3.62.8 \(\int \frac {e^{12}+e^6 (9-3 e^5)+18 x}{e^{12}} \, dx\)

Optimal. Leaf size=19 \[ x+\frac {1}{4} \left (-3+e^5-\frac {6 x}{e^6}\right )^2 \]

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Rubi [A]  time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.42, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {9} \begin {gather*} \frac {\left (18 x+e^6 \left (9-3 e^5+e^6\right )\right )^2}{36 e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^12 + E^6*(9 - 3*E^5) + 18*x)/E^12,x]

[Out]

(E^6*(9 - 3*E^5 + E^6) + 18*x)^2/(36*E^12)

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 {gather*} \begin {aligned} \text {integral} &=\frac {\left (e^6 \left (9-3 e^5+e^6\right )+18 x\right )^2}{36 e^{12}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.16 \begin {gather*} x+\frac {9 x}{e^6}-\frac {3 x}{e}+\frac {9 x^2}{e^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^12 + E^6*(9 - 3*E^5) + 18*x)/E^12,x]

[Out]

x + (9*x)/E^6 - (3*x)/E + (9*x^2)/E^12

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fricas [A]  time = 0.63, size = 23, normalized size = 1.21 \begin {gather*} {\left (9 \, x^{2} + x e^{12} - 3 \, x e^{11} + 9 \, x e^{6}\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(6)^2+(-3*exp(5)+9)*exp(6)+18*x)/exp(6)^2,x, algorithm="fricas")

[Out]

(9*x^2 + x*e^12 - 3*x*e^11 + 9*x*e^6)*e^(-12)

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giac [A]  time = 0.35, size = 24, normalized size = 1.26 \begin {gather*} -{\left (3 \, x {\left (e^{5} - 3\right )} e^{6} - 9 \, x^{2} - x e^{12}\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(6)^2+(-3*exp(5)+9)*exp(6)+18*x)/exp(6)^2,x, algorithm="giac")

[Out]

-(3*x*(e^5 - 3)*e^6 - 9*x^2 - x*e^12)*e^(-12)

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maple [A]  time = 0.04, size = 20, normalized size = 1.05

method	result	size
risch	\(x -3 \,{\mathrm e}^{-1} x +9 \,{\mathrm e}^{-6} x +9 \,{\mathrm e}^{-12} x^{2}\)	\(20\)
norman	\(\left (\left (-3 \,{\mathrm e}^{5}+{\mathrm e}^{6}+9\right ) x +9 \,{\mathrm e}^{-6} x^{2}\right ) {\mathrm e}^{-6}\)	\(26\)
gosper	\(-x \left (3 \,{\mathrm e}^{6} {\mathrm e}^{5}-{\mathrm e}^{12}-9 \,{\mathrm e}^{6}-9 x \right ) {\mathrm e}^{-12}\)	\(28\)
default	\({\mathrm e}^{-12} \left (x \,{\mathrm e}^{12}+\left (-3 \,{\mathrm e}^{5}+9\right ) {\mathrm e}^{6} x +9 x^{2}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6)^2+(-3*exp(5)+9)*exp(6)+18*x)/exp(6)^2,x,method=_RETURNVERBOSE)

[Out]

x-3*exp(-1)*x+9*exp(-6)*x+9*exp(-12)*x^2

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maxima [A]  time = 0.44, size = 24, normalized size = 1.26 \begin {gather*} -{\left (3 \, x {\left (e^{5} - 3\right )} e^{6} - 9 \, x^{2} - x e^{12}\right )} e^{\left (-12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(6)^2+(-3*exp(5)+9)*exp(6)+18*x)/exp(6)^2,x, algorithm="maxima")

[Out]

-(3*x*(e^5 - 3)*e^6 - 9*x^2 - x*e^12)*e^(-12)

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mupad [B]  time = 0.25, size = 22, normalized size = 1.16 \begin {gather*} \frac {{\mathrm {e}}^{-12}\,{\left (18\,x+{\mathrm {e}}^{12}-{\mathrm {e}}^6\,\left (3\,{\mathrm {e}}^5-9\right )\right )}^2}{36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-12)*(18*x + exp(12) - exp(6)*(3*exp(5) - 9)),x)

[Out]

(exp(-12)*(18*x + exp(12) - exp(6)*(3*exp(5) - 9))^2)/36

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sympy [A]  time = 0.06, size = 22, normalized size = 1.16 \begin {gather*} \frac {9 x^{2}}{e^{12}} + \frac {x \left (- 3 e^{5} + 9 + e^{6}\right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(6)**2+(-3*exp(5)+9)*exp(6)+18*x)/exp(6)**2,x)

[Out]

9*x**2*exp(-12) + x*(-3*exp(5) + 9 + exp(6))*exp(-6)

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