3.65.33 \(\int -\frac {1}{e^6 (-1152+2880 x-2880 x^2+1440 x^3-360 x^4+36 x^5)} \, dx\) [6433]

Optimal. Leaf size=14 \[ \frac {1}{9 e^6 (-4+2 x)^4} \]

[Out]

1/9/(2*x-4)^4/exp(3)^2

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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {12, 2083, 32} \begin {gather*} \frac {1}{144 e^6 (2-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(1/(E^6*(-1152 + 2880*x - 2880*x^2 + 1440*x^3 - 360*x^4 + 36*x^5))),x]

[Out]

1/(144*E^6*(2 - x)^4)

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 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\int \frac {1}{-1152+2880 x-2880 x^2+1440 x^3-360 x^4+36 x^5} \, dx}{e^6}\\ &=-\frac {\int \frac {1}{36 (-2+x)^5} \, dx}{e^6}\\ &=-\frac {\int \frac {1}{(-2+x)^5} \, dx}{36 e^6}\\ &=\frac {1}{144 e^6 (2-x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{144 e^6 (-2+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(1/(E^6*(-1152 + 2880*x - 2880*x^2 + 1440*x^3 - 360*x^4 + 36*x^5))),x]

[Out]

1/(144*E^6*(-2 + x)^4)

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Maple [A]
time = 0.34, size = 12, normalized size = 0.86

method	result	size
default	\(\frac {{\mathrm e}^{-6}}{144 \left (x -2\right )^{4}}\)	\(12\)
norman	\(\frac {{\mathrm e}^{-6}}{144 \left (x -2\right )^{4}}\)	\(12\)
risch	\(\frac {{\mathrm e}^{-6}}{144 x^{4}-1152 x^{3}+3456 x^{2}-4608 x +2304}\)	\(25\)
gosper	\(\frac {{\mathrm e}^{-6}}{144 x^{4}-1152 x^{3}+3456 x^{2}-4608 x +2304}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(36*x^5-360*x^4+1440*x^3-2880*x^2+2880*x-1152)/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

1/144/exp(3)^2/(x-2)^4

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (9) = 18\).
time = 0.27, size = 24, normalized size = 1.71 \begin {gather*} \frac {e^{\left (-6\right )}}{144 \, {\left (x^{4} - 8 \, x^{3} + 24 \, x^{2} - 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(36*x^5-360*x^4+1440*x^3-2880*x^2+2880*x-1152)/exp(3)^2,x, algorithm="maxima")

[Out]

1/144*e^(-6)/(x^4 - 8*x^3 + 24*x^2 - 32*x + 16)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (9) = 18\).
time = 0.38, size = 24, normalized size = 1.71 \begin {gather*} \frac {e^{\left (-6\right )}}{144 \, {\left (x^{4} - 8 \, x^{3} + 24 \, x^{2} - 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(36*x^5-360*x^4+1440*x^3-2880*x^2+2880*x-1152)/exp(3)^2,x, algorithm="fricas")

[Out]

1/144*e^(-6)/(x^4 - 8*x^3 + 24*x^2 - 32*x + 16)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (12) = 24\).
time = 0.11, size = 37, normalized size = 2.64 \begin {gather*} \frac {1}{144 x^{4} e^{6} - 1152 x^{3} e^{6} + 3456 x^{2} e^{6} - 4608 x e^{6} + 2304 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(36*x**5-360*x**4+1440*x**3-2880*x**2+2880*x-1152)/exp(3)**2,x)

[Out]

1/(144*x**4*exp(6) - 1152*x**3*exp(6) + 3456*x**2*exp(6) - 4608*x*exp(6) + 2304*exp(6))

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Giac [A]
time = 0.39, size = 9, normalized size = 0.64 \begin {gather*} \frac {e^{\left (-6\right )}}{144 \, {\left (x - 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/(36*x^5-360*x^4+1440*x^3-2880*x^2+2880*x-1152)/exp(3)^2,x, algorithm="giac")

[Out]

1/144*e^(-6)/(x - 2)^4

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Mupad [B]
time = 0.08, size = 9, normalized size = 0.64 \begin {gather*} \frac {{\mathrm {e}}^{-6}}{144\,{\left (x-2\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-6)/(2880*x - 2880*x^2 + 1440*x^3 - 360*x^4 + 36*x^5 - 1152),x)

[Out]

exp(-6)/(144*(x - 2)^4)

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