3.70.85 \(\int \frac {-16 x-2 x^2-8 x^4-4 x^5+(-8 x-2 x^2+8 x^4+2 x^5) \log (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4})}{(-4-x+4 x^3+x^4) \log ^2(\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4})} \, dx\)

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

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Rubi [F]  time = 2.17, 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 {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x - 2*x^2 - 8*x^4 - 4*x^5 + (-8*x - 2*x^2 + 8*x^4 + 2*x^5)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*
x^5 + 16*x^6 + 8*x^7 + x^8)/x^4])/((-4 - x + 4*x^3 + x^4)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x
^6 + 8*x^7 + x^8)/x^4]^2),x]

[Out]

8*Defer[Int][Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^(-2), x] - ((4*I)*Defer[Int][1/((-1 + I*Sqrt[3] - 2*x)*Log[(-4
- x + 4*x^3 + x^4)^2/x^4]^2), x])/Sqrt[3] - 2*Defer[Int][1/((-1 + x)*Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^2), x]
- 4*Defer[Int][x/Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^2, x] - 32*Defer[Int][1/((4 + x)*Log[(-4 - x + 4*x^3 + x^4)
^2/x^4]^2), x] + (2*(3 + I*Sqrt[3])*Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^2),
x])/3 - ((4*I)*Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^2), x])/Sqrt[3] + (2*(3 -
I*Sqrt[3])*Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*Log[(-4 - x + 4*x^3 + x^4)^2/x^4]^2), x])/3 + 2*Defer[Int][x/L
og[(-4 - x + 4*x^3 + x^4)^2/x^4], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (8+x+4 x^3+2 x^4-\left (-4-x+4 x^3+x^4\right ) \log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )\right )}{\left (4+x-4 x^3-x^4\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=2 \int \frac {x \left (8+x+4 x^3+2 x^4-\left (-4-x+4 x^3+x^4\right ) \log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )\right )}{\left (4+x-4 x^3-x^4\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=2 \int \left (-\frac {x \left (8+x+4 x^3+2 x^4\right )}{(-1+x) (4+x) \left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (8+x+4 x^3+2 x^4\right )}{(-1+x) (4+x) \left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \left (-\frac {4}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {2 x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {16}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1-x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )-2 \int \frac {1-x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \left (\frac {1}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}-\frac {x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )-2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )-2 \int \frac {1}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \left (\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )+2 \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1-\frac {i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx-2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ &=-\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-\frac {(4 i) \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx}{\sqrt {3}}+\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^2}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x - 2*x^2 - 8*x^4 - 4*x^5 + (-8*x - 2*x^2 + 8*x^4 + 2*x^5)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^
4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8)/x^4])/((-4 - x + 4*x^3 + x^4)*Log[(16 + 8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5
+ 16*x^6 + 8*x^7 + x^8)/x^4]^2),x]

[Out]

x^2/Log[(-4 - x + 4*x^3 + x^4)^2/x^4]

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fricas [B]  time = 0.62, size = 47, normalized size = 2.14 \begin {gather*} \frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^
2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="fricas")

[Out]

x^2/log((x^8 + 8*x^7 + 16*x^6 - 2*x^5 - 16*x^4 - 32*x^3 + x^2 + 8*x + 16)/x^4)

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giac [B]  time = 0.43, size = 47, normalized size = 2.14 \begin {gather*} \frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^
2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="giac")

[Out]

x^2/log((x^8 + 8*x^7 + 16*x^6 - 2*x^5 - 16*x^4 - 32*x^3 + x^2 + 8*x + 16)/x^4)

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maple [B]  time = 0.08, size = 48, normalized size = 2.18

method	result	size
norman	\(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\)	\(48\)
risch	\(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5+8*x^4-2*x^2-8*x)*ln((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^2-16*x)
/(x^4+4*x^3-x-4)/ln((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x,method=_RETURNVERBOSE)

[Out]

x^2/ln((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)

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maxima [A]  time = 0.42, size = 27, normalized size = 1.23 \begin {gather*} \frac {x^{2}}{2 \, {\left (\log \left (x^{2} + x + 1\right ) + \log \left (x + 4\right ) + \log \left (x - 1\right ) - 2 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5+8*x^4-2*x^2-8*x)*log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)-4*x^5-8*x^4-2*x^
2-16*x)/(x^4+4*x^3-x-4)/log((x^8+8*x^7+16*x^6-2*x^5-16*x^4-32*x^3+x^2+8*x+16)/x^4)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.47, size = 160, normalized size = 7.27 \begin {gather*} x+\frac {x^2+\frac {x^2\,\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )\,\left (-x^4-4\,x^3+x+4\right )}{2\,x^4+4\,x^3+x+8}}{\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )}-\frac {-\frac {13\,x^3}{4}+\frac {9\,x^2}{2}+3\,x-8}{x^4+2\,x^3+\frac {x}{2}+4}+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4)*(8*x + 2*x^2 - 8*x^4 -
2*x^5) + 2*x^2 + 8*x^4 + 4*x^5)/(log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4)^2*
(x - 4*x^3 - x^4 + 4)),x)

[Out]

x + (x^2 + (x^2*log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4)*(x - 4*x^3 - x^4 +
4))/(x + 4*x^3 + 2*x^4 + 8))/log((8*x + x^2 - 32*x^3 - 16*x^4 - 2*x^5 + 16*x^6 + 8*x^7 + x^8 + 16)/x^4) - (3*x
+ (9*x^2)/2 - (13*x^3)/4 - 8)/(x/2 + 2*x^3 + x^4 + 4) + x^2/2

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sympy [B]  time = 0.20, size = 44, normalized size = 2.00 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {x^{8} + 8 x^{7} + 16 x^{6} - 2 x^{5} - 16 x^{4} - 32 x^{3} + x^{2} + 8 x + 16}{x^{4}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5+8*x**4-2*x**2-8*x)*ln((x**8+8*x**7+16*x**6-2*x**5-16*x**4-32*x**3+x**2+8*x+16)/x**4)-4*x**5
-8*x**4-2*x**2-16*x)/(x**4+4*x**3-x-4)/ln((x**8+8*x**7+16*x**6-2*x**5-16*x**4-32*x**3+x**2+8*x+16)/x**4)**2,x)

[Out]

x**2/log((x**8 + 8*x**7 + 16*x**6 - 2*x**5 - 16*x**4 - 32*x**3 + x**2 + 8*x + 16)/x**4)

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