3.93.60 \(\int \frac {-5+(-10 x^2+10 x^3) \log (\frac {-1+x}{2 x})+(-10+10 x) \log (\frac {-1+x}{2 x}) \log (x)}{(-x+x^2) \log (\frac {-1+x}{2 x})} \, dx\)

Optimal. Leaf size=26 \[ 5 \left (x^2+\log ^2(x)-\log \left (5 \log \left (\frac {-1+x}{2 x}\right )\right )\right ) \]

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Rubi [F]  time = 0.59, 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 {-5+\left (-10 x^2+10 x^3\right ) \log \left (\frac {-1+x}{2 x}\right )+(-10+10 x) \log \left (\frac {-1+x}{2 x}\right ) \log (x)}{\left (-x+x^2\right ) \log \left (\frac {-1+x}{2 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + (-10*x^2 + 10*x^3)*Log[(-1 + x)/(2*x)] + (-10 + 10*x)*Log[(-1 + x)/(2*x)]*Log[x])/((-x + x^2)*Log[(-
1 + x)/(2*x)]),x]

[Out]

5*x^2 + 5*Log[x]^2 - 5*Defer[Int][1/((-1 + x)*Log[1/2 - 1/(2*x)]), x] + 5*Defer[Int][1/(x*Log[1/2 - 1/(2*x)]),
x]

Rubi steps

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

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Mathematica [A]  time = 5.02, size = 24, normalized size = 0.92 \begin {gather*} 5 \left (x^2+\log ^2(x)-\log \left (\log \left (\frac {-1+x}{2 x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + (-10*x^2 + 10*x^3)*Log[(-1 + x)/(2*x)] + (-10 + 10*x)*Log[(-1 + x)/(2*x)]*Log[x])/((-x + x^2)*
Log[(-1 + x)/(2*x)]),x]

[Out]

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

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fricas [A]  time = 0.50, size = 24, normalized size = 0.92 \begin {gather*} 5 \, x^{2} + 5 \, \log \relax (x)^{2} - 5 \, \log \left (\log \left (\frac {x - 1}{2 \, x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-10)*log(1/2*(x-1)/x)*log(x)+(10*x^3-10*x^2)*log(1/2*(x-1)/x)-5)/(x^2-x)/log(1/2*(x-1)/x),x, a
lgorithm="fricas")

[Out]

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

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giac [A]  time = 0.25, size = 28, normalized size = 1.08 \begin {gather*} 5 \, x^{2} + 5 \, \log \relax (x)^{2} - 5 \, \log \left (-\log \relax (2) + \log \left (x - 1\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-10)*log(1/2*(x-1)/x)*log(x)+(10*x^3-10*x^2)*log(1/2*(x-1)/x)-5)/(x^2-x)/log(1/2*(x-1)/x),x, a
lgorithm="giac")

[Out]

5*x^2 + 5*log(x)^2 - 5*log(-log(2) + log(x - 1) - log(x))

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maple [A]  time = 0.58, size = 29, normalized size = 1.12

method	result	size
default	\(5 \ln \relax (x )^{2}+5 x^{2}-5 \ln \left (\ln \relax (2)-\ln \left (\frac {x -1}{x}\right )\right )\)	\(29\)
risch	\(5 x^{2}+5 \ln \relax (x )^{2}-5 \ln \left (\ln \left (x -1\right )+\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{x}\right ) \mathrm {csgn}\left (i \left (x -1\right )\right )-\pi \mathrm {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (x -1\right )}{x}\right )^{2} \mathrm {csgn}\left (i \left (x -1\right )\right )+2 i \ln \relax (2)+2 i \ln \relax (x )\right )}{2}\right )\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x-10)*ln(1/2*(x-1)/x)*ln(x)+(10*x^3-10*x^2)*ln(1/2*(x-1)/x)-5)/(x^2-x)/ln(1/2*(x-1)/x),x,method=_RETU
RNVERBOSE)

[Out]

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

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maxima [A]  time = 0.53, size = 28, normalized size = 1.08 \begin {gather*} 5 \, x^{2} + 5 \, \log \relax (x)^{2} - 5 \, \log \left (-\log \relax (2) + \log \left (x - 1\right ) - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-10)*log(1/2*(x-1)/x)*log(x)+(10*x^3-10*x^2)*log(1/2*(x-1)/x)-5)/(x^2-x)/log(1/2*(x-1)/x),x, a
lgorithm="maxima")

[Out]

5*x^2 + 5*log(x)^2 - 5*log(-log(2) + log(x - 1) - log(x))

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mupad [B]  time = 7.48, size = 25, normalized size = 0.96 \begin {gather*} 5\,{\ln \relax (x)}^2-5\,\ln \left (\ln \left (\frac {\frac {x}{2}-\frac {1}{2}}{x}\right )\right )+5\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((x/2 - 1/2)/x)*(10*x^2 - 10*x^3) - log((x/2 - 1/2)/x)*log(x)*(10*x - 10) + 5)/(log((x/2 - 1/2)/x)*(x
- x^2)),x)

[Out]

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

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sympy [A]  time = 0.31, size = 24, normalized size = 0.92 \begin {gather*} 5 x^{2} + 5 \log {\relax (x )}^{2} - 5 \log {\left (\log {\left (\frac {\frac {x}{2} - \frac {1}{2}}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x-10)*ln(1/2*(x-1)/x)*ln(x)+(10*x**3-10*x**2)*ln(1/2*(x-1)/x)-5)/(x**2-x)/ln(1/2*(x-1)/x),x)

[Out]

5*x**2 + 5*log(x)**2 - 5*log(log((x/2 - 1/2)/x))

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