3.2.20 \(\int \frac {324-432 e^x}{4624+144 e^8+144 e^{2 x}+e^x (-1632-288 e^4-216 x)+1224 x+81 x^2+e^4 (1632+216 x)} \, dx\)

Optimal. Leaf size=27 \[ -1+\frac {3}{-e^4+e^x+\frac {1}{4} \left (-\frac {68}{3}+x\right )-x} \]

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Rubi [A]  time = 0.14, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 3, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6688, 12, 6686} \begin {gather*} \frac {36}{-9 x+12 e^x-4 \left (17+3 e^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(324 - 432*E^x)/(4624 + 144*E^8 + 144*E^(2*x) + E^x*(-1632 - 288*E^4 - 216*x) + 1224*x + 81*x^2 + E^4*(163
2 + 216*x)),x]

[Out]

36/(12*E^x - 4*(17 + 3*E^4) - 9*x)

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {108 \left (3-4 e^x\right )}{\left (12 e^x-68 \left (1+\frac {3 e^4}{17}\right )-9 x\right )^2} \, dx\\ &=108 \int \frac {3-4 e^x}{\left (12 e^x-68 \left (1+\frac {3 e^4}{17}\right )-9 x\right )^2} \, dx\\ &=\frac {36}{12 e^x-4 \left (17+3 e^4\right )-9 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.70 \begin {gather*} -\frac {36}{68+12 e^4-12 e^x+9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(324 - 432*E^x)/(4624 + 144*E^8 + 144*E^(2*x) + E^x*(-1632 - 288*E^4 - 216*x) + 1224*x + 81*x^2 + E^
4*(1632 + 216*x)),x]

[Out]

-36/(68 + 12*E^4 - 12*E^x + 9*x)

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fricas [A]  time = 0.78, size = 17, normalized size = 0.63 \begin {gather*} -\frac {36}{9 \, x + 12 \, e^{4} - 12 \, e^{x} + 68} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*exp(x)+324)/(144*exp(x)^2+(-288*exp(4)-216*x-1632)*exp(x)+144*exp(4)^2+(216*x+1632)*exp(4)+81*
x^2+1224*x+4624),x, algorithm="fricas")

[Out]

-36/(9*x + 12*e^4 - 12*e^x + 68)

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giac [A]  time = 0.39, size = 17, normalized size = 0.63 \begin {gather*} -\frac {36}{9 \, x + 12 \, e^{4} - 12 \, e^{x} + 68} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*exp(x)+324)/(144*exp(x)^2+(-288*exp(4)-216*x-1632)*exp(x)+144*exp(4)^2+(216*x+1632)*exp(4)+81*
x^2+1224*x+4624),x, algorithm="giac")

[Out]

-36/(9*x + 12*e^4 - 12*e^x + 68)

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maple [A]  time = 0.10, size = 18, normalized size = 0.67

method	result	size
norman	\(-\frac {36}{9 x +68+12 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{x}}\)	\(18\)
risch	\(-\frac {36}{9 x +68+12 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-432*exp(x)+324)/(144*exp(x)^2+(-288*exp(4)-216*x-1632)*exp(x)+144*exp(4)^2+(216*x+1632)*exp(4)+81*x^2+12
24*x+4624),x,method=_RETURNVERBOSE)

[Out]

-36/(9*x+68+12*exp(4)-12*exp(x))

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maxima [A]  time = 0.95, size = 17, normalized size = 0.63 \begin {gather*} -\frac {36}{9 \, x + 12 \, e^{4} - 12 \, e^{x} + 68} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*exp(x)+324)/(144*exp(x)^2+(-288*exp(4)-216*x-1632)*exp(x)+144*exp(4)^2+(216*x+1632)*exp(4)+81*
x^2+1224*x+4624),x, algorithm="maxima")

[Out]

-36/(9*x + 12*e^4 - 12*e^x + 68)

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mupad [B]  time = 0.33, size = 33, normalized size = 1.22 \begin {gather*} \frac {27\,\left (3\,x-4\,{\mathrm {e}}^x\right )}{\left (3\,{\mathrm {e}}^4+17\right )\,\left (9\,x+12\,{\mathrm {e}}^4-12\,{\mathrm {e}}^x+68\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(432*exp(x) - 324)/(1224*x + 144*exp(2*x) + 144*exp(8) - exp(x)*(216*x + 288*exp(4) + 1632) + 81*x^2 + ex
p(4)*(216*x + 1632) + 4624),x)

[Out]

(27*(3*x - 4*exp(x)))/((3*exp(4) + 17)*(9*x + 12*exp(4) - 12*exp(x) + 68))

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sympy [A]  time = 0.13, size = 15, normalized size = 0.56 \begin {gather*} \frac {36}{- 9 x + 12 e^{x} - 12 e^{4} - 68} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-432*exp(x)+324)/(144*exp(x)**2+(-288*exp(4)-216*x-1632)*exp(x)+144*exp(4)**2+(216*x+1632)*exp(4)+8
1*x**2+1224*x+4624),x)

[Out]

36/(-9*x + 12*exp(x) - 12*exp(4) - 68)

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