3.97.80 \(\int \frac {612-657 x+162 x^2+(108-54 x) \log (2-x)}{-578 x^3+697 x^4-276 x^5+36 x^6+(-204 x^3+174 x^4-36 x^5) \log (2-x)+(-18 x^3+9 x^4) \log ^2(2-x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {3}{x^2 \left (\frac {17}{3}-2 x+\log (2-x)\right )} \]

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Rubi [F]  time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {612-657 x+162 x^2+(108-54 x) \log (2-x)}{-578 x^3+697 x^4-276 x^5+36 x^6+\left (-204 x^3+174 x^4-36 x^5\right ) \log (2-x)+\left (-18 x^3+9 x^4\right ) \log ^2(2-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(612 - 657*x + 162*x^2 + (108 - 54*x)*Log[2 - x])/(-578*x^3 + 697*x^4 - 276*x^5 + 36*x^6 + (-204*x^3 + 174
*x^4 - 36*x^5)*Log[2 - x] + (-18*x^3 + 9*x^4)*Log[2 - x]^2),x]

[Out]

(-27*Defer[Int][1/((-2 + x)*(-17 + 6*x - 3*Log[2 - x])^2), x])/4 + (135*Defer[Int][1/(x^2*(-17 + 6*x - 3*Log[2
- x])^2), x])/2 + (27*Defer[Int][1/(x*(-17 + 6*x - 3*Log[2 - x])^2), x])/4 + 18*Defer[Int][1/(x^3*(-17 + 6*x
- 3*Log[2 - x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 \left (-68+73 x-18 x^2+6 (-2+x) \log (2-x)\right )}{(2-x) x^3 (17-6 x+3 \log (2-x))^2} \, dx\\ &=9 \int \frac {-68+73 x-18 x^2+6 (-2+x) \log (2-x)}{(2-x) x^3 (17-6 x+3 \log (2-x))^2} \, dx\\ &=9 \int \left (\frac {3 (-5+2 x)}{(-2+x) x^2 (-17+6 x-3 \log (2-x))^2}+\frac {2}{x^3 (-17+6 x-3 \log (2-x))}\right ) \, dx\\ &=18 \int \frac {1}{x^3 (-17+6 x-3 \log (2-x))} \, dx+27 \int \frac {-5+2 x}{(-2+x) x^2 (-17+6 x-3 \log (2-x))^2} \, dx\\ &=18 \int \frac {1}{x^3 (-17+6 x-3 \log (2-x))} \, dx+27 \int \left (-\frac {1}{4 (-2+x) (-17+6 x-3 \log (2-x))^2}+\frac {5}{2 x^2 (-17+6 x-3 \log (2-x))^2}+\frac {1}{4 x (-17+6 x-3 \log (2-x))^2}\right ) \, dx\\ &=-\left (\frac {27}{4} \int \frac {1}{(-2+x) (-17+6 x-3 \log (2-x))^2} \, dx\right )+\frac {27}{4} \int \frac {1}{x (-17+6 x-3 \log (2-x))^2} \, dx+18 \int \frac {1}{x^3 (-17+6 x-3 \log (2-x))} \, dx+\frac {135}{2} \int \frac {1}{x^2 (-17+6 x-3 \log (2-x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} \frac {9}{x^2 (17-6 x+3 \log (2-x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(612 - 657*x + 162*x^2 + (108 - 54*x)*Log[2 - x])/(-578*x^3 + 697*x^4 - 276*x^5 + 36*x^6 + (-204*x^3
+ 174*x^4 - 36*x^5)*Log[2 - x] + (-18*x^3 + 9*x^4)*Log[2 - x]^2),x]

[Out]

9/(x^2*(17 - 6*x + 3*Log[2 - x]))

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fricas [A]  time = 0.43, size = 26, normalized size = 1.30 \begin {gather*} -\frac {9}{6 \, x^{3} - 3 \, x^{2} \log \left (-x + 2\right ) - 17 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x+108)*log(2-x)+162*x^2-657*x+612)/((9*x^4-18*x^3)*log(2-x)^2+(-36*x^5+174*x^4-204*x^3)*log(2-
x)+36*x^6-276*x^5+697*x^4-578*x^3),x, algorithm="fricas")

[Out]

-9/(6*x^3 - 3*x^2*log(-x + 2) - 17*x^2)

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giac [A]  time = 0.17, size = 26, normalized size = 1.30 \begin {gather*} -\frac {9}{6 \, x^{3} - 3 \, x^{2} \log \left (-x + 2\right ) - 17 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x+108)*log(2-x)+162*x^2-657*x+612)/((9*x^4-18*x^3)*log(2-x)^2+(-36*x^5+174*x^4-204*x^3)*log(2-
x)+36*x^6-276*x^5+697*x^4-578*x^3),x, algorithm="giac")

[Out]

-9/(6*x^3 - 3*x^2*log(-x + 2) - 17*x^2)

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maple [A]  time = 0.07, size = 21, normalized size = 1.05

method	result	size
norman	\(-\frac {9}{x^{2} \left (-17+6 x -3 \ln \left (2-x \right )\right )}\)	\(21\)
risch	\(-\frac {9}{x^{2} \left (-17+6 x -3 \ln \left (2-x \right )\right )}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-54*x+108)*ln(2-x)+162*x^2-657*x+612)/((9*x^4-18*x^3)*ln(2-x)^2+(-36*x^5+174*x^4-204*x^3)*ln(2-x)+36*x^6
-276*x^5+697*x^4-578*x^3),x,method=_RETURNVERBOSE)

[Out]

-9/x^2/(-17+6*x-3*ln(2-x))

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maxima [A]  time = 0.42, size = 26, normalized size = 1.30 \begin {gather*} -\frac {9}{6 \, x^{3} - 3 \, x^{2} \log \left (-x + 2\right ) - 17 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x+108)*log(2-x)+162*x^2-657*x+612)/((9*x^4-18*x^3)*log(2-x)^2+(-36*x^5+174*x^4-204*x^3)*log(2-
x)+36*x^6-276*x^5+697*x^4-578*x^3),x, algorithm="maxima")

[Out]

-9/(6*x^3 - 3*x^2*log(-x + 2) - 17*x^2)

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mupad [B]  time = 6.11, size = 20, normalized size = 1.00 \begin {gather*} \frac {9}{x^2\,\left (3\,\ln \left (2-x\right )-6\,x+17\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((657*x + log(2 - x)*(54*x - 108) - 162*x^2 - 612)/(log(2 - x)*(204*x^3 - 174*x^4 + 36*x^5) + log(2 - x)^2*
(18*x^3 - 9*x^4) + 578*x^3 - 697*x^4 + 276*x^5 - 36*x^6),x)

[Out]

9/(x^2*(3*log(2 - x) - 6*x + 17))

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sympy [A]  time = 0.21, size = 20, normalized size = 1.00 \begin {gather*} \frac {9}{- 6 x^{3} + 3 x^{2} \log {\left (2 - x \right )} + 17 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x+108)*ln(2-x)+162*x**2-657*x+612)/((9*x**4-18*x**3)*ln(2-x)**2+(-36*x**5+174*x**4-204*x**3)*l
n(2-x)+36*x**6-276*x**5+697*x**4-578*x**3),x)

[Out]

9/(-6*x**3 + 3*x**2*log(2 - x) + 17*x**2)

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