left (12 x^{2}-104 x \right )-4 x^{2}\right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-18*x+156)*exp(2)^2+(24*x-104)*exp(2)-8*x)/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

-(-2+3*exp(2))*x*(3*exp(2)*x-52*exp(2)-2*x)/exp(2)^2

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maxima [A]  time = 0.37, size = 36, normalized size = 1.44 \begin {gather*} -{\left (4 \, x^{2} + 3 \, {\left (3 \, x^{2} - 52 \, x\right )} e^{4} - 4 \, {\left (3 \, x^{2} - 26 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x+156)*exp(2)^2+(24*x-104)*exp(2)-8*x)/exp(2)^2,x, algorithm="maxima")

[Out]

-(4*x^2 + 3*(3*x^2 - 52*x)*e^4 - 4*(3*x^2 - 26*x)*e^2)*e^(-4)

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mupad [B]  time = 0.07, size = 27, normalized size = 1.08 \begin {gather*} 52\,x\,{\mathrm {e}}^{-2}\,\left (3\,{\mathrm {e}}^2-2\right )-x^2\,{\mathrm {e}}^{-4}\,{\left (3\,{\mathrm {e}}^2-2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-4)*(8*x - exp(2)*(24*x - 104) + exp(4)*(18*x - 156)),x)

[Out]

52*x*exp(-2)*(3*exp(2) - 2) - x^2*exp(-4)*(3*exp(2) - 2)^2

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sympy [A]  time = 0.06, size = 29, normalized size = 1.16 \begin {gather*} \frac {x^{2} \left (- 9 e^{4} - 4 + 12 e^{2}\right )}{e^{4}} + \frac {x \left (-104 + 156 e^{2}\right )}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x+156)*exp(2)**2+(24*x-104)*exp(2)-8*x)/exp(2)**2,x)

[Out]

x**2*(-9*exp(4) - 4 + 12*exp(2))*exp(-4) + x*(-104 + 156*exp(2))*exp(-2)

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