3.38.44 \(\int -\frac {12}{324+e^2-36 x+x^2+e (-36+2 x)} \, dx\)

Optimal. Leaf size=10 \[ 4+\frac {12}{-18+e+x} \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 1981, 27, 32} \begin {gather*} -\frac {12}{-x-e+18} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-12/(324 + E^2 - 36*x + x^2 + E*(-36 + 2*x)),x]

[Out]

-12/(18 - E - 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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (12 \int \frac {1}{324+e^2-36 x+x^2+e (-36+2 x)} \, dx\right )\\ &=-\left (12 \int \frac {1}{(-18+e)^2-2 (18-e) x+x^2} \, dx\right )\\ &=-\left (12 \int \frac {1}{(-18+e+x)^2} \, dx\right )\\ &=-\frac {12}{18-e-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.80 \begin {gather*} \frac {12}{-18+e+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-12/(324 + E^2 - 36*x + x^2 + E*(-36 + 2*x)),x]

[Out]

12/(-18 + E + x)

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fricas [A]  time = 0.52, size = 9, normalized size = 0.90 \begin {gather*} \frac {12}{x + e - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(exp(1)^2+(2*x-36)*exp(1)+x^2-36*x+324),x, algorithm="fricas")

[Out]

12/(x + e - 18)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(exp(1)^2+(2*x-36)*exp(1)+x^2-36*x+324),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: -24*1/2/sqrt(-exp(1)^2+exp(2))*atan((sag
eVARx+exp(1)-18)/sqrt(-exp(1)^2+exp(2)))

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maple [A]  time = 0.15, size = 10, normalized size = 1.00

method	result	size
gosper	\(\frac {12}{{\mathrm e}+x -18}\)	\(10\)
norman	\(\frac {12}{{\mathrm e}+x -18}\)	\(10\)
risch	\(\frac {12}{{\mathrm e}+x -18}\)	\(10\)
meijerg	\(\frac {12 x}{\left ({\mathrm e}-18\right ) \left (-{\mathrm e}+18\right ) \left (1-\frac {x}{-{\mathrm e}+18}\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-12/(exp(1)^2+(2*x-36)*exp(1)+x^2-36*x+324),x,method=_RETURNVERBOSE)

[Out]

12/(exp(1)+x-18)

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maxima [A]  time = 0.35, size = 9, normalized size = 0.90 \begin {gather*} \frac {12}{x + e - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(exp(1)^2+(2*x-36)*exp(1)+x^2-36*x+324),x, algorithm="maxima")

[Out]

12/(x + e - 18)

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mupad [B]  time = 0.10, size = 9, normalized size = 0.90 \begin {gather*} \frac {12}{x+\mathrm {e}-18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-12/(exp(2) - 36*x + x^2 + exp(1)*(2*x - 36) + 324),x)

[Out]

12/(x + exp(1) - 18)

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sympy [A]  time = 0.14, size = 7, normalized size = 0.70 \begin {gather*} \frac {12}{x - 18 + e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-12/(exp(1)**2+(2*x-36)*exp(1)+x**2-36*x+324),x)

[Out]

12/(x - 18 + E)

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