3.57.96 \(\int \frac {8-14 x+7 x^2+(-16 x+4 x^3) \log (x)+(4 x-x^3) \log (\frac {1}{4 x-4 x^2+x^3})}{8-4 x+(-8 x^2+4 x^3) \log (x)+(2 x^2-x^3) \log (\frac {1}{4 x-4 x^2+x^3})} \, dx\)

Optimal. Leaf size=26 \[ x+\log \left (4+x^2 \left (-4 \log (x)+\log \left (\frac {x}{\left (-2 x+x^2\right )^2}\right )\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.91, 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 {8-14 x+7 x^2+\left (-16 x+4 x^3\right ) \log (x)+\left (4 x-x^3\right ) \log \left (\frac {1}{4 x-4 x^2+x^3}\right )}{8-4 x+\left (-8 x^2+4 x^3\right ) \log (x)+\left (2 x^2-x^3\right ) \log \left (\frac {1}{4 x-4 x^2+x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 - 14*x + 7*x^2 + (-16*x + 4*x^3)*Log[x] + (4*x - x^3)*Log[(4*x - 4*x^2 + x^3)^(-1)])/(8 - 4*x + (-8*x^2
+ 4*x^3)*Log[x] + (2*x^2 - x^3)*Log[(4*x - 4*x^2 + x^3)^(-1)]),x]

[Out]

x + 2*Log[x] - 4*Defer[Int][(4 + x^2*Log[1/((-2 + x)^2*x)] - 4*x^2*Log[x])^(-1), x] - 8*Defer[Int][1/((-2 + x)
*(4 + x^2*Log[1/((-2 + x)^2*x)] - 4*x^2*Log[x])), x] - 8*Defer[Int][1/(x*(4 + x^2*Log[1/((-2 + x)^2*x)] - 4*x^
2*Log[x])), x] - 7*Defer[Int][x/(4 + x^2*Log[1/((-2 + x)^2*x)] - 4*x^2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-14 x+7 x^2-x \left (-4+x^2\right ) \log \left (\frac {1}{(-2+x)^2 x}\right )+4 x \left (-4+x^2\right ) \log (x)}{(2-x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {2+x}{x}+\frac {16-8 x+10 x^2-7 x^3}{(-2+x) x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}\right ) \, dx\\ &=\int \frac {2+x}{x} \, dx+\int \frac {16-8 x+10 x^2-7 x^3}{(-2+x) x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ &=\int \left (1+\frac {2}{x}\right ) \, dx+\int \left (-\frac {4}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)}-\frac {8}{(-2+x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}-\frac {8}{x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )}-\frac {7 x}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)}\right ) \, dx\\ &=x+2 \log (x)-4 \int \frac {1}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)} \, dx-7 \int \frac {x}{4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)} \, dx-8 \int \frac {1}{(-2+x) \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx-8 \int \frac {1}{x \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.49, size = 26, normalized size = 1.00 \begin {gather*} x+\log \left (4+x^2 \log \left (\frac {1}{(-2+x)^2 x}\right )-4 x^2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 14*x + 7*x^2 + (-16*x + 4*x^3)*Log[x] + (4*x - x^3)*Log[(4*x - 4*x^2 + x^3)^(-1)])/(8 - 4*x + (
-8*x^2 + 4*x^3)*Log[x] + (2*x^2 - x^3)*Log[(4*x - 4*x^2 + x^3)^(-1)]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 40, normalized size = 1.54 \begin {gather*} x + 2 \, \log \relax (x) + \log \left (\frac {4 \, x^{2} \log \relax (x) - x^{2} \log \left (\frac {1}{x^{3} - 4 \, x^{2} + 4 \, x}\right ) - 4}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-16*x)*log(x)+(-x^3+4*x)*log(1/(x^3-4*x^2+4*x))+7*x^2-14*x+8)/((4*x^3-8*x^2)*log(x)+(-x^3+2*x
^2)*log(1/(x^3-4*x^2+4*x))-4*x+8),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.31, size = 25, normalized size = 0.96 \begin {gather*} x + \log \left (x^{2} \log \left (x^{2} - 4 \, x + 4\right ) + 5 \, x^{2} \log \relax (x) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-16*x)*log(x)+(-x^3+4*x)*log(1/(x^3-4*x^2+4*x))+7*x^2-14*x+8)/((4*x^3-8*x^2)*log(x)+(-x^3+2*x
^2)*log(1/(x^3-4*x^2+4*x))-4*x+8),x, algorithm="giac")

[Out]

x + log(x^2*log(x^2 - 4*x + 4) + 5*x^2*log(x) - 4)

________________________________________________________________________________________

maple [C]  time = 0.12, size = 201, normalized size = 7.73

method	result	size
risch	\(2 \ln \relax (x )+x +\ln \left (\ln \left (x -2\right )-\frac {i \left (\pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2}-\pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+\pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right )^{2} \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )-2 \pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )\right ) \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{2}+\pi \,x^{2} \mathrm {csgn}\left (i \left (x -2\right )^{2}\right )^{3}-\pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{3}+\pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x \left (x -2\right )^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+10 i x^{2} \ln \relax (x )-8 i\right )}{4 x^{2}}\right )\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-16*x)*ln(x)+(-x^3+4*x)*ln(1/(x^3-4*x^2+4*x))+7*x^2-14*x+8)/((4*x^3-8*x^2)*ln(x)+(-x^3+2*x^2)*ln(1/
(x^3-4*x^2+4*x))-4*x+8),x,method=_RETURNVERBOSE)

[Out]

2*ln(x)+x+ln(ln(x-2)-1/4*I*(Pi*x^2*csgn(I/(x-2)^2)*csgn(I/x/(x-2)^2)^2-Pi*x^2*csgn(I/(x-2)^2)*csgn(I/x/(x-2)^2
)*csgn(I/x)+Pi*x^2*csgn(I*(x-2))^2*csgn(I*(x-2)^2)-2*Pi*x^2*csgn(I*(x-2))*csgn(I*(x-2)^2)^2+Pi*x^2*csgn(I*(x-2
)^2)^3-Pi*x^2*csgn(I/x/(x-2)^2)^3+Pi*x^2*csgn(I/x/(x-2)^2)^2*csgn(I/x)+10*I*x^2*ln(x)-8*I)/x^2)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 30, normalized size = 1.15 \begin {gather*} x + 2 \, \log \relax (x) + \log \left (\frac {2 \, x^{2} \log \left (x - 2\right ) + 5 \, x^{2} \log \relax (x) - 4}{2 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-16*x)*log(x)+(-x^3+4*x)*log(1/(x^3-4*x^2+4*x))+7*x^2-14*x+8)/((4*x^3-8*x^2)*log(x)+(-x^3+2*x
^2)*log(1/(x^3-4*x^2+4*x))-4*x+8),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.21, size = 34, normalized size = 1.31 \begin {gather*} x+\ln \left (\frac {1}{x^2}\right )+2\,\ln \relax (x)+\ln \left (x^2\,\ln \left (\frac {1}{x\,{\left (x-2\right )}^2}\right )-4\,x^2\,\ln \relax (x)+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7*x^2 - log(x)*(16*x - 4*x^3) - 14*x + log(1/(4*x - 4*x^2 + x^3))*(4*x - x^3) + 8)/(4*x + log(x)*(8*x^2
- 4*x^3) - log(1/(4*x - 4*x^2 + x^3))*(2*x^2 - x^3) - 8),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.45, size = 36, normalized size = 1.38 \begin {gather*} x + 2 \log {\relax (x )} + \log {\left (\log {\left (\frac {1}{x^{3} - 4 x^{2} + 4 x} \right )} + \frac {- 4 x^{2} \log {\relax (x )} + 4}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-16*x)*ln(x)+(-x**3+4*x)*ln(1/(x**3-4*x**2+4*x))+7*x**2-14*x+8)/((4*x**3-8*x**2)*ln(x)+(-x**
3+2*x**2)*ln(1/(x**3-4*x**2+4*x))-4*x+8),x)

[Out]

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

________________________________________________________________________________________