3.32.31 \(\int (-2 e^{2 x}-2 x) \, dx\)

Optimal. Leaf size=19 \[ -\frac {41}{4}+e^3-e^{2 x}-x^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2194} \begin {gather*} -x^2-e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2*E^(2*x) - 2*x,x]

[Out]

-E^(2*x) - x^2

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x^2-2 \int e^{2 x} \, dx\\ &=-e^{2 x}-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} -2 \left (\frac {e^{2 x}}{2}+\frac {x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2*E^(2*x) - 2*x,x]

[Out]

-2*(E^(2*x)/2 + x^2/2)

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fricas [A]  time = 0.48, size = 12, normalized size = 0.63 \begin {gather*} -x^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - e^(2*x)

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giac [A]  time = 0.17, size = 12, normalized size = 0.63 \begin {gather*} -x^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - e^(2*x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.68

method	result	size
default	\(-x^{2}-{\mathrm e}^{2 x}\)	\(13\)
norman	\(-x^{2}-{\mathrm e}^{2 x}\)	\(13\)
risch	\(-x^{2}-{\mathrm e}^{2 x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2-exp(x)^2

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maxima [A]  time = 0.51, size = 12, normalized size = 0.63 \begin {gather*} -x^{2} - e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 - e^(2*x)

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mupad [B]  time = 1.80, size = 12, normalized size = 0.63 \begin {gather*} -{\mathrm {e}}^{2\,x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 2*x - 2*exp(2*x),x)

[Out]

- exp(2*x) - x^2

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sympy [A]  time = 0.07, size = 8, normalized size = 0.42 \begin {gather*} - x^{2} - e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*exp(x)**2-2*x,x)

[Out]

-x**2 - exp(2*x)

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