3.90.80 \(\int \frac {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log (1-2 x+x^2)}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+(-150+610 x-626 x^2+182 x^3-16 x^4) \log (1-2 x+x^2)+(-25+35 x-11 x^2+x^3) \log ^2(1-2 x+x^2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {4 x}{(-5+x) \left (-3+8 x-\log \left ((1-x)^2\right )\right )} \]

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Rubi [F]  time = 1.33, 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 {-60+20 x+40 x^2-32 x^3+(-20+20 x) \log \left (1-2 x+x^2\right )}{-225+1515 x-3379 x^2+2777 x^3-752 x^4+64 x^5+\left (-150+610 x-626 x^2+182 x^3-16 x^4\right ) \log \left (1-2 x+x^2\right )+\left (-25+35 x-11 x^2+x^3\right ) \log ^2\left (1-2 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-60 + 20*x + 40*x^2 - 32*x^3 + (-20 + 20*x)*Log[1 - 2*x + x^2])/(-225 + 1515*x - 3379*x^2 + 2777*x^3 - 75
2*x^4 + 64*x^5 + (-150 + 610*x - 626*x^2 + 182*x^3 - 16*x^4)*Log[1 - 2*x + x^2] + (-25 + 35*x - 11*x^2 + x^3)*
Log[1 - 2*x + x^2]^2),x]

[Out]

-32*Defer[Int][(-3 + 8*x - Log[(-1 + x)^2])^(-2), x] - 150*Defer[Int][1/((-5 + x)*(-3 + 8*x - Log[(-1 + x)^2])
^2), x] - 2*Defer[Int][1/((-1 + x)*(-3 + 8*x - Log[(-1 + x)^2])^2), x] - 20*Defer[Int][1/((-5 + x)^2*(-3 + 8*x
- Log[(-1 + x)^2])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (15-5 x-10 x^2+8 x^3-5 (-1+x) \log \left ((-1+x)^2\right )\right )}{(1-x) (5-x)^2 \left (3-8 x+\log \left ((-1+x)^2\right )\right )^2} \, dx\\ &=4 \int \frac {15-5 x-10 x^2+8 x^3-5 (-1+x) \log \left ((-1+x)^2\right )}{(1-x) (5-x)^2 \left (3-8 x+\log \left ((-1+x)^2\right )\right )^2} \, dx\\ &=4 \int \left (-\frac {2 x (-5+4 x)}{(-5+x) (-1+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2}-\frac {5}{(-5+x)^2 \left (-3+8 x-\log \left ((-1+x)^2\right )\right )}\right ) \, dx\\ &=-\left (8 \int \frac {x (-5+4 x)}{(-5+x) (-1+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2} \, dx\right )-20 \int \frac {1}{(-5+x)^2 \left (-3+8 x-\log \left ((-1+x)^2\right )\right )} \, dx\\ &=-\left (8 \int \left (\frac {4}{\left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2}+\frac {75}{4 (-5+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2}+\frac {1}{4 (-1+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2}\right ) \, dx\right )-20 \int \frac {1}{(-5+x)^2 \left (-3+8 x-\log \left ((-1+x)^2\right )\right )} \, dx\\ &=-\left (2 \int \frac {1}{(-1+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2} \, dx\right )-20 \int \frac {1}{(-5+x)^2 \left (-3+8 x-\log \left ((-1+x)^2\right )\right )} \, dx-32 \int \frac {1}{\left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2} \, dx-150 \int \frac {1}{(-5+x) \left (-3+8 x-\log \left ((-1+x)^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.70, size = 23, normalized size = 0.92 \begin {gather*} -\frac {4 x}{(-5+x) \left (-5-8 (-1+x)+\log \left ((-1+x)^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-60 + 20*x + 40*x^2 - 32*x^3 + (-20 + 20*x)*Log[1 - 2*x + x^2])/(-225 + 1515*x - 3379*x^2 + 2777*x^
3 - 752*x^4 + 64*x^5 + (-150 + 610*x - 626*x^2 + 182*x^3 - 16*x^4)*Log[1 - 2*x + x^2] + (-25 + 35*x - 11*x^2 +
x^3)*Log[1 - 2*x + x^2]^2),x]

[Out]

(-4*x)/((-5 + x)*(-5 - 8*(-1 + x) + Log[(-1 + x)^2]))

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fricas [A]  time = 0.78, size = 29, normalized size = 1.16 \begin {gather*} \frac {4 \, x}{8 \, x^{2} - {\left (x - 5\right )} \log \left (x^{2} - 2 \, x + 1\right ) - 43 \, x + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25)*log(x^2-2*x+1)^2+(-16*x^4+182
*x^3-626*x^2+610*x-150)*log(x^2-2*x+1)+64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="fricas")

[Out]

4*x/(8*x^2 - (x - 5)*log(x^2 - 2*x + 1) - 43*x + 15)

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giac [A]  time = 0.24, size = 38, normalized size = 1.52 \begin {gather*} \frac {4 \, x}{8 \, x^{2} - x \log \left (x^{2} - 2 \, x + 1\right ) - 43 \, x + 5 \, \log \left (x^{2} - 2 \, x + 1\right ) + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25)*log(x^2-2*x+1)^2+(-16*x^4+182
*x^3-626*x^2+610*x-150)*log(x^2-2*x+1)+64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="giac")

[Out]

4*x/(8*x^2 - x*log(x^2 - 2*x + 1) - 43*x + 5*log(x^2 - 2*x + 1) + 15)

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maple [A]  time = 0.12, size = 27, normalized size = 1.08

method	result	size
risch	\(\frac {4 x}{\left (x -5\right ) \left (8 x -\ln \left (x^{2}-2 x +1\right )-3\right )}\)	\(27\)
norman	\(\frac {4 x}{8 x^{2}-\ln \left (x^{2}-2 x +1\right ) x -43 x +5 \ln \left (x^{2}-2 x +1\right )+15}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x-20)*ln(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25)*ln(x^2-2*x+1)^2+(-16*x^4+182*x^3-626
*x^2+610*x-150)*ln(x^2-2*x+1)+64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x,method=_RETURNVERBOSE)

[Out]

4*x/(x-5)/(8*x-ln(x^2-2*x+1)-3)

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maxima [A]  time = 0.39, size = 24, normalized size = 0.96 \begin {gather*} \frac {4 \, x}{8 \, x^{2} - 2 \, {\left (x - 5\right )} \log \left (x - 1\right ) - 43 \, x + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x-20)*log(x^2-2*x+1)-32*x^3+40*x^2+20*x-60)/((x^3-11*x^2+35*x-25)*log(x^2-2*x+1)^2+(-16*x^4+182
*x^3-626*x^2+610*x-150)*log(x^2-2*x+1)+64*x^5-752*x^4+2777*x^3-3379*x^2+1515*x-225),x, algorithm="maxima")

[Out]

4*x/(8*x^2 - 2*(x - 5)*log(x - 1) - 43*x + 15)

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mupad [B]  time = 8.26, size = 24, normalized size = 0.96 \begin {gather*} -\frac {4\,x}{\left (x-5\right )\,\left (\ln \left (x^2-2\,x+1\right )-8\,x+3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x + log(x^2 - 2*x + 1)*(20*x - 20) + 40*x^2 - 32*x^3 - 60)/(1515*x + log(x^2 - 2*x + 1)^2*(35*x - 11*x
^2 + x^3 - 25) - log(x^2 - 2*x + 1)*(626*x^2 - 610*x - 182*x^3 + 16*x^4 + 150) - 3379*x^2 + 2777*x^3 - 752*x^4
+ 64*x^5 - 225),x)

[Out]

-(4*x)/((x - 5)*(log(x^2 - 2*x + 1) - 8*x + 3))

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sympy [A]  time = 0.17, size = 27, normalized size = 1.08 \begin {gather*} - \frac {4 x}{- 8 x^{2} + 43 x + \left (x - 5\right ) \log {\left (x^{2} - 2 x + 1 \right )} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x-20)*ln(x**2-2*x+1)-32*x**3+40*x**2+20*x-60)/((x**3-11*x**2+35*x-25)*ln(x**2-2*x+1)**2+(-16*x*
*4+182*x**3-626*x**2+610*x-150)*ln(x**2-2*x+1)+64*x**5-752*x**4+2777*x**3-3379*x**2+1515*x-225),x)

[Out]

-4*x/(-8*x**2 + 43*x + (x - 5)*log(x**2 - 2*x + 1) - 15)

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