3.4.6 \(\int \frac {20480000 e^{e^5-4 x}}{-27+e^{3 e^5-12 x}-9 e^{2 e^5-8 x}+27 e^{e^5-4 x}} \, dx\)

Optimal. Leaf size=15 \[ \frac {2560000}{\left (-3+e^{e^5-4 x}\right )^2} \]

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Rubi [A]  time = 0.10, antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {12, 2282, 32} \begin {gather*} \frac {2560000}{\left (3-e^{e^5-4 x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20480000*E^(E^5 - 4*x))/(-27 + E^(3*E^5 - 12*x) - 9*E^(2*E^5 - 8*x) + 27*E^(E^5 - 4*x)),x]

[Out]

2560000/(3 - E^(E^5 - 4*x))^2

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=20480000 \int \frac {e^{e^5-4 x}}{-27+e^{3 e^5-12 x}-9 e^{2 e^5-8 x}+27 e^{e^5-4 x}} \, dx\\ &=-\left (5120000 \operatorname {Subst}\left (\int \frac {1}{(-3+x)^3} \, dx,x,e^{e^5-4 x}\right )\right )\\ &=\frac {2560000}{\left (3-e^{e^5-4 x}\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 1.60 \begin {gather*} \frac {2560000 e^{8 x}}{\left (-e^{e^5}+3 e^{4 x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20480000*E^(E^5 - 4*x))/(-27 + E^(3*E^5 - 12*x) - 9*E^(2*E^5 - 8*x) + 27*E^(E^5 - 4*x)),x]

[Out]

(2560000*E^(8*x))/(-E^E^5 + 3*E^(4*x))^2

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fricas [A]  time = 0.89, size = 26, normalized size = 1.73 \begin {gather*} -\frac {2560000}{6 \, e^{\left (-4 \, x + e^{5}\right )} - e^{\left (-8 \, x + 2 \, e^{5}\right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20480000*exp(exp(5)-4*x)/(exp(exp(5)-4*x)^3-9*exp(exp(5)-4*x)^2+27*exp(exp(5)-4*x)-27),x, algorithm=
"fricas")

[Out]

-2560000/(6*e^(-4*x + e^5) - e^(-8*x + 2*e^5) - 9)

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giac [B]  time = 0.23, size = 31, normalized size = 2.07 \begin {gather*} \frac {2560000 \, {\left (6 \, e^{\left (4 \, x\right )} - e^{\left (e^{5}\right )}\right )} e^{\left (e^{5}\right )}}{9 \, {\left (3 \, e^{\left (4 \, x\right )} - e^{\left (e^{5}\right )}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20480000*exp(exp(5)-4*x)/(exp(exp(5)-4*x)^3-9*exp(exp(5)-4*x)^2+27*exp(exp(5)-4*x)-27),x, algorithm=
"giac")

[Out]

2560000/9*(6*e^(4*x) - e^(e^5))*e^(e^5)/(3*e^(4*x) - e^(e^5))^2

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maple [A]  time = 0.02, size = 14, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {2560000}{\left ({\mathrm e}^{{\mathrm e}^{5}-4 x}-3\right )^{2}}\)	\(14\)
default	\(\frac {2560000}{\left ({\mathrm e}^{{\mathrm e}^{5}-4 x}-3\right )^{2}}\)	\(14\)
norman	\(\frac {2560000}{\left ({\mathrm e}^{{\mathrm e}^{5}-4 x}-3\right )^{2}}\)	\(14\)
risch	\(\frac {2560000}{\left ({\mathrm e}^{{\mathrm e}^{5}-4 x}-3\right )^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(20480000*exp(exp(5)-4*x)/(exp(exp(5)-4*x)^3-9*exp(exp(5)-4*x)^2+27*exp(exp(5)-4*x)-27),x,method=_RETURNVER
BOSE)

[Out]

2560000/(exp(exp(5)-4*x)-3)^2

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maxima [A]  time = 0.39, size = 26, normalized size = 1.73 \begin {gather*} -\frac {2560000}{6 \, e^{\left (-4 \, x + e^{5}\right )} - e^{\left (-8 \, x + 2 \, e^{5}\right )} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20480000*exp(exp(5)-4*x)/(exp(exp(5)-4*x)^3-9*exp(exp(5)-4*x)^2+27*exp(exp(5)-4*x)-27),x, algorithm=
"maxima")

[Out]

-2560000/(6*e^(-4*x + e^5) - e^(-8*x + 2*e^5) - 9)

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mupad [B]  time = 0.17, size = 31, normalized size = 2.07 \begin {gather*} \frac {2560000\,{\mathrm {e}}^{{\mathrm {e}}^5}\,\left (6\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{{\mathrm {e}}^5}\right )}{9\,{\left (3\,{\mathrm {e}}^{4\,x}-{\mathrm {e}}^{{\mathrm {e}}^5}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20480000*exp(exp(5) - 4*x))/(27*exp(exp(5) - 4*x) - 9*exp(2*exp(5) - 8*x) + exp(3*exp(5) - 12*x) - 27),x)

[Out]

(2560000*exp(exp(5))*(6*exp(4*x) - exp(exp(5))))/(9*(3*exp(4*x) - exp(exp(5)))^2)

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sympy [A]  time = 0.10, size = 22, normalized size = 1.47 \begin {gather*} \frac {2560000}{e^{- 8 x + 2 e^{5}} - 6 e^{- 4 x + e^{5}} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(20480000*exp(exp(5)-4*x)/(exp(exp(5)-4*x)**3-9*exp(exp(5)-4*x)**2+27*exp(exp(5)-4*x)-27),x)

[Out]

2560000/(exp(-8*x + 2*exp(5)) - 6*exp(-4*x + exp(5)) + 9)

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