3.102.10 \(\int \frac {-2+e^8 (-14-4 x)}{e^8} \, dx\)

Optimal. Leaf size=19 \[ \frac {2 (3-x) x \left (7+\frac {1}{e^8}+x\right )}{-3+x} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.95, 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 {1}{2} (2 x+7)^2-\frac {2 x}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + E^8*(-14 - 4*x))/E^8,x]

[Out]

(-2*x)/E^8 - (7 + 2*x)^2/2

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 (-2+e^8 (-14-4 x)\right ) \, dx}{e^8}\\ &=-\frac {2 x}{e^8}-\frac {1}{2} (7+2 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 \begin {gather*} -14 x-\frac {2 x}{e^8}-2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + E^8*(-14 - 4*x))/E^8,x]

[Out]

-14*x - (2*x)/E^8 - 2*x^2

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fricas [A]  time = 0.50, size = 16, normalized size = 0.84 \begin {gather*} -2 \, {\left ({\left (x^{2} + 7 \, x\right )} e^{8} + x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-14)*exp(4)^2-2)/exp(4)^2,x, algorithm="fricas")

[Out]

-2*((x^2 + 7*x)*e^8 + x)*e^(-8)

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giac [A]  time = 0.12, size = 16, normalized size = 0.84 \begin {gather*} -2 \, {\left ({\left (x^{2} + 7 \, x\right )} e^{8} + x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-14)*exp(4)^2-2)/exp(4)^2,x, algorithm="giac")

[Out]

-2*((x^2 + 7*x)*e^8 + x)*e^(-8)

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maple [A]  time = 0.02, size = 15, normalized size = 0.79

method	result	size
risch	\(-2 x^{2}-14 x -2 x \,{\mathrm e}^{-8}\)	\(15\)
gosper	\(-2 x \left (x \,{\mathrm e}^{8}+7 \,{\mathrm e}^{8}+1\right ) {\mathrm e}^{-8}\)	\(22\)
default	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} \left (-2 x^{2}-14 x \right )-2 x \right )\)	\(24\)
norman	\(\left (-2 x^{2} {\mathrm e}^{4}-2 \left (7 \,{\mathrm e}^{8}+1\right ) {\mathrm e}^{-4} x \right ) {\mathrm e}^{-4}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x-14)*exp(4)^2-2)/exp(4)^2,x,method=_RETURNVERBOSE)

[Out]

-2*x^2-14*x-2*x*exp(-8)

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maxima [A]  time = 0.34, size = 16, normalized size = 0.84 \begin {gather*} -2 \, {\left ({\left (x^{2} + 7 \, x\right )} e^{8} + x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-14)*exp(4)^2-2)/exp(4)^2,x, algorithm="maxima")

[Out]

-2*((x^2 + 7*x)*e^8 + x)*e^(-8)

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mupad [B]  time = 0.49, size = 16, normalized size = 0.84 \begin {gather*} -\frac {{\mathrm {e}}^{-16}\,{\left ({\mathrm {e}}^8\,\left (4\,x+14\right )+2\right )}^2}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-8)*(exp(8)*(4*x + 14) + 2),x)

[Out]

-(exp(-16)*(exp(8)*(4*x + 14) + 2)^2)/8

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sympy [A]  time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} - 2 x^{2} + \frac {x \left (- 14 e^{8} - 2\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x-14)*exp(4)**2-2)/exp(4)**2,x)

[Out]

-2*x**2 + x*(-14*exp(8) - 2)*exp(-8)

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