3.91.22 \(\int \frac {-6+27 x-28 x^2+8 x^3+4 x^2 \log (\frac {1}{4} (-2+x))+(-24 x+44 x^2-16 x^3) \log ^2(\frac {1}{4} (-2+x))+(-16 x^2+8 x^3) \log ^4(\frac {1}{4} (-2+x))+(12-46 x+52 x^2-16 x^3+(40 x-84 x^2+32 x^3) \log ^2(\frac {1}{4} (-2+x))+(32 x^2-16 x^3) \log ^4(\frac {1}{4} (-2+x))) \log (\frac {-2 x+4 x^2-4 x^2 \log ^2(\frac {1}{4} (-2+x))}{3-4 x+4 x \log ^2(\frac {1}{4} (-2+x))})}{-6 x^3+23 x^4-26 x^5+8 x^6+(-20 x^4+42 x^5-16 x^6) \log ^2(\frac {1}{4} (-2+x))+(-16 x^5+8 x^6) \log ^4(\frac {1}{4} (-2+x))} \, dx\)

Optimal. Leaf size=32 \[ \frac {\log \left (-x+\frac {x}{3-x \left (4-4 \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}\right )}{x^2} \]

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Rubi [F]  time = 31.63, 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 {-6+27 x-28 x^2+8 x^3+4 x^2 \log \left (\frac {1}{4} (-2+x)\right )+\left (-24 x+44 x^2-16 x^3\right ) \log ^2\left (\frac {1}{4} (-2+x)\right )+\left (-16 x^2+8 x^3\right ) \log ^4\left (\frac {1}{4} (-2+x)\right )+\left (12-46 x+52 x^2-16 x^3+\left (40 x-84 x^2+32 x^3\right ) \log ^2\left (\frac {1}{4} (-2+x)\right )+\left (32 x^2-16 x^3\right ) \log ^4\left (\frac {1}{4} (-2+x)\right )\right ) \log \left (\frac {-2 x+4 x^2-4 x^2 \log ^2\left (\frac {1}{4} (-2+x)\right )}{3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )}\right )}{-6 x^3+23 x^4-26 x^5+8 x^6+\left (-20 x^4+42 x^5-16 x^6\right ) \log ^2\left (\frac {1}{4} (-2+x)\right )+\left (-16 x^5+8 x^6\right ) \log ^4\left (\frac {1}{4} (-2+x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 27*x - 28*x^2 + 8*x^3 + 4*x^2*Log[(-2 + x)/4] + (-24*x + 44*x^2 - 16*x^3)*Log[(-2 + x)/4]^2 + (-16*x
^2 + 8*x^3)*Log[(-2 + x)/4]^4 + (12 - 46*x + 52*x^2 - 16*x^3 + (40*x - 84*x^2 + 32*x^3)*Log[(-2 + x)/4]^2 + (3
2*x^2 - 16*x^3)*Log[(-2 + x)/4]^4)*Log[(-2*x + 4*x^2 - 4*x^2*Log[(-2 + x)/4]^2)/(3 - 4*x + 4*x*Log[(-2 + x)/4]
^2)])/(-6*x^3 + 23*x^4 - 26*x^5 + 8*x^6 + (-20*x^4 + 42*x^5 - 16*x^6)*Log[(-2 + x)/4]^2 + (-16*x^5 + 8*x^6)*Lo
g[(-2 + x)/4]^4),x]

[Out]

1/(2*x^2) + Log[(-2*x*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2))/(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)]/x^2 - Defer[Int][(
x*Log[-1/2 + x/4])/(-3 + 4*x - 4*x*Log[(-2 + x)/4]^2), x] + 24*Defer[Int][Log[-1/2 + x/4]^2/(x^2*(-3 + 4*x - 4
*x*Log[(-2 + x)/4]^2)), x] + 16*Defer[Int][Log[-1/2 + x/4]^4/(x*(-3 + 4*x - 4*x*Log[(-2 + x)/4]^2)), x] - Defe
r[Int][Log[-1/2 + x/4]/(-1 + 2*x - 2*x*Log[(-2 + x)/4]^2), x] - 2*Defer[Int][Log[-1/2 + x/4]/(x*(-1 + 2*x - 2*
x*Log[(-2 + x)/4]^2)), x] - Defer[Int][(x*Log[-1/2 + x/4])/(-1 + 2*x - 2*x*Log[(-2 + x)/4]^2), x]/2 + 16*Defer
[Int][Log[-1/2 + x/4]^2/(x*(-1 + 2*x - 2*x*Log[(-2 + x)/4]^2)), x] + 4*Defer[Int][1/(x^3*(1 - 2*x + 2*x*Log[(-
2 + x)/4]^2)), x] - 12*Defer[Int][1/(x^2*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2)), x] + 8*Defer[Int][1/(x*(1 - 2*x +
2*x*Log[(-2 + x)/4]^2)), x] + 2*Log[4]*Defer[Int][1/(x*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2)), x] - Defer[Int][Lo
g[-1/2 + x/4]/(1 - 2*x + 2*x*Log[(-2 + x)/4]^2), x] - Defer[Int][(x*Log[-1/2 + x/4])/(1 - 2*x + 2*x*Log[(-2 +
x)/4]^2), x]/2 + 12*Defer[Int][Log[-1/2 + x/4]^2/(x^2*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2)), x] + 8*Defer[Int][Lo
g[-1/2 + x/4]^4/(x*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2)), x] - Log[16]*Defer[Int][(3 - 4*x + 4*x*Log[(-2 + x)/4]^
2)^(-1), x] + 2*Log[16]*Defer[Int][1/((-2 + x)*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x] - 9*Defer[Int][1/(x^3*(3
- 4*x + 4*x*Log[(-2 + x)/4]^2)), x] + 24*Defer[Int][1/(x^2*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x] - 16*Defer[
Int][1/(x*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x] - 2*Log[16]*Defer[Int][1/(x*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)
), x] - 2*Defer[Int][Log[-1/2 + x/4]/(3 - 4*x + 4*x*Log[(-2 + x)/4]^2), x] - 4*Defer[Int][Log[-1/2 + x/4]/((2
- x)*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x] - 4*Defer[Int][Log[-1/2 + x/4]/(x*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2
)), x] - Defer[Int][(x*Log[-1/2 + x/4])/(3 - 4*x + 4*x*Log[(-2 + x)/4]^2), x] + 32*Defer[Int][Log[-1/2 + x/4]^
2/(x*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x] - 2*Defer[Int][Log[-2 + x]/(x*(1 - 2*x + 2*x*Log[(-2 + x)/4]^2)),
x] + 2*Defer[Int][Log[-2 + x]/(3 - 4*x + 4*x*Log[(-2 + x)/4]^2), x] - 4*Defer[Int][Log[-2 + x]/((-2 + x)*(3 -
4*x + 4*x*Log[(-2 + x)/4]^2)), x] + 4*Defer[Int][Log[-2 + x]/(x*(3 - 4*x + 4*x*Log[(-2 + x)/4]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8}{(-2+x) \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}-\frac {6}{(-2+x) x^3 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}+\frac {27}{(-2+x) x^2 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}-\frac {28}{(-2+x) x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}+\frac {4 \log \left (-\frac {1}{2}+\frac {x}{4}\right )}{(-2+x) x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}+\frac {4 (3-4 x) \log ^2\left (-\frac {1}{2}+\frac {x}{4}\right )}{x^2 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}+\frac {8 \log ^4\left (-\frac {1}{2}+\frac {x}{4}\right )}{x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}-\frac {2 \log \left (-\frac {2 x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}{3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )}\right )}{x^3}\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (-\frac {2 x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}{3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )}\right )}{x^3} \, dx\right )+4 \int \frac {\log \left (-\frac {1}{2}+\frac {x}{4}\right )}{(-2+x) x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx+4 \int \frac {(3-4 x) \log ^2\left (-\frac {1}{2}+\frac {x}{4}\right )}{x^2 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx-6 \int \frac {1}{(-2+x) x^3 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx+8 \int \frac {1}{(-2+x) \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx+8 \int \frac {\log ^4\left (-\frac {1}{2}+\frac {x}{4}\right )}{x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx+27 \int \frac {1}{(-2+x) x^2 \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx-28 \int \frac {1}{(-2+x) x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right ) \left (3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.76, size = 46, normalized size = 1.44 \begin {gather*} \frac {\log \left (-\frac {2 x \left (1-2 x+2 x \log ^2\left (\frac {1}{4} (-2+x)\right )\right )}{3-4 x+4 x \log ^2\left (\frac {1}{4} (-2+x)\right )}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 27*x - 28*x^2 + 8*x^3 + 4*x^2*Log[(-2 + x)/4] + (-24*x + 44*x^2 - 16*x^3)*Log[(-2 + x)/4]^2 +
(-16*x^2 + 8*x^3)*Log[(-2 + x)/4]^4 + (12 - 46*x + 52*x^2 - 16*x^3 + (40*x - 84*x^2 + 32*x^3)*Log[(-2 + x)/4]^
2 + (32*x^2 - 16*x^3)*Log[(-2 + x)/4]^4)*Log[(-2*x + 4*x^2 - 4*x^2*Log[(-2 + x)/4]^2)/(3 - 4*x + 4*x*Log[(-2 +
x)/4]^2)])/(-6*x^3 + 23*x^4 - 26*x^5 + 8*x^6 + (-20*x^4 + 42*x^5 - 16*x^6)*Log[(-2 + x)/4]^2 + (-16*x^5 + 8*x
^6)*Log[(-2 + x)/4]^4),x]

[Out]

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

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fricas [A]  time = 0.69, size = 45, normalized size = 1.41 \begin {gather*} \frac {\log \left (-\frac {2 \, {\left (2 \, x^{2} \log \left (\frac {1}{4} \, x - \frac {1}{2}\right )^{2} - 2 \, x^{2} + x\right )}}{4 \, x \log \left (\frac {1}{4} \, x - \frac {1}{2}\right )^{2} - 4 \, x + 3}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2)*log(1/4*x-1/2)^4+(32*x^3-84*x^2+40*x)*log(1/4*x-1/2)^2-16*x^3+52*x^2-46*x+12)*log
((-4*x^2*log(1/4*x-1/2)^2+4*x^2-2*x)/(4*x*log(1/4*x-1/2)^2+3-4*x))+(8*x^3-16*x^2)*log(1/4*x-1/2)^4+(-16*x^3+44
*x^2-24*x)*log(1/4*x-1/2)^2+4*x^2*log(1/4*x-1/2)+8*x^3-28*x^2+27*x-6)/((8*x^6-16*x^5)*log(1/4*x-1/2)^4+(-16*x^
6+42*x^5-20*x^4)*log(1/4*x-1/2)^2+8*x^6-26*x^5+23*x^4-6*x^3),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2)*log(1/4*x-1/2)^4+(32*x^3-84*x^2+40*x)*log(1/4*x-1/2)^2-16*x^3+52*x^2-46*x+12)*log
((-4*x^2*log(1/4*x-1/2)^2+4*x^2-2*x)/(4*x*log(1/4*x-1/2)^2+3-4*x))+(8*x^3-16*x^2)*log(1/4*x-1/2)^4+(-16*x^3+44
*x^2-24*x)*log(1/4*x-1/2)^2+4*x^2*log(1/4*x-1/2)+8*x^3-28*x^2+27*x-6)/((8*x^6-16*x^5)*log(1/4*x-1/2)^4+(-16*x^
6+42*x^5-20*x^4)*log(1/4*x-1/2)^2+8*x^6-26*x^5+23*x^4-6*x^3),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.90, size = 568, normalized size = 17.75

method	result	size
risch	\(-\frac {\ln \left (\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{x^{2}}+\frac {-2 i \pi \mathrm {csgn}\left (\frac {i x \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right ) \mathrm {csgn}\left (\frac {i x \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )+i \pi \,\mathrm {csgn}\left (i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{3}+i \pi \,\mathrm {csgn}\left (\frac {i \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right ) \mathrm {csgn}\left (\frac {i x \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i x \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{\frac {3}{4}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x}\right )^{3}+2 i \pi +2 \ln \relax (x )+2 \ln \left (\frac {1}{2}+\left (\ln \left (\frac {x}{4}-\frac {1}{2}\right )^{2}-1\right ) x \right )}{2 x^{2}}\)	\(568\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*x^3+32*x^2)*ln(1/4*x-1/2)^4+(32*x^3-84*x^2+40*x)*ln(1/4*x-1/2)^2-16*x^3+52*x^2-46*x+12)*ln((-4*x^2*
ln(1/4*x-1/2)^2+4*x^2-2*x)/(4*x*ln(1/4*x-1/2)^2+3-4*x))+(8*x^3-16*x^2)*ln(1/4*x-1/2)^4+(-16*x^3+44*x^2-24*x)*l
n(1/4*x-1/2)^2+4*x^2*ln(1/4*x-1/2)+8*x^3-28*x^2+27*x-6)/((8*x^6-16*x^5)*ln(1/4*x-1/2)^4+(-16*x^6+42*x^5-20*x^4
)*ln(1/4*x-1/2)^2+8*x^6-26*x^5+23*x^4-6*x^3),x,method=_RETURNVERBOSE)

[Out]

-1/x^2*ln(3/4+(ln(1/4*x-1/2)^2-1)*x)+1/2*(-2*I*Pi*csgn(I*x/(3/4+(ln(1/4*x-1/2)^2-1)*x)*(1/2+(ln(1/4*x-1/2)^2-1
)*x))^2-I*Pi*csgn(I*x)*csgn(I*(1/2+(ln(1/4*x-1/2)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))*csgn(I*x/(3/4+(ln(1/4*x
-1/2)^2-1)*x)*(1/2+(ln(1/4*x-1/2)^2-1)*x))+I*Pi*csgn(I*x)*csgn(I*x/(3/4+(ln(1/4*x-1/2)^2-1)*x)*(1/2+(ln(1/4*x-
1/2)^2-1)*x))^2-I*Pi*csgn(I*(1/2+(ln(1/4*x-1/2)^2-1)*x))*csgn(I/(3/4+(ln(1/4*x-1/2)^2-1)*x))*csgn(I*(1/2+(ln(1
/4*x-1/2)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))+I*Pi*csgn(I*(1/2+(ln(1/4*x-1/2)^2-1)*x))*csgn(I*(1/2+(ln(1/4*x-
1/2)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))^2+I*Pi*csgn(I/(3/4+(ln(1/4*x-1/2)^2-1)*x))*csgn(I*(1/2+(ln(1/4*x-1/2
)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))^2-I*Pi*csgn(I*(1/2+(ln(1/4*x-1/2)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))^
3+I*Pi*csgn(I*(1/2+(ln(1/4*x-1/2)^2-1)*x)/(3/4+(ln(1/4*x-1/2)^2-1)*x))*csgn(I*x/(3/4+(ln(1/4*x-1/2)^2-1)*x)*(1
/2+(ln(1/4*x-1/2)^2-1)*x))^2+I*Pi*csgn(I*x/(3/4+(ln(1/4*x-1/2)^2-1)*x)*(1/2+(ln(1/4*x-1/2)^2-1)*x))^3+2*I*Pi+2
*ln(x)+2*ln(1/2+(ln(1/4*x-1/2)^2-1)*x))/x^2

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maxima [C]  time = 0.52, size = 75, normalized size = 2.34 \begin {gather*} -\frac {-i \, \pi - \log \relax (2) + \log \left (4 \, {\left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) \log \left (x - 2\right ) + \log \left (x - 2\right )^{2} - 1\right )} x + 3\right ) - \log \left (2 \, {\left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) \log \left (x - 2\right ) + \log \left (x - 2\right )^{2} - 1\right )} x + 1\right ) - \log \relax (x)}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3+32*x^2)*log(1/4*x-1/2)^4+(32*x^3-84*x^2+40*x)*log(1/4*x-1/2)^2-16*x^3+52*x^2-46*x+12)*log
((-4*x^2*log(1/4*x-1/2)^2+4*x^2-2*x)/(4*x*log(1/4*x-1/2)^2+3-4*x))+(8*x^3-16*x^2)*log(1/4*x-1/2)^4+(-16*x^3+44
*x^2-24*x)*log(1/4*x-1/2)^2+4*x^2*log(1/4*x-1/2)+8*x^3-28*x^2+27*x-6)/((8*x^6-16*x^5)*log(1/4*x-1/2)^4+(-16*x^
6+42*x^5-20*x^4)*log(1/4*x-1/2)^2+8*x^6-26*x^5+23*x^4-6*x^3),x, algorithm="maxima")

[Out]

-(-I*pi - log(2) + log(4*(4*log(2)^2 - 4*log(2)*log(x - 2) + log(x - 2)^2 - 1)*x + 3) - log(2*(4*log(2)^2 - 4*
log(2)*log(x - 2) + log(x - 2)^2 - 1)*x + 1) - log(x))/x^2

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mupad [B]  time = 7.60, size = 45, normalized size = 1.41 \begin {gather*} \frac {\ln \left (-\frac {2\,\left (2\,x^2\,{\ln \left (\frac {x}{4}-\frac {1}{2}\right )}^2-2\,x^2+x\right )}{4\,x\,{\ln \left (\frac {x}{4}-\frac {1}{2}\right )}^2-4\,x+3}\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(27*x - log(x/4 - 1/2)^2*(24*x - 44*x^2 + 16*x^3) + log(-(2*x + 4*x^2*log(x/4 - 1/2)^2 - 4*x^2)/(4*x*log(
x/4 - 1/2)^2 - 4*x + 3))*(log(x/4 - 1/2)^2*(40*x - 84*x^2 + 32*x^3) - 46*x + log(x/4 - 1/2)^4*(32*x^2 - 16*x^3
) + 52*x^2 - 16*x^3 + 12) - log(x/4 - 1/2)^4*(16*x^2 - 8*x^3) - 28*x^2 + 8*x^3 + 4*x^2*log(x/4 - 1/2) - 6)/(lo
g(x/4 - 1/2)^4*(16*x^5 - 8*x^6) + log(x/4 - 1/2)^2*(20*x^4 - 42*x^5 + 16*x^6) + 6*x^3 - 23*x^4 + 26*x^5 - 8*x^
6),x)

[Out]

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

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sympy [A]  time = 1.04, size = 46, normalized size = 1.44 \begin {gather*} \frac {\log {\left (\frac {- 4 x^{2} \log {\left (\frac {x}{4} - \frac {1}{2} \right )}^{2} + 4 x^{2} - 2 x}{4 x \log {\left (\frac {x}{4} - \frac {1}{2} \right )}^{2} - 4 x + 3} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x**3+32*x**2)*ln(1/4*x-1/2)**4+(32*x**3-84*x**2+40*x)*ln(1/4*x-1/2)**2-16*x**3+52*x**2-46*x+1
2)*ln((-4*x**2*ln(1/4*x-1/2)**2+4*x**2-2*x)/(4*x*ln(1/4*x-1/2)**2+3-4*x))+(8*x**3-16*x**2)*ln(1/4*x-1/2)**4+(-
16*x**3+44*x**2-24*x)*ln(1/4*x-1/2)**2+4*x**2*ln(1/4*x-1/2)+8*x**3-28*x**2+27*x-6)/((8*x**6-16*x**5)*ln(1/4*x-
1/2)**4+(-16*x**6+42*x**5-20*x**4)*ln(1/4*x-1/2)**2+8*x**6-26*x**5+23*x**4-6*x**3),x)

[Out]

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

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