3.11.33 \(\int (4-8 e^{2 x}) \, dx\)

Optimal. Leaf size=12 \[ 4 \left (2-e^{2 x}+x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2194} \begin {gather*} 4 x-4 e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 - 8*E^(2*x),x]

[Out]

-4*E^(2*x) + 4*x

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} &=4 x-8 \int e^{2 x} \, dx\\ &=-4 e^{2 x}+4 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} -4 e^{2 x}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 - 8*E^(2*x),x]

[Out]

-4*E^(2*x) + 4*x

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fricas [A]  time = 1.07, size = 10, normalized size = 0.83 \begin {gather*} 4 \, x - 4 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - 4*e^(2*x)

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giac [A]  time = 0.23, size = 10, normalized size = 0.83 \begin {gather*} 4 \, x - 4 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - 4*e^(2*x)

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maple [A]  time = 0.02, size = 11, normalized size = 0.92

method	result	size
default	\(4 x -4 \,{\mathrm e}^{2 x}\)	\(11\)
norman	\(4 x -4 \,{\mathrm e}^{2 x}\)	\(11\)
risch	\(4 x -4 \,{\mathrm e}^{2 x}\)	\(11\)
derivativedivides	\(-4 \,{\mathrm e}^{2 x}+2 \ln \left ({\mathrm e}^{2 x}\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x-4*exp(2*x)

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maxima [A]  time = 0.42, size = 10, normalized size = 0.83 \begin {gather*} 4 \, x - 4 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - 4*e^(2*x)

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mupad [B]  time = 0.05, size = 10, normalized size = 0.83 \begin {gather*} 4\,x-4\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4 - 8*exp(2*x),x)

[Out]

4*x - 4*exp(2*x)

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sympy [A]  time = 0.07, size = 8, normalized size = 0.67 \begin {gather*} 4 x - 4 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - 4*exp(2*x)

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