3.32.92 \(\int \frac {e^{\frac {x^2}{22+e^x}} (132 x+e^x (6 x-3 x^2))}{-7260+484 e^5+e^{2 x} (-15+e^5)+e^x (-660+44 e^5)} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^{\frac {x^2}{22+e^x}}}{-5+\frac {e^5}{3}} \]

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Rubi [A]  time = 0.64, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6688, 12, 6706} \begin {gather*} -\frac {3 e^{\frac {x^2}{e^x+22}}}{15-e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(x^2/(22 + E^x))*(132*x + E^x*(6*x - 3*x^2)))/(-7260 + 484*E^5 + E^(2*x)*(-15 + E^5) + E^x*(-660 + 44*E
^5)),x]

[Out]

(-3*E^(x^2/(22 + E^x)))/(15 - E^5)

Rule 12

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

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{\frac {x^2}{22+e^x}} \left (-44+e^x (-2+x)\right ) x}{\left (15-e^5\right ) \left (22+e^x\right )^2} \, dx\\ &=\frac {3 \int \frac {e^{\frac {x^2}{22+e^x}} \left (-44+e^x (-2+x)\right ) x}{\left (22+e^x\right )^2} \, dx}{15-e^5}\\ &=-\frac {3 e^{\frac {x^2}{22+e^x}}}{15-e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 22, normalized size = 0.88 \begin {gather*} \frac {3 e^{\frac {x^2}{22+e^x}}}{-15+e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(x^2/(22 + E^x))*(132*x + E^x*(6*x - 3*x^2)))/(-7260 + 484*E^5 + E^(2*x)*(-15 + E^5) + E^x*(-660
+ 44*E^5)),x]

[Out]

(3*E^(x^2/(22 + E^x)))/(-15 + E^5)

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fricas [A]  time = 0.68, size = 19, normalized size = 0.76 \begin {gather*} \frac {3 \, e^{\left (\frac {x^{2}}{e^{x} + 22}\right )}}{e^{5} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x)+132*x)*exp(x^2/(exp(x)+22))/((exp(5)-15)*exp(x)^2+(44*exp(5)-660)*exp(x)+484*ex
p(5)-7260),x, algorithm="fricas")

[Out]

3*e^(x^2/(e^x + 22))/(e^5 - 15)

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giac [A]  time = 0.23, size = 19, normalized size = 0.76 \begin {gather*} \frac {3 \, e^{\left (\frac {x^{2}}{e^{x} + 22}\right )}}{e^{5} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x)+132*x)*exp(x^2/(exp(x)+22))/((exp(5)-15)*exp(x)^2+(44*exp(5)-660)*exp(x)+484*ex
p(5)-7260),x, algorithm="giac")

[Out]

3*e^(x^2/(e^x + 22))/(e^5 - 15)

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maple [A]  time = 0.22, size = 20, normalized size = 0.80

method	result	size
risch	\(\frac {3 \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{x}+22}}}{{\mathrm e}^{5}-15}\)	\(20\)
norman	\(\frac {\frac {66 \,{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{x}+22}}}{{\mathrm e}^{5}-15}+\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{x}+22}}}{{\mathrm e}^{5}-15}}{{\mathrm e}^{x}+22}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2+6*x)*exp(x)+132*x)*exp(x^2/(exp(x)+22))/((exp(5)-15)*exp(x)^2+(44*exp(5)-660)*exp(x)+484*exp(5)-7
260),x,method=_RETURNVERBOSE)

[Out]

3/(exp(5)-15)*exp(x^2/(exp(x)+22))

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maxima [A]  time = 0.58, size = 19, normalized size = 0.76 \begin {gather*} \frac {3 \, e^{\left (\frac {x^{2}}{e^{x} + 22}\right )}}{e^{5} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+6*x)*exp(x)+132*x)*exp(x^2/(exp(x)+22))/((exp(5)-15)*exp(x)^2+(44*exp(5)-660)*exp(x)+484*ex
p(5)-7260),x, algorithm="maxima")

[Out]

3*e^(x^2/(e^x + 22))/(e^5 - 15)

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mupad [B]  time = 0.30, size = 19, normalized size = 0.76 \begin {gather*} \frac {3\,{\mathrm {e}}^{\frac {x^2}{{\mathrm {e}}^x+22}}}{{\mathrm {e}}^5-15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2/(exp(x) + 22))*(132*x + exp(x)*(6*x - 3*x^2)))/(484*exp(5) + exp(2*x)*(exp(5) - 15) + exp(x)*(44*
exp(5) - 660) - 7260),x)

[Out]

(3*exp(x^2/(exp(x) + 22)))/(exp(5) - 15)

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sympy [A]  time = 0.28, size = 15, normalized size = 0.60 \begin {gather*} \frac {3 e^{\frac {x^{2}}{e^{x} + 22}}}{-15 + e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2+6*x)*exp(x)+132*x)*exp(x**2/(exp(x)+22))/((exp(5)-15)*exp(x)**2+(44*exp(5)-660)*exp(x)+484
*exp(5)-7260),x)

[Out]

3*exp(x**2/(exp(x) + 22))/(-15 + exp(5))

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