3.12.34 \(\int \frac {297+e^3+297 \log (2 x)}{e^3} \, dx\) [1134]

Optimal. Leaf size=12 \[ x+\frac {297 x \log (2 x)}{e^3} \]

[Out]

x+297/exp(3)*x*ln(2*x)

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(12)=24\).
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 2.25, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 2332} \begin {gather*} \frac {\left (297+e^3\right ) x}{e^3}-\frac {297 x}{e^3}+\frac {297 x \log (2 x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(297 + E^3 + 297*Log[2*x])/E^3,x]

[Out]

(-297*x)/E^3 + ((297 + E^3)*x)/E^3 + (297*x*Log[2*x])/E^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (297+e^3+297 \log (2 x)\right ) \, dx}{e^3}\\ &=\frac {\left (297+e^3\right ) x}{e^3}+\frac {297 \int \log (2 x) \, dx}{e^3}\\ &=-\frac {297 x}{e^3}+\frac {\left (297+e^3\right ) x}{e^3}+\frac {297 x \log (2 x)}{e^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.42 \begin {gather*} \frac {e^3 x+297 x \log (2 x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(297 + E^3 + 297*Log[2*x])/E^3,x]

[Out]

(E^3*x + 297*x*Log[2*x])/E^3

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Maple [A]
time = 0.09, size = 18, normalized size = 1.50

method	result	size
risch	\(x +297 \,{\mathrm e}^{-3} x \ln \left (2 x \right )\)	\(12\)
norman	\(x +297 \,{\mathrm e}^{-3} x \ln \left (2 x \right )\)	\(14\)
default	\({\mathrm e}^{-3} \left (297 x \ln \left (2 x \right )+x \,{\mathrm e}^{3}\right )\)	\(18\)
derivativedivides	\(\frac {{\mathrm e}^{-3} \left (594 x \ln \left (2 x \right )+2 x \,{\mathrm e}^{3}\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((297*ln(2*x)+exp(3)+297)/exp(3),x,method=_RETURNVERBOSE)

[Out]

1/exp(3)*(297*x*ln(2*x)+x*exp(3))

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Maxima [A]
time = 0.26, size = 15, normalized size = 1.25 \begin {gather*} {\left (x e^{3} + 297 \, x \log \left (2 \, x\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((297*log(2*x)+exp(3)+297)/exp(3),x, algorithm="maxima")

[Out]

(x*e^3 + 297*x*log(2*x))*e^(-3)

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Fricas [A]
time = 0.34, size = 15, normalized size = 1.25 \begin {gather*} {\left (x e^{3} + 297 \, x \log \left (2 \, x\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((297*log(2*x)+exp(3)+297)/exp(3),x, algorithm="fricas")

[Out]

(x*e^3 + 297*x*log(2*x))*e^(-3)

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Sympy [A]
time = 0.04, size = 12, normalized size = 1.00 \begin {gather*} \frac {297 x \log {\left (2 x \right )}}{e^{3}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((297*ln(2*x)+exp(3)+297)/exp(3),x)

[Out]

297*x*exp(-3)*log(2*x) + x

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Giac [A]
time = 0.39, size = 15, normalized size = 1.25 \begin {gather*} {\left (x e^{3} + 297 \, x \log \left (2 \, x\right )\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((297*log(2*x)+exp(3)+297)/exp(3),x, algorithm="giac")

[Out]

(x*e^3 + 297*x*log(2*x))*e^(-3)

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Mupad [B]
time = 0.79, size = 12, normalized size = 1.00 \begin {gather*} x\,\left (297\,\ln \left (2\,x\right )\,{\mathrm {e}}^{-3}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-3)*(297*log(2*x) + exp(3) + 297),x)

[Out]

x*(297*log(2*x)*exp(-3) + 1)

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