3.31.79 \(\int \frac {-8+4 x-26 x^2+28 x^3-6 x^4+(4-4 x-3 x^2+4 x^3-x^4) \log (\frac {e^{\frac {x}{-2+x}} (1-2 x^2+x^4)}{x})}{(-4 x^2+4 x^3+3 x^4-4 x^5+x^6) \log ^3(\frac {e^{\frac {x}{-2+x}} (1-2 x^2+x^4)}{x})} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{x \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \]

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Rubi [F]  time = 2.55, 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+4 x-26 x^2+28 x^3-6 x^4+\left (4-4 x-3 x^2+4 x^3-x^4\right ) \log \left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )}{\left (-4 x^2+4 x^3+3 x^4-4 x^5+x^6\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 + 4*x - 26*x^2 + 28*x^3 - 6*x^4 + (4 - 4*x - 3*x^2 + 4*x^3 - x^4)*Log[(E^(x/(-2 + x))*(1 - 2*x^2 + x^4
))/x])/((-4*x^2 + 4*x^3 + 3*x^4 - 4*x^5 + x^6)*Log[(E^(x/(-2 + x))*(1 - 2*x^2 + x^4))/x]^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-4 x+26 x^2-28 x^3+6 x^4-\left (4-4 x-3 x^2+4 x^3-x^4\right ) \log \left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )}{(2-x)^2 x^2 \left (1-x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (1-2 x^2+x^4\right )}{x}\right )} \, dx\\ &=\int \left (-\frac {2 \left (4-2 x+13 x^2-14 x^3+3 x^4\right )}{(-2+x)^2 x^2 \left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {4-2 x+13 x^2-14 x^3+3 x^4}{(-2+x)^2 x^2 \left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\right )-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=-\left (2 \int \left (-\frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}+\frac {1}{2 (-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{2 x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}+\frac {4}{\left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx\right )-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-8 \int \frac {1}{\left (-1+x^2\right ) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-8 \int \left (\frac {1}{2 (-1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}-\frac {1}{2 (1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )}\right ) \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ &=2 \int \frac {1}{(-2+x)^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+2 \int \frac {1}{x^2 \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-4 \int \frac {1}{(-1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+4 \int \frac {1}{(1+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{(-2+x) \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx+\int \frac {1}{x \log ^3\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx-\int \frac {1}{x^2 \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 27, normalized size = 1.00 \begin {gather*} \frac {1}{x \log ^2\left (\frac {e^{\frac {x}{-2+x}} \left (-1+x^2\right )^2}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 4*x - 26*x^2 + 28*x^3 - 6*x^4 + (4 - 4*x - 3*x^2 + 4*x^3 - x^4)*Log[(E^(x/(-2 + x))*(1 - 2*x^2
+ x^4))/x])/((-4*x^2 + 4*x^3 + 3*x^4 - 4*x^5 + x^6)*Log[(E^(x/(-2 + x))*(1 - 2*x^2 + x^4))/x]^3),x]

[Out]

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

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fricas [A]  time = 0.64, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{x \log \left (\frac {{\left (x^{4} - 2 \, x^{2} + 1\right )} e^{\left (\frac {x}{x - 2}\right )}}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+4*x^3-3*x^2-4*x+4)*log((x^4-2*x^2+1)*exp(x/(x-2))/x)-6*x^4+28*x^3-26*x^2+4*x-8)/(x^6-4*x^5+3*
x^4+4*x^3-4*x^2)/log((x^4-2*x^2+1)*exp(x/(x-2))/x)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.31, size = 331, normalized size = 12.26 \begin {gather*} \frac {3 \, x^{6} - 26 \, x^{5} + 81 \, x^{4} - 110 \, x^{3} + 64 \, x^{2} - 24 \, x + 16}{3 \, x^{7} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 6 \, x^{7} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 26 \, x^{6} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 3 \, x^{7} - 40 \, x^{6} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 81 \, x^{5} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} - 14 \, x^{6} + 82 \, x^{5} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 110 \, x^{4} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 13 \, x^{5} - 56 \, x^{4} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 64 \, x^{3} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} - 2 \, x^{4} + 16 \, x^{3} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) - 24 \, x^{2} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2} + 4 \, x^{3} - 16 \, x^{2} \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right ) + 16 \, x \log \left (\frac {x^{4} - 2 \, x^{2} + 1}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+4*x^3-3*x^2-4*x+4)*log((x^4-2*x^2+1)*exp(x/(x-2))/x)-6*x^4+28*x^3-26*x^2+4*x-8)/(x^6-4*x^5+3*
x^4+4*x^3-4*x^2)/log((x^4-2*x^2+1)*exp(x/(x-2))/x)^3,x, algorithm="giac")

[Out]

(3*x^6 - 26*x^5 + 81*x^4 - 110*x^3 + 64*x^2 - 24*x + 16)/(3*x^7*log((x^4 - 2*x^2 + 1)/x)^2 + 6*x^7*log((x^4 -
2*x^2 + 1)/x) - 26*x^6*log((x^4 - 2*x^2 + 1)/x)^2 + 3*x^7 - 40*x^6*log((x^4 - 2*x^2 + 1)/x) + 81*x^5*log((x^4
- 2*x^2 + 1)/x)^2 - 14*x^6 + 82*x^5*log((x^4 - 2*x^2 + 1)/x) - 110*x^4*log((x^4 - 2*x^2 + 1)/x)^2 + 13*x^5 - 5
6*x^4*log((x^4 - 2*x^2 + 1)/x) + 64*x^3*log((x^4 - 2*x^2 + 1)/x)^2 - 2*x^4 + 16*x^3*log((x^4 - 2*x^2 + 1)/x) -
24*x^2*log((x^4 - 2*x^2 + 1)/x)^2 + 4*x^3 - 16*x^2*log((x^4 - 2*x^2 + 1)/x) + 16*x*log((x^4 - 2*x^2 + 1)/x)^2
)

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maple [C]  time = 0.38, size = 393, normalized size = 14.56

method	result	size
risch	\(-\frac {4}{x \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x^{2}-1\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )\right ) \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )-\pi \,\mathrm {csgn}\left (i \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{\frac {x}{x -2}} \left (x^{2}-1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2}-1\right )^{2} {\mathrm e}^{\frac {x}{x -2}}}{x}\right )^{3}-2 i \ln \relax (x )+4 i \ln \left (x^{2}-1\right )+2 i \ln \left ({\mathrm e}^{\frac {x}{x -2}}\right )\right )^{2}}\)	\(393\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4+4*x^3-3*x^2-4*x+4)*ln((x^4-2*x^2+1)*exp(x/(x-2))/x)-6*x^4+28*x^3-26*x^2+4*x-8)/(x^6-4*x^5+3*x^4+4*x
^3-4*x^2)/ln((x^4-2*x^2+1)*exp(x/(x-2))/x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.94, size = 165, normalized size = 6.11 \begin {gather*} \frac {x^{2} - 4 \, x + 4}{x^{3} + 4 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x + 1\right )^{2} + 4 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x - 1\right )^{2} + {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)^{2} + 4 \, {\left (x^{3} - 2 \, x^{2} + 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (x - 1\right ) - {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)\right )} \log \left (x + 1\right ) + 4 \, {\left (x^{3} - 2 \, x^{2} - {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \relax (x)\right )} \log \left (x - 1\right ) - 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4+4*x^3-3*x^2-4*x+4)*log((x^4-2*x^2+1)*exp(x/(x-2))/x)-6*x^4+28*x^3-26*x^2+4*x-8)/(x^6-4*x^5+3*
x^4+4*x^3-4*x^2)/log((x^4-2*x^2+1)*exp(x/(x-2))/x)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.33, size = 29, normalized size = 1.07 \begin {gather*} \frac {1}{x\,{\ln \left (\frac {{\mathrm {e}}^{\frac {x}{x-2}}\,\left (x^4-2\,x^2+1\right )}{x}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((exp(x/(x - 2))*(x^4 - 2*x^2 + 1))/x)*(4*x + 3*x^2 - 4*x^3 + x^4 - 4) - 4*x + 26*x^2 - 28*x^3 + 6*x^
4 + 8)/(log((exp(x/(x - 2))*(x^4 - 2*x^2 + 1))/x)^3*(4*x^3 - 4*x^2 + 3*x^4 - 4*x^5 + x^6)),x)

[Out]

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

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sympy [A]  time = 0.45, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{x \log {\left (\frac {\left (x^{4} - 2 x^{2} + 1\right ) e^{\frac {x}{x - 2}}}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4+4*x**3-3*x**2-4*x+4)*ln((x**4-2*x**2+1)*exp(x/(x-2))/x)-6*x**4+28*x**3-26*x**2+4*x-8)/(x**6-
4*x**5+3*x**4+4*x**3-4*x**2)/ln((x**4-2*x**2+1)*exp(x/(x-2))/x)**3,x)

[Out]

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

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