\(\int \frac {2 e^4+6 x \log (9)}{e^4} \, dx\) [10016]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=-5+2 x+\frac {3 x^2 \log (9)}{e^4} \]

[Out]

6*ln(3)*x^2/exp(2)^2-5+2*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {9} \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=\frac {\left (3 x \log (9)+e^4\right )^2}{3 e^4 \log (9)} \]

[In]

Int[(2*E^4 + 6*x*Log[9])/E^4,x]

[Out]

(E^4 + 3*x*Log[9])^2/(3*E^4*Log[9])

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[a*((b + c*x)^2/(2*c)), x] /; FreeQ[{a, b, c}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (e^4+3 x \log (9)\right )^2}{3 e^4 \log (9)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=2 x+\frac {3 x^2 \log (9)}{e^4} \]

[In]

Integrate[(2*E^4 + 6*x*Log[9])/E^4,x]

[Out]

2*x + (3*x^2*Log[9])/E^4

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method result size
risch \(2 x +6 \ln \left (3\right ) x^{2} {\mathrm e}^{-4}\) \(14\)
gosper \(2 x \left ({\mathrm e}^{4}+3 x \ln \left (3\right )\right ) {\mathrm e}^{-4}\) \(18\)
default \(2 \,{\mathrm e}^{-4} \left (3 x^{2} \ln \left (3\right )+x \,{\mathrm e}^{4}\right )\) \(21\)
parallelrisch \({\mathrm e}^{-4} \left (2 x \,{\mathrm e}^{4}+6 x^{2} \ln \left (3\right )\right )\) \(21\)
norman \(\left (2 \,{\mathrm e}^{2} x +6 \,{\mathrm e}^{-2} \ln \left (3\right ) x^{2}\right ) {\mathrm e}^{-2}\) \(23\)

[In]

int((12*x*ln(3)+2*exp(2)^2)/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

2*x+6*ln(3)*x^2*exp(-4)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=2 \, {\left (3 \, x^{2} \log \left (3\right ) + x e^{4}\right )} e^{\left (-4\right )} \]

[In]

integrate((12*x*log(3)+2*exp(2)^2)/exp(2)^2,x, algorithm="fricas")

[Out]

2*(3*x^2*log(3) + x*e^4)*e^(-4)

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=\frac {6 x^{2} \log {\left (3 \right )}}{e^{4}} + 2 x \]

[In]

integrate((12*x*ln(3)+2*exp(2)**2)/exp(2)**2,x)

[Out]

6*x**2*exp(-4)*log(3) + 2*x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=2 \, {\left (3 \, x^{2} \log \left (3\right ) + x e^{4}\right )} e^{\left (-4\right )} \]

[In]

integrate((12*x*log(3)+2*exp(2)^2)/exp(2)^2,x, algorithm="maxima")

[Out]

2*(3*x^2*log(3) + x*e^4)*e^(-4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=2 \, {\left (3 \, x^{2} \log \left (3\right ) + x e^{4}\right )} e^{\left (-4\right )} \]

[In]

integrate((12*x*log(3)+2*exp(2)^2)/exp(2)^2,x, algorithm="giac")

[Out]

2*(3*x^2*log(3) + x*e^4)*e^(-4)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {2 e^4+6 x \log (9)}{e^4} \, dx=\frac {{\mathrm {e}}^{-4}\,{\left ({\mathrm {e}}^4+6\,x\,\ln \left (3\right )\right )}^2}{6\,\ln \left (3\right )} \]

[In]

int(exp(-4)*(2*exp(4) + 12*x*log(3)),x)

[Out]

(exp(-4)*(exp(4) + 6*x*log(3))^2)/(6*log(3))