3.68.58 \(\int \frac {x+(4+8 x) \log (\frac {2+4 x}{x})}{(16+32 x) \log (\frac {2+4 x}{x})+(8 x+16 x^2) \log (\frac {2+4 x}{x}) \log (\log (\frac {2+4 x}{x}))+(x^2+2 x^3) \log (\frac {2+4 x}{x}) \log ^2(\log (\frac {2+4 x}{x}))} \, dx\)

Optimal. Leaf size=19 \[ -1+\frac {x}{4+x \log \left (\log \left (4+\frac {2}{x}\right )\right )} \]

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Rubi [F]  time = 0.88, 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 {x+(4+8 x) \log \left (\frac {2+4 x}{x}\right )}{(16+32 x) \log \left (\frac {2+4 x}{x}\right )+\left (8 x+16 x^2\right ) \log \left (\frac {2+4 x}{x}\right ) \log \left (\log \left (\frac {2+4 x}{x}\right )\right )+\left (x^2+2 x^3\right ) \log \left (\frac {2+4 x}{x}\right ) \log ^2\left (\log \left (\frac {2+4 x}{x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.04, size = 17, normalized size = 0.89 \begin {gather*} \frac {x}{4+x \log \left (\log \left (4+\frac {2}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(4 + x*Log[Log[4 + 2/x]])

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fricas [A]  time = 0.74, size = 20, normalized size = 1.05 \begin {gather*} \frac {x}{x \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x*log(log(2*(2*x + 1)/x)) + 4)

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giac [B]  time = 0.46, size = 345, normalized size = 18.16 \begin {gather*} \frac {8 \, x^{2} \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) - 8 \, x^{2} \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) + x^{2} \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) + 4 \, x \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) - 4 \, x \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )}{8 \, x^{2} \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) - 8 \, x^{2} \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) + x^{2} \log \left (4 \, x + 2\right ) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) - x^{2} \log \relax (x) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) + 4 \, x \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) - 4 \, x \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) \log \left (\log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )\right ) + 32 \, x \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) - 32 \, x \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) + 4 \, x \log \left (4 \, x + 2\right ) - 4 \, x \log \relax (x) + 16 \, \log \left (4 \, x + 2\right ) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right ) - 16 \, \log \relax (x) \log \left (\frac {2 \, {\left (2 \, x + 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*x^2*log(4*x + 2)*log(2*(2*x + 1)/x) - 8*x^2*log(x)*log(2*(2*x + 1)/x) + x^2*log(2*(2*x + 1)/x) + 4*x*log(4*
x + 2)*log(2*(2*x + 1)/x) - 4*x*log(x)*log(2*(2*x + 1)/x))/(8*x^2*log(4*x + 2)*log(2*(2*x + 1)/x)*log(log(2*(2
*x + 1)/x)) - 8*x^2*log(x)*log(2*(2*x + 1)/x)*log(log(2*(2*x + 1)/x)) + x^2*log(4*x + 2)*log(log(2*(2*x + 1)/x
)) - x^2*log(x)*log(log(2*(2*x + 1)/x)) + 4*x*log(4*x + 2)*log(2*(2*x + 1)/x)*log(log(2*(2*x + 1)/x)) - 4*x*lo
g(x)*log(2*(2*x + 1)/x)*log(log(2*(2*x + 1)/x)) + 32*x*log(4*x + 2)*log(2*(2*x + 1)/x) - 32*x*log(x)*log(2*(2*
x + 1)/x) + 4*x*log(4*x + 2) - 4*x*log(x) + 16*log(4*x + 2)*log(2*(2*x + 1)/x) - 16*log(x)*log(2*(2*x + 1)/x))

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (8 x +4\right ) \ln \left (\frac {4 x +2}{x}\right )+x}{\left (2 x^{3}+x^{2}\right ) \ln \left (\frac {4 x +2}{x}\right ) \ln \left (\ln \left (\frac {4 x +2}{x}\right )\right )^{2}+\left (16 x^{2}+8 x \right ) \ln \left (\frac {4 x +2}{x}\right ) \ln \left (\ln \left (\frac {4 x +2}{x}\right )\right )+\left (32 x +16\right ) \ln \left (\frac {4 x +2}{x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x+4)*ln((4*x+2)/x)+x)/((2*x^3+x^2)*ln((4*x+2)/x)*ln(ln((4*x+2)/x))^2+(16*x^2+8*x)*ln((4*x+2)/x)*ln(ln(
(4*x+2)/x))+(32*x+16)*ln((4*x+2)/x)),x)

[Out]

int(((8*x+4)*ln((4*x+2)/x)+x)/((2*x^3+x^2)*ln((4*x+2)/x)*ln(ln((4*x+2)/x))^2+(16*x^2+8*x)*ln((4*x+2)/x)*ln(ln(
(4*x+2)/x))+(32*x+16)*ln((4*x+2)/x)),x)

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maxima [A]  time = 0.53, size = 22, normalized size = 1.16 \begin {gather*} \frac {x}{x \log \left (\log \relax (2) + \log \left (2 \, x + 1\right ) - \log \relax (x)\right ) + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.17, size = 167, normalized size = 8.79 \begin {gather*} \frac {x\,{\left (\ln \left (\frac {4\,x+2}{x}\right )+2\,x\,\ln \left (\frac {4\,x+2}{x}\right )\right )}^2\,\left (x+4\,\ln \left (\frac {4\,x+2}{x}\right )+8\,x\,\ln \left (\frac {4\,x+2}{x}\right )\right )}{\ln \left (\frac {4\,x+2}{x}\right )\,\left (2\,x+1\right )\,\left (x\,\ln \left (\ln \left (\frac {4\,x+2}{x}\right )\right )+4\right )\,\left (16\,x^2\,{\ln \left (\frac {4\,x+2}{x}\right )}^2+2\,x^2\,\ln \left (\frac {4\,x+2}{x}\right )+16\,x\,{\ln \left (\frac {4\,x+2}{x}\right )}^2+x\,\ln \left (\frac {4\,x+2}{x}\right )+4\,{\ln \left (\frac {4\,x+2}{x}\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(log((4*x + 2)/x) + 2*x*log((4*x + 2)/x))^2*(x + 4*log((4*x + 2)/x) + 8*x*log((4*x + 2)/x)))/(log((4*x + 2)
/x)*(2*x + 1)*(x*log(log((4*x + 2)/x)) + 4)*(16*x*log((4*x + 2)/x)^2 + 2*x^2*log((4*x + 2)/x) + 4*log((4*x + 2
)/x)^2 + 16*x^2*log((4*x + 2)/x)^2 + x*log((4*x + 2)/x)))

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sympy [A]  time = 0.31, size = 14, normalized size = 0.74 \begin {gather*} \frac {x}{x \log {\left (\log {\left (\frac {4 x + 2}{x} \right )} \right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+4)*ln((4*x+2)/x)+x)/((2*x**3+x**2)*ln((4*x+2)/x)*ln(ln((4*x+2)/x))**2+(16*x**2+8*x)*ln((4*x+2)
/x)*ln(ln((4*x+2)/x))+(32*x+16)*ln((4*x+2)/x)),x)

[Out]

x/(x*log(log((4*x + 2)/x)) + 4)

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