3.67.65 \(\int \frac {5}{32-64 x+32 x^2+(32-32 x) \log (16)+8 \log ^2(16)} \, dx\)

Optimal. Leaf size=13 \[ \frac {5}{16 (2-2 x+\log (16))} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 1981, 27, 32} \begin {gather*} \frac {5}{16 (-2 x+2+\log (16))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[5/(32 - 64*x + 32*x^2 + (32 - 32*x)*Log[16] + 8*Log[16]^2),x]

[Out]

5/(16*(2 - 2*x + Log[16]))

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} &=5 \int \frac {1}{32-64 x+32 x^2+(32-32 x) \log (16)+8 \log ^2(16)} \, dx\\ &=5 \int \frac {1}{32 x^2-32 x (2+\log (16))+8 (2+\log (16))^2} \, dx\\ &=5 \int \frac {1}{8 (-2+2 x-\log (16))^2} \, dx\\ &=\frac {5}{8} \int \frac {1}{(-2+2 x-\log (16))^2} \, dx\\ &=\frac {5}{16 (2-2 x+\log (16))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {5}{8 (4-4 x+\log (256))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[5/(32 - 64*x + 32*x^2 + (32 - 32*x)*Log[16] + 8*Log[16]^2),x]

[Out]

5/(8*(4 - 4*x + Log[256]))

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fricas [A]  time = 0.53, size = 11, normalized size = 0.85 \begin {gather*} -\frac {5}{32 \, {\left (x - 2 \, \log \relax (2) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(128*log(2)^2+4*(-32*x+32)*log(2)+32*x^2-64*x+32),x, algorithm="fricas")

[Out]

-5/32/(x - 2*log(2) - 1)

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giac [A]  time = 0.13, size = 11, normalized size = 0.85 \begin {gather*} -\frac {5}{32 \, {\left (x - 2 \, \log \relax (2) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(128*log(2)^2+4*(-32*x+32)*log(2)+32*x^2-64*x+32),x, algorithm="giac")

[Out]

-5/32/(x - 2*log(2) - 1)

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maple [A]  time = 0.13, size = 12, normalized size = 0.92

method	result	size
default	\(-\frac {5}{32 \left (-2 \ln \relax (2)+x -1\right )}\)	\(12\)
risch	\(\frac {5}{64 \left (\ln \relax (2)-\frac {x}{2}+\frac {1}{2}\right )}\)	\(12\)
gosper	\(\frac {5}{32 \left (2 \ln \relax (2)-x +1\right )}\)	\(14\)
norman	\(\frac {5}{32 \left (2 \ln \relax (2)-x +1\right )}\)	\(14\)
meijerg	\(-\frac {5 x}{32 \left (-1-2 \ln \relax (2)\right ) \left (1-\frac {x}{1+2 \ln \relax (2)}\right ) \left (1+2 \ln \relax (2)\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/(128*ln(2)^2+4*(-32*x+32)*ln(2)+32*x^2-64*x+32),x,method=_RETURNVERBOSE)

[Out]

-5/32/(-2*ln(2)+x-1)

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maxima [A]  time = 0.36, size = 11, normalized size = 0.85 \begin {gather*} -\frac {5}{32 \, {\left (x - 2 \, \log \relax (2) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(128*log(2)^2+4*(-32*x+32)*log(2)+32*x^2-64*x+32),x, algorithm="maxima")

[Out]

-5/32/(x - 2*log(2) - 1)

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mupad [B]  time = 0.08, size = 13, normalized size = 1.00 \begin {gather*} \frac {5}{32\,\left (\ln \relax (4)-x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/(128*log(2)^2 - 4*log(2)*(32*x - 32) - 64*x + 32*x^2 + 32),x)

[Out]

5/(32*(log(4) - x + 1))

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sympy [A]  time = 0.15, size = 12, normalized size = 0.92 \begin {gather*} - \frac {5}{32 x - 64 \log {\relax (2 )} - 32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/(128*ln(2)**2+4*(-32*x+32)*ln(2)+32*x**2-64*x+32),x)

[Out]

-5/(32*x - 64*log(2) - 32)

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