3.54.5 \(\int \frac {1}{2} (-1-80 e^{2 x}) \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.52, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 2194} \begin {gather*} -\frac {x}{2}-20 e^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 80*E^(2*x))/2,x]

[Out]

-20*E^(2*x) - x/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{2} \int \left (-1-80 e^{2 x}\right ) \, dx\\ &=-\frac {x}{2}-40 \int e^{2 x} \, dx\\ &=-20 e^{2 x}-\frac {x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.52 \begin {gather*} -20 e^{2 x}-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 80*E^(2*x))/2,x]

[Out]

-20*E^(2*x) - x/2

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fricas [A]  time = 0.99, size = 10, normalized size = 0.40 \begin {gather*} -\frac {1}{2} \, x - 20 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x)^2-1/2,x, algorithm="fricas")

[Out]

-1/2*x - 20*e^(2*x)

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giac [A]  time = 1.97, size = 10, normalized size = 0.40 \begin {gather*} -\frac {1}{2} \, x - 20 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x)^2-1/2,x, algorithm="giac")

[Out]

-1/2*x - 20*e^(2*x)

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maple [A]  time = 0.03, size = 11, normalized size = 0.44

method	result	size
default	\(-\frac {x}{2}-20 \,{\mathrm e}^{2 x}\)	\(11\)
norman	\(-\frac {x}{2}-20 \,{\mathrm e}^{2 x}\)	\(11\)
risch	\(-\frac {x}{2}-20 \,{\mathrm e}^{2 x}\)	\(11\)
derivativedivides	\(-20 \,{\mathrm e}^{2 x}-\frac {\ln \left ({\mathrm e}^{x}\right )}{2}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-40*exp(x)^2-1/2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x-20*exp(x)^2

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maxima [A]  time = 0.36, size = 10, normalized size = 0.40 \begin {gather*} -\frac {1}{2} \, x - 20 \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x)^2-1/2,x, algorithm="maxima")

[Out]

-1/2*x - 20*e^(2*x)

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mupad [B]  time = 0.05, size = 10, normalized size = 0.40 \begin {gather*} -\frac {x}{2}-20\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 40*exp(2*x) - 1/2,x)

[Out]

- x/2 - 20*exp(2*x)

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sympy [A]  time = 0.07, size = 10, normalized size = 0.40 \begin {gather*} - \frac {x}{2} - 20 e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-40*exp(x)**2-1/2,x)

[Out]

-x/2 - 20*exp(2*x)

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