3.68.26 \(\int \frac {1}{2} (-3+2 e^{2 x}) \, dx\)

Optimal. Leaf size=21 \[ 6+\frac {1}{2} \left (-\frac {5}{4}+e^{2 x}-x\right )-x \]

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 0.71, 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 {e^{2 x}}{2}-\frac {3 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 2*E^(2*x))/2,x]

[Out]

E^(2*x)/2 - (3*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 (-3+2 e^{2 x}\right ) \, dx\\ &=-\frac {3 x}{2}+\int e^{2 x} \, dx\\ &=\frac {e^{2 x}}{2}-\frac {3 x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.71 \begin {gather*} \frac {e^{2 x}}{2}-\frac {3 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 2*E^(2*x))/2,x]

[Out]

E^(2*x)/2 - (3*x)/2

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fricas [A]  time = 0.51, size = 10, normalized size = 0.48 \begin {gather*} -\frac {3}{2} \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)-3/2,x, algorithm="fricas")

[Out]

-3/2*x + 1/2*e^(2*x)

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giac [A]  time = 0.15, size = 10, normalized size = 0.48 \begin {gather*} -\frac {3}{2} \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)-3/2,x, algorithm="giac")

[Out]

-3/2*x + 1/2*e^(2*x)

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maple [A]  time = 0.02, size = 11, normalized size = 0.52

method	result	size
default	\(-\frac {3 x}{2}+\frac {{\mathrm e}^{2 x}}{2}\)	\(11\)
norman	\(-\frac {3 x}{2}+\frac {{\mathrm e}^{2 x}}{2}\)	\(11\)
risch	\(-\frac {3 x}{2}+\frac {{\mathrm e}^{2 x}}{2}\)	\(11\)
derivativedivides	\(\frac {{\mathrm e}^{2 x}}{2}-\frac {3 \ln \left ({\mathrm e}^{2 x}\right )}{4}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)-3/2,x,method=_RETURNVERBOSE)

[Out]

-3/2*x+1/2*exp(2*x)

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maxima [A]  time = 0.36, size = 10, normalized size = 0.48 \begin {gather*} -\frac {3}{2} \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)-3/2,x, algorithm="maxima")

[Out]

-3/2*x + 1/2*e^(2*x)

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mupad [B]  time = 0.04, size = 10, normalized size = 0.48 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{2}-\frac {3\,x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x) - 3/2,x)

[Out]

exp(2*x)/2 - (3*x)/2

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sympy [A]  time = 0.06, size = 10, normalized size = 0.48 \begin {gather*} - \frac {3 x}{2} + \frac {e^{2 x}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)-3/2,x)

[Out]

-3*x/2 + exp(2*x)/2

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