3.58.30 \(\int \frac {-20 x^3-10 x^5+(20 x+10 x^3) \log (x)+(-10+5 x^2) \log ^2(x)+(40 x^2+20 x^4+(-20-10 x^2) \log (x)+(-10-5 x^2) \log ^2(x)) \log (\frac {2+x^2}{x})+(-20 x-10 x^3) \log ^2(\frac {2+x^2}{x})}{2 x^2+x^4+(-4 x-2 x^3) \log (\frac {2+x^2}{x})+(2+x^2) \log ^2(\frac {2+x^2}{x})} \, dx\)

Optimal. Leaf size=29 \[ 5 \left (-4+x \left (-x+\frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )}\right )\right ) \]

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Rubi [F]  time = 1.41, 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 {-20 x^3-10 x^5+\left (20 x+10 x^3\right ) \log (x)+\left (-10+5 x^2\right ) \log ^2(x)+\left (40 x^2+20 x^4+\left (-20-10 x^2\right ) \log (x)+\left (-10-5 x^2\right ) \log ^2(x)\right ) \log \left (\frac {2+x^2}{x}\right )+\left (-20 x-10 x^3\right ) \log ^2\left (\frac {2+x^2}{x}\right )}{2 x^2+x^4+\left (-4 x-2 x^3\right ) \log \left (\frac {2+x^2}{x}\right )+\left (2+x^2\right ) \log ^2\left (\frac {2+x^2}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20*x^3 - 10*x^5 + (20*x + 10*x^3)*Log[x] + (-10 + 5*x^2)*Log[x]^2 + (40*x^2 + 20*x^4 + (-20 - 10*x^2)*Lo
g[x] + (-10 - 5*x^2)*Log[x]^2)*Log[(2 + x^2)/x] + (-20*x - 10*x^3)*Log[(2 + x^2)/x]^2)/(2*x^2 + x^4 + (-4*x -
2*x^3)*Log[(2 + x^2)/x] + (2 + x^2)*Log[(2 + x^2)/x]^2),x]

[Out]

-5*x^2 + 5*Defer[Int][Log[x]^2/(x - Log[2/x + x])^2, x] - (5*I)*Sqrt[2]*Defer[Int][Log[x]^2/((I*Sqrt[2] - x)*(
x - Log[2/x + x])^2), x] - 5*Defer[Int][(x*Log[x]^2)/(x - Log[2/x + x])^2, x] - (5*I)*Sqrt[2]*Defer[Int][Log[x
]^2/((I*Sqrt[2] + x)*(x - Log[2/x + x])^2), x] + 10*Defer[Int][Log[x]/(x - Log[2/x + x]), x] + 5*Defer[Int][Lo
g[x]^2/(x - Log[2/x + x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-10 x+\frac {10 \log (x)}{x-\log \left (\frac {2}{x}+x\right )}-\frac {5 \log ^2(x) \left (2-x^2+\left (2+x^2\right ) \log \left (\frac {2}{x}+x\right )\right )}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2}\right ) \, dx\\ &=-5 x^2-5 \int \frac {\log ^2(x) \left (2-x^2+\left (2+x^2\right ) \log \left (\frac {2}{x}+x\right )\right )}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx\\ &=-5 x^2-5 \int \left (\frac {\left (2+2 x-x^2+x^3\right ) \log ^2(x)}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2}-\frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )}\right ) \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx\\ &=-5 x^2-5 \int \frac {\left (2+2 x-x^2+x^3\right ) \log ^2(x)}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx+5 \int \frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx\\ &=-5 x^2-5 \int \left (-\frac {\log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2}+\frac {x \log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2}+\frac {4 \log ^2(x)}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2}\right ) \, dx+5 \int \frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx\\ &=-5 x^2+5 \int \frac {\log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx-5 \int \frac {x \log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx+5 \int \frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx-20 \int \frac {\log ^2(x)}{\left (2+x^2\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx\\ &=-5 x^2+5 \int \frac {\log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx-5 \int \frac {x \log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx+5 \int \frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx-20 \int \left (\frac {i \log ^2(x)}{2 \sqrt {2} \left (i \sqrt {2}-x\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2}+\frac {i \log ^2(x)}{2 \sqrt {2} \left (i \sqrt {2}+x\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2}\right ) \, dx\\ &=-5 x^2+5 \int \frac {\log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx-5 \int \frac {x \log ^2(x)}{\left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx+5 \int \frac {\log ^2(x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx+10 \int \frac {\log (x)}{x-\log \left (\frac {2}{x}+x\right )} \, dx-\left (5 i \sqrt {2}\right ) \int \frac {\log ^2(x)}{\left (i \sqrt {2}-x\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx-\left (5 i \sqrt {2}\right ) \int \frac {\log ^2(x)}{\left (i \sqrt {2}+x\right ) \left (x-\log \left (\frac {2}{x}+x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 27, normalized size = 0.93 \begin {gather*} -5 x^2-\frac {5 x \log ^2(x)}{-x+\log \left (\frac {2}{x}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*x^3 - 10*x^5 + (20*x + 10*x^3)*Log[x] + (-10 + 5*x^2)*Log[x]^2 + (40*x^2 + 20*x^4 + (-20 - 10*x
^2)*Log[x] + (-10 - 5*x^2)*Log[x]^2)*Log[(2 + x^2)/x] + (-20*x - 10*x^3)*Log[(2 + x^2)/x]^2)/(2*x^2 + x^4 + (-
4*x - 2*x^3)*Log[(2 + x^2)/x] + (2 + x^2)*Log[(2 + x^2)/x]^2),x]

[Out]

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

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fricas [A]  time = 0.67, size = 44, normalized size = 1.52 \begin {gather*} -\frac {5 \, {\left (x^{3} - x \log \relax (x)^{2} - x^{2} \log \left (\frac {x^{2} + 2}{x}\right )\right )}}{x - \log \left (\frac {x^{2} + 2}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3-20*x)*log((x^2+2)/x)^2+((-5*x^2-10)*log(x)^2+(-10*x^2-20)*log(x)+20*x^4+40*x^2)*log((x^2+2
)/x)+(5*x^2-10)*log(x)^2+(10*x^3+20*x)*log(x)-10*x^5-20*x^3)/((x^2+2)*log((x^2+2)/x)^2+(-2*x^3-4*x)*log((x^2+2
)/x)+x^4+2*x^2),x, algorithm="fricas")

[Out]

-5*(x^3 - x*log(x)^2 - x^2*log((x^2 + 2)/x))/(x - log((x^2 + 2)/x))

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giac [A]  time = 0.36, size = 27, normalized size = 0.93 \begin {gather*} -5 \, x^{2} + \frac {5 \, x \log \relax (x)^{2}}{x - \log \left (x^{2} + 2\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3-20*x)*log((x^2+2)/x)^2+((-5*x^2-10)*log(x)^2+(-10*x^2-20)*log(x)+20*x^4+40*x^2)*log((x^2+2
)/x)+(5*x^2-10)*log(x)^2+(10*x^3+20*x)*log(x)-10*x^5-20*x^3)/((x^2+2)*log((x^2+2)/x)^2+(-2*x^3-4*x)*log((x^2+2
)/x)+x^4+2*x^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.13, size = 134, normalized size = 4.62

method	result	size
risch	\(-5 x^{2}+\frac {10 x \ln \relax (x )^{2}}{-i \pi \,\mathrm {csgn}\left (i \left (x^{2}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+2\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x^{2}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+2\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+2\right )}{x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+2\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+2 x +2 \ln \relax (x )-2 \ln \left (x^{2}+2\right )}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^3-20*x)*ln((x^2+2)/x)^2+((-5*x^2-10)*ln(x)^2+(-10*x^2-20)*ln(x)+20*x^4+40*x^2)*ln((x^2+2)/x)+(5*x^
2-10)*ln(x)^2+(10*x^3+20*x)*ln(x)-10*x^5-20*x^3)/((x^2+2)*ln((x^2+2)/x)^2+(-2*x^3-4*x)*ln((x^2+2)/x)+x^4+2*x^2
),x,method=_RETURNVERBOSE)

[Out]

-5*x^2+10*x*ln(x)^2/(-I*Pi*csgn(I*(x^2+2))*csgn(I/x*(x^2+2))^2+I*Pi*csgn(I*(x^2+2))*csgn(I/x*(x^2+2))*csgn(I/x
)+I*Pi*csgn(I/x*(x^2+2))^3-I*Pi*csgn(I/x*(x^2+2))^2*csgn(I/x)+2*x+2*ln(x)-2*ln(x^2+2))

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maxima [A]  time = 0.39, size = 44, normalized size = 1.52 \begin {gather*} -\frac {5 \, {\left (x^{3} - x^{2} \log \left (x^{2} + 2\right ) + x^{2} \log \relax (x) - x \log \relax (x)^{2}\right )}}{x - \log \left (x^{2} + 2\right ) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3-20*x)*log((x^2+2)/x)^2+((-5*x^2-10)*log(x)^2+(-10*x^2-20)*log(x)+20*x^4+40*x^2)*log((x^2+2
)/x)+(5*x^2-10)*log(x)^2+(10*x^3+20*x)*log(x)-10*x^5-20*x^3)/((x^2+2)*log((x^2+2)/x)^2+(-2*x^3-4*x)*log((x^2+2
)/x)+x^4+2*x^2),x, algorithm="maxima")

[Out]

-5*(x^3 - x^2*log(x^2 + 2) + x^2*log(x) - x*log(x)^2)/(x - log(x^2 + 2) + log(x))

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mupad [B]  time = 3.91, size = 171, normalized size = 5.90 \begin {gather*} 10\,\ln \relax (x)-\frac {\frac {5\,x\,\left (2\,x^3\,\ln \relax (x)+x^2\,{\ln \relax (x)}^2+4\,x\,\ln \relax (x)-2\,{\ln \relax (x)}^2\right )}{x^3-x^2+2\,x+2}-\frac {5\,x\,\ln \left (\frac {x^2+2}{x}\right )\,\left (x^2+2\right )\,\left ({\ln \relax (x)}^2+2\,\ln \relax (x)\right )}{x^3-x^2+2\,x+2}}{x-\ln \left (\frac {x^2+2}{x}\right )}+{\ln \relax (x)}^2\,\left (\frac {5\,x^2-10}{x^3-x^2+2\,x+2}+5\right )-5\,x^2+\frac {\ln \relax (x)\,\left (10\,x^2-20\right )}{x^3-x^2+2\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((x^2 + 2)/x)^2*(20*x + 10*x^3) + log((x^2 + 2)/x)*(log(x)^2*(5*x^2 + 10) - 40*x^2 - 20*x^4 + log(x)*
(10*x^2 + 20)) - log(x)^2*(5*x^2 - 10) - log(x)*(20*x + 10*x^3) + 20*x^3 + 10*x^5)/(log((x^2 + 2)/x)^2*(x^2 +
2) - log((x^2 + 2)/x)*(4*x + 2*x^3) + 2*x^2 + x^4),x)

[Out]

10*log(x) - ((5*x*(2*x^3*log(x) - 2*log(x)^2 + x^2*log(x)^2 + 4*x*log(x)))/(2*x - x^2 + x^3 + 2) - (5*x*log((x
^2 + 2)/x)*(x^2 + 2)*(2*log(x) + log(x)^2))/(2*x - x^2 + x^3 + 2))/(x - log((x^2 + 2)/x)) + log(x)^2*((5*x^2 -
10)/(2*x - x^2 + x^3 + 2) + 5) - 5*x^2 + (log(x)*(10*x^2 - 20))/(2*x - x^2 + x^3 + 2)

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sympy [A]  time = 0.31, size = 24, normalized size = 0.83 \begin {gather*} - 5 x^{2} - \frac {5 x \log {\relax (x )}^{2}}{- x + \log {\left (\frac {x^{2} + 2}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**3-20*x)*ln((x**2+2)/x)**2+((-5*x**2-10)*ln(x)**2+(-10*x**2-20)*ln(x)+20*x**4+40*x**2)*ln((x
**2+2)/x)+(5*x**2-10)*ln(x)**2+(10*x**3+20*x)*ln(x)-10*x**5-20*x**3)/((x**2+2)*ln((x**2+2)/x)**2+(-2*x**3-4*x)
*ln((x**2+2)/x)+x**4+2*x**2),x)

[Out]

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

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