3.97.61 \(\int \frac {(x+x^2-x^3) \log (4+4 x-4 x^2)+\log (-1+x) (-x+3 x^2-2 x^3+(1-2 x^2+x^3) \log (4+4 x-4 x^2))}{(1-2 x^2+x^3) \log ^2(-1+x) \log ^2(4+4 x-4 x^2)} \, dx\)

Optimal. Leaf size=22 \[ \frac {x}{\log (-1+x) \log \left (4+4 \left (x-x^2\right )\right )} \]

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Rubi [F]  time = 2.52, 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 {\left (x+x^2-x^3\right ) \log \left (4+4 x-4 x^2\right )+\log (-1+x) \left (-x+3 x^2-2 x^3+\left (1-2 x^2+x^3\right ) \log \left (4+4 x-4 x^2\right )\right )}{\left (1-2 x^2+x^3\right ) \log ^2(-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((x + x^2 - x^3)*Log[4 + 4*x - 4*x^2] + Log[-1 + x]*(-x + 3*x^2 - 2*x^3 + (1 - 2*x^2 + x^3)*Log[4 + 4*x -
4*x^2]))/((1 - 2*x^2 + x^3)*Log[-1 + x]^2*Log[4 + 4*x - 4*x^2]^2),x]

[Out]

-2*Defer[Int][1/(Log[-1 + x]*Log[4 + 4*x - 4*x^2]^2), x] + (4*Defer[Int][1/((1 + Sqrt[5] - 2*x)*Log[-1 + x]*Lo
g[4 + 4*x - 4*x^2]^2), x])/Sqrt[5] - ((5 + Sqrt[5])*Defer[Int][1/((-1 - Sqrt[5] + 2*x)*Log[-1 + x]*Log[4 + 4*x
- 4*x^2]^2), x])/5 + (4*Defer[Int][1/((-1 + Sqrt[5] + 2*x)*Log[-1 + x]*Log[4 + 4*x - 4*x^2]^2), x])/Sqrt[5] -
((5 - Sqrt[5])*Defer[Int][1/((-1 + Sqrt[5] + 2*x)*Log[-1 + x]*Log[4 + 4*x - 4*x^2]^2), x])/5 - Defer[Int][1/(
Log[-1 + x]^2*Log[4 + 4*x - 4*x^2]), x] - Defer[Int][1/((-1 + x)*Log[-1 + x]^2*Log[4 + 4*x - 4*x^2]), x] + Def
er[Int][1/(Log[-1 + x]*Log[4 + 4*x - 4*x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {x (-1+2 x)}{\left (1+x-x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {-x-\log (-1+x)+x \log (-1+x)}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx\\ &=\int \frac {x (-1+2 x)}{\left (1+x-x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx+\int \frac {-x-\log (-1+x)+x \log (-1+x)}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=\int \left (-\frac {2}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {-2-x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx+\int \left (-\frac {x}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}-\frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {x}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\int \frac {-2-x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {x}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {x}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\int \left (-\frac {2}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}-\frac {x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx-\int \left (\frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx+\int \left (\frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )}+\frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )}\right ) \, dx-\int \frac {1}{(-1+x) \log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )-2 \int \frac {1}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {x}{\left (-1-x+x^2\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \left (-\frac {2}{\sqrt {5} \left (1+\sqrt {5}-2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}-\frac {2}{\sqrt {5} \left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx\right )-2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \left (\frac {1+\frac {1}{\sqrt {5}}}{\left (-1-\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}+\frac {1-\frac {1}{\sqrt {5}}}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )}\right ) \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ &=-\left (2 \int \frac {1}{\log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx\right )+\frac {4 \int \frac {1}{\left (1+\sqrt {5}-2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx}{\sqrt {5}}+\frac {4 \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx}{\sqrt {5}}-\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {1}{\left (-1+\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {1}{\left (-1-\sqrt {5}+2 x\right ) \log (-1+x) \log ^2\left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{\log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx-\int \frac {1}{(-1+x) \log ^2(-1+x) \log \left (4+4 x-4 x^2\right )} \, dx+\int \frac {1}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log (-1+x) \log \left (4+4 x-4 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((x + x^2 - x^3)*Log[4 + 4*x - 4*x^2] + Log[-1 + x]*(-x + 3*x^2 - 2*x^3 + (1 - 2*x^2 + x^3)*Log[4 +
4*x - 4*x^2]))/((1 - 2*x^2 + x^3)*Log[-1 + x]^2*Log[4 + 4*x - 4*x^2]^2),x]

[Out]

x/(Log[-1 + x]*Log[4 + 4*x - 4*x^2])

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fricas [A]  time = 0.63, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log \left (-4 \, x^{2} + 4 \, x + 4\right ) \log \left (x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2+1)*log(-4*x^2+4*x+4)-2*x^3+3*x^2-x)*log(x-1)+(-x^3+x^2+x)*log(-4*x^2+4*x+4))/(x^3-2*x^2
+1)/log(-4*x^2+4*x+4)^2/log(x-1)^2,x, algorithm="fricas")

[Out]

x/(log(-4*x^2 + 4*x + 4)*log(x - 1))

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giac [A]  time = 0.26, size = 21, normalized size = 0.95 \begin {gather*} \frac {x}{\log \left (-4 \, x^{2} + 4 \, x + 4\right ) \log \left (x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2+1)*log(-4*x^2+4*x+4)-2*x^3+3*x^2-x)*log(x-1)+(-x^3+x^2+x)*log(-4*x^2+4*x+4))/(x^3-2*x^2
+1)/log(-4*x^2+4*x+4)^2/log(x-1)^2,x, algorithm="giac")

[Out]

x/(log(-4*x^2 + 4*x + 4)*log(x - 1))

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maple [A]  time = 0.05, size = 22, normalized size = 1.00

method	result	size
risch	\(\frac {x}{\ln \left (x -1\right ) \ln \left (-4 x^{2}+4 x +4\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3-2*x^2+1)*ln(-4*x^2+4*x+4)-2*x^3+3*x^2-x)*ln(x-1)+(-x^3+x^2+x)*ln(-4*x^2+4*x+4))/(x^3-2*x^2+1)/ln(-4
*x^2+4*x+4)^2/ln(x-1)^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.52, size = 32, normalized size = 1.45 \begin {gather*} \frac {x}{{\left (i \, \pi + 2 \, \log \relax (2)\right )} \log \left (x - 1\right ) + \log \left (x^{2} - x - 1\right ) \log \left (x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3-2*x^2+1)*log(-4*x^2+4*x+4)-2*x^3+3*x^2-x)*log(x-1)+(-x^3+x^2+x)*log(-4*x^2+4*x+4))/(x^3-2*x^2
+1)/log(-4*x^2+4*x+4)^2/log(x-1)^2,x, algorithm="maxima")

[Out]

x/((I*pi + 2*log(2))*log(x - 1) + log(x^2 - x - 1)*log(x - 1))

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mupad [B]  time = 6.96, size = 349, normalized size = 15.86 \begin {gather*} \frac {x}{4}-\frac {5\,\ln \left (x-1\right )}{8}+\frac {\frac {x}{\ln \left (x-1\right )}-\frac {\ln \left (-4\,x^2+4\,x+4\right )\,\left (-x^2+x+1\right )\,\left (x+\ln \left (x-1\right )-x\,\ln \left (x-1\right )\right )}{{\ln \left (x-1\right )}^2\,\left (2\,x-1\right )\,\left (x-1\right )}}{\ln \left (-4\,x^2+4\,x+4\right )}+\frac {\frac {\ln \left (x-1\right )\,\left (4\,x^4-11\,x^3+10\,x^2-2\right )}{2\,{\left (2\,x-1\right )}^2\,\left (x-1\right )}+\frac {x\,\left (-x^2+x+1\right )}{\left (2\,x-1\right )\,\left (x-1\right )}-\frac {{\ln \left (x-1\right )}^2\,\left (x-1\right )\,\left (2\,x^2-2\,x+3\right )}{2\,{\left (2\,x-1\right )}^2}}{{\ln \left (x-1\right )}^2}+\frac {\frac {13\,x^3}{16}-\frac {39\,x^2}{32}+\frac {3\,x}{8}+\frac {3}{32}}{x^4-\frac {5\,x^3}{2}+\frac {9\,x^2}{4}-\frac {7\,x}{8}+\frac {1}{8}}+\frac {\frac {x^3-6\,x^2+4\,x}{2\,{\left (2\,x-1\right )}^2\,\left (x-1\right )}+\frac {{\ln \left (x-1\right )}^2\,\left (x-1\right )\,\left (-4\,x^3+6\,x^2+2\,x-7\right )}{2\,{\left (2\,x-1\right )}^3}+\frac {\ln \left (x-1\right )\,\left (-8\,x^3+7\,x^2+4\,x-4\right )}{2\,{\left (2\,x-1\right )}^3\,\left (x-1\right )}}{\ln \left (x-1\right )}+\frac {\ln \left (x-1\right )\,\left (\frac {x^4}{4}-\frac {11\,x^2}{16}+\frac {33\,x}{32}-\frac {33}{64}\right )}{x^3-\frac {3\,x^2}{2}+\frac {3\,x}{4}-\frac {1}{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*x - 4*x^2 + 4)*(x + x^2 - x^3) - log(x - 1)*(x - log(4*x - 4*x^2 + 4)*(x^3 - 2*x^2 + 1) - 3*x^2 + 2
*x^3))/(log(x - 1)^2*log(4*x - 4*x^2 + 4)^2*(x^3 - 2*x^2 + 1)),x)

[Out]

x/4 - (5*log(x - 1))/8 + (x/log(x - 1) - (log(4*x - 4*x^2 + 4)*(x - x^2 + 1)*(x + log(x - 1) - x*log(x - 1)))/
(log(x - 1)^2*(2*x - 1)*(x - 1)))/log(4*x - 4*x^2 + 4) + ((log(x - 1)*(10*x^2 - 11*x^3 + 4*x^4 - 2))/(2*(2*x -
1)^2*(x - 1)) + (x*(x - x^2 + 1))/((2*x - 1)*(x - 1)) - (log(x - 1)^2*(x - 1)*(2*x^2 - 2*x + 3))/(2*(2*x - 1)
^2))/log(x - 1)^2 + ((3*x)/8 - (39*x^2)/32 + (13*x^3)/16 + 3/32)/((9*x^2)/4 - (7*x)/8 - (5*x^3)/2 + x^4 + 1/8)
+ ((4*x - 6*x^2 + x^3)/(2*(2*x - 1)^2*(x - 1)) + (log(x - 1)^2*(x - 1)*(2*x + 6*x^2 - 4*x^3 - 7))/(2*(2*x - 1
)^3) + (log(x - 1)*(4*x + 7*x^2 - 8*x^3 - 4))/(2*(2*x - 1)^3*(x - 1)))/log(x - 1) + (log(x - 1)*((33*x)/32 - (
11*x^2)/16 + x^4/4 - 33/64))/((3*x)/4 - (3*x^2)/2 + x^3 - 1/8)

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sympy [A]  time = 0.30, size = 17, normalized size = 0.77 \begin {gather*} \frac {x}{\log {\left (x - 1 \right )} \log {\left (- 4 x^{2} + 4 x + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3-2*x**2+1)*ln(-4*x**2+4*x+4)-2*x**3+3*x**2-x)*ln(x-1)+(-x**3+x**2+x)*ln(-4*x**2+4*x+4))/(x**3
-2*x**2+1)/ln(-4*x**2+4*x+4)**2/ln(x-1)**2,x)

[Out]

x/(log(x - 1)*log(-4*x**2 + 4*x + 4))

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