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

Optimal. Leaf size=28 \[ e^4+x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \]

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Rubi [F]  time = 2.60, 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 {-1+\left (1+8 x^3\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^3\right ) \log ^2\left (\frac {1}{x}\right )+\left (\left (-1-8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (-2-16 x^2\right ) \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (\left (1+8 x^2\right ) \log \left (\frac {1}{x}\right )+\left (2+16 x^2\right ) \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (2 x \log \left (\frac {1}{x}\right )+4 x \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )}{4 x^2 \log \left (\frac {1}{x}\right )+8 x^2 \log ^2\left (\frac {1}{x}\right )+\left (-4 x \log \left (\frac {1}{x}\right )-8 x \log ^2\left (\frac {1}{x}\right )\right ) \log (x)+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2(x)+\left (4 x \log \left (\frac {1}{x}\right )+8 x \log ^2\left (\frac {1}{x}\right )+\left (-2 \log \left (\frac {1}{x}\right )-4 \log ^2\left (\frac {1}{x}\right )\right ) \log (x)\right ) \log \left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )+\left (\log \left (\frac {1}{x}\right )+2 \log ^2\left (\frac {1}{x}\right )\right ) \log ^2\left (\frac {1+2 \log \left (\frac {1}{x}\right )}{\log \left (\frac {1}{x}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 25, normalized size = 0.89 \begin {gather*} x^2+\frac {x}{2 x-\log (x)+\log \left (2+\frac {1}{\log \left (\frac {1}{x}\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.80, size = 62, normalized size = 2.21 \begin {gather*} \frac {2 \, x^{3} + x^{2} \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + x^{2} \log \left (\frac {1}{x}\right ) + x}{2 \, x + \log \left (\frac {2 \, \log \left (\frac {1}{x}\right ) + 1}{\log \left (\frac {1}{x}\right )}\right ) + \log \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.96, size = 28, normalized size = 1.00 \begin {gather*} x^{2} + \frac {x}{2 \, x - \log \relax (x) + \log \left (2 \, \log \relax (x) - 1\right ) - \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.32, size = 136, normalized size = 4.86

method	result	size
risch	\(x^{2}+\frac {2 x}{-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-\frac {1}{2}\right )}{\ln \relax (x )}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-\frac {1}{2}\right )}{\ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-\frac {1}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-\frac {1}{2}\right )}{\ln \relax (x )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \relax (x )-\frac {1}{2}\right )}{\ln \relax (x )}\right )^{3}+2 \ln \relax (2)+4 x -2 \ln \relax (x )-2 \ln \left (\ln \relax (x )\right )+2 \ln \left (\ln \relax (x )-\frac {1}{2}\right )}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.66, size = 56, normalized size = 2.00 \begin {gather*} \frac {2 \, x^{3} - x^{2} \log \relax (x) + x^{2} \log \left (2 \, \log \relax (x) - 1\right ) - x^{2} \log \left (\log \relax (x)\right ) + x}{2 \, x - \log \relax (x) + \log \left (2 \, \log \relax (x) - 1\right ) - \log \left (\log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+\ln \left (\frac {1}{x}\right )\,\left (8\,x^3+1\right )+\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (\ln \left (\frac {1}{x}\right )\,\left (8\,x^2+1\right )-\ln \relax (x)\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^2+2\right )\right )-\ln \relax (x)\,\left (\left (16\,x^2+2\right )\,{\ln \left (\frac {1}{x}\right )}^2+\left (8\,x^2+1\right )\,\ln \left (\frac {1}{x}\right )\right )+{\ln \relax (x)}^2\,\left (4\,x\,{\ln \left (\frac {1}{x}\right )}^2+2\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \left (\frac {1}{x}\right )}^2\,\left (16\,x^3+2\right )-1}{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )\,\left (4\,x\,\ln \left (\frac {1}{x}\right )-\ln \relax (x)\,\left (4\,{\ln \left (\frac {1}{x}\right )}^2+2\,\ln \left (\frac {1}{x}\right )\right )+8\,x\,{\ln \left (\frac {1}{x}\right )}^2\right )-\ln \relax (x)\,\left (8\,x\,{\ln \left (\frac {1}{x}\right )}^2+4\,x\,\ln \left (\frac {1}{x}\right )\right )+{\ln \relax (x)}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+4\,x^2\,\ln \left (\frac {1}{x}\right )+{\ln \left (\frac {2\,\ln \left (\frac {1}{x}\right )+1}{\ln \left (\frac {1}{x}\right )}\right )}^2\,\left (2\,{\ln \left (\frac {1}{x}\right )}^2+\ln \left (\frac {1}{x}\right )\right )+8\,x^2\,{\ln \left (\frac {1}{x}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.41, size = 24, normalized size = 0.86 \begin {gather*} x^{2} + \frac {x}{2 x - \log {\relax (x )} + \log {\left (- \frac {1 - 2 \log {\relax (x )}}{\log {\relax (x )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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