3.56.33 \(\int \frac {5}{1-2 x+x^2} \, dx\)

Optimal. Leaf size=24 \[ 10+\frac {5}{1-x}-\frac {3}{1+e^2-\log (5)} \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 0.38, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 27, 32} \begin {gather*} \frac {5}{1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[5/(1 - 2*x + x^2),x]

[Out]

5/(1 - 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 \int \frac {1}{1-2 x+x^2} \, dx\\ &=5 \int \frac {1}{(-1+x)^2} \, dx\\ &=\frac {5}{1-x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 0.29 \begin {gather*} -\frac {5}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5/(1 - 2*x + x^2),x]

[Out]

-5/(-1 + x)

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fricas [A]  time = 1.02, size = 7, normalized size = 0.29 \begin {gather*} -\frac {5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(x^2-2*x+1),x, algorithm="fricas")

[Out]

-5/(x - 1)

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giac [A]  time = 0.12, size = 7, normalized size = 0.29 \begin {gather*} -\frac {5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(x^2-2*x+1),x, algorithm="giac")

[Out]

-5/(x - 1)

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maple [A]  time = 0.29, size = 8, normalized size = 0.33

method	result	size
gosper	\(-\frac {5}{x -1}\)	\(8\)
default	\(-\frac {5}{x -1}\)	\(8\)
norman	\(-\frac {5}{x -1}\)	\(8\)
risch	\(-\frac {5}{x -1}\)	\(8\)
meijerg	\(\frac {5 x}{1-x}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/(x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

-5/(x-1)

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maxima [A]  time = 0.37, size = 7, normalized size = 0.29 \begin {gather*} -\frac {5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(x^2-2*x+1),x, algorithm="maxima")

[Out]

-5/(x - 1)

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mupad [B]  time = 0.04, size = 7, normalized size = 0.29 \begin {gather*} -\frac {5}{x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/(x^2 - 2*x + 1),x)

[Out]

-5/(x - 1)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.21 \begin {gather*} - \frac {5}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(x**2-2*x+1),x)

[Out]

-5/(x - 1)

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