3.71.96 \(\int \frac {99-198 x+(132+66 x-66 x^2) \log ^2(-2-x+x^2)}{(-2-x+x^2) \log ^2(-2-x+x^2)} \, dx\)

Optimal. Leaf size=19 \[ -33 \left (2 x-\frac {3}{\log \left (-2-x+x^2\right )}\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6728, 6686} \begin {gather*} \frac {99}{\log \left (x^2-x-2\right )}-66 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(99 - 198*x + (132 + 66*x - 66*x^2)*Log[-2 - x + x^2]^2)/((-2 - x + x^2)*Log[-2 - x + x^2]^2),x]

[Out]

-66*x + 99/Log[-2 - x + x^2]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-66-\frac {99 (-1+2 x)}{(-2+x) (1+x) \log ^2\left (-2-x+x^2\right )}\right ) \, dx\\ &=-66 x-99 \int \frac {-1+2 x}{(-2+x) (1+x) \log ^2\left (-2-x+x^2\right )} \, dx\\ &=-66 x+\frac {99}{\log \left (-2-x+x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} -33 \left (2 x-\frac {3}{\log \left (-2-x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(99 - 198*x + (132 + 66*x - 66*x^2)*Log[-2 - x + x^2]^2)/((-2 - x + x^2)*Log[-2 - x + x^2]^2),x]

[Out]

-33*(2*x - 3/Log[-2 - x + x^2])

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fricas [A]  time = 1.38, size = 27, normalized size = 1.42 \begin {gather*} -\frac {33 \, {\left (2 \, x \log \left (x^{2} - x - 2\right ) - 3\right )}}{\log \left (x^{2} - x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-66*x^2+66*x+132)*log(x^2-x-2)^2-198*x+99)/(x^2-x-2)/log(x^2-x-2)^2,x, algorithm="fricas")

[Out]

-33*(2*x*log(x^2 - x - 2) - 3)/log(x^2 - x - 2)

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giac [A]  time = 0.24, size = 17, normalized size = 0.89 \begin {gather*} -66 \, x + \frac {99}{\log \left (x^{2} - x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-66*x^2+66*x+132)*log(x^2-x-2)^2-198*x+99)/(x^2-x-2)/log(x^2-x-2)^2,x, algorithm="giac")

[Out]

-66*x + 99/log(x^2 - x - 2)

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maple [A]  time = 0.14, size = 18, normalized size = 0.95

method	result	size
default	\(-66 x +\frac {99}{\ln \left (x^{2}-x -2\right )}\)	\(18\)
risch	\(-66 x +\frac {99}{\ln \left (x^{2}-x -2\right )}\)	\(18\)
norman	\(\frac {99-66 \ln \left (x^{2}-x -2\right ) x}{\ln \left (x^{2}-x -2\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-66*x^2+66*x+132)*ln(x^2-x-2)^2-198*x+99)/(x^2-x-2)/ln(x^2-x-2)^2,x,method=_RETURNVERBOSE)

[Out]

-66*x+99/ln(x^2-x-2)

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maxima [A]  time = 0.41, size = 29, normalized size = 1.53 \begin {gather*} -\frac {33 \, {\left (2 \, x \log \left (x + 1\right ) + 2 \, x \log \left (x - 2\right ) - 3\right )}}{\log \left (x + 1\right ) + \log \left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-66*x^2+66*x+132)*log(x^2-x-2)^2-198*x+99)/(x^2-x-2)/log(x^2-x-2)^2,x, algorithm="maxima")

[Out]

-33*(2*x*log(x + 1) + 2*x*log(x - 2) - 3)/(log(x + 1) + log(x - 2))

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mupad [B]  time = 4.33, size = 17, normalized size = 0.89 \begin {gather*} \frac {99}{\ln \left (x^2-x-2\right )}-66\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2 - x - 2)^2*(66*x - 66*x^2 + 132) - 198*x + 99)/(log(x^2 - x - 2)^2*(x - x^2 + 2)),x)

[Out]

99/log(x^2 - x - 2) - 66*x

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sympy [A]  time = 0.17, size = 12, normalized size = 0.63 \begin {gather*} - 66 x + \frac {99}{\log {\left (x^{2} - x - 2 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-66*x**2+66*x+132)*ln(x**2-x-2)**2-198*x+99)/(x**2-x-2)/ln(x**2-x-2)**2,x)

[Out]

-66*x + 99/log(x**2 - x - 2)

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