3.61.25 \(\int \frac {(\frac {16}{e^{25}}+x) \log (\frac {16}{e^{25}}+x)+(\frac {16 x}{e^{25}}+x^2) \log (x) \log ^2(\frac {16}{e^{25}}+x)+(-x \log (x)+(-\frac {16}{e^{25}}-x) \log (x) \log (\frac {16}{e^{25}}+x)) \log (\log (x))}{(\frac {16 x^2}{e^{25}}+x^3) \log (x) \log ^2(\frac {16}{e^{25}}+x)} \, dx\)

Optimal. Leaf size=20 \[ \log \left (x \sqrt [x \log \left (\frac {16}{e^{25}}+x\right )]{\log (x)}\right ) \]

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Rubi [F]  time = 1.72, 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 {\left (\frac {16}{e^{25}}+x\right ) \log \left (\frac {16}{e^{25}}+x\right )+\left (\frac {16 x}{e^{25}}+x^2\right ) \log (x) \log ^2\left (\frac {16}{e^{25}}+x\right )+\left (-x \log (x)+\left (-\frac {16}{e^{25}}-x\right ) \log (x) \log \left (\frac {16}{e^{25}}+x\right )\right ) \log (\log (x))}{\left (\frac {16 x^2}{e^{25}}+x^3\right ) \log (x) \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((16/E^25 + x)*Log[16/E^25 + x] + ((16*x)/E^25 + x^2)*Log[x]*Log[16/E^25 + x]^2 + (-(x*Log[x]) + (-16/E^25
- x)*Log[x]*Log[16/E^25 + x])*Log[Log[x]])/(((16*x^2)/E^25 + x^3)*Log[x]*Log[16/E^25 + x]^2),x]

[Out]

Log[x] + Defer[Int][1/(x^2*Log[x]*Log[16/E^25 + x]), x] - (E^25*Defer[Int][Log[Log[x]]/(x*Log[16/E^25 + x]^2),
x])/16 + (E^50*Defer[Int][Log[Log[x]]/((16 + E^25*x)*Log[16/E^25 + x]^2), x])/16 - Defer[Int][Log[Log[x]]/(x^
2*Log[16/E^25 + x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (\frac {16}{e^{25}}+x\right ) \log \left (\frac {16}{e^{25}}+x\right )+\left (\frac {16 x}{e^{25}}+x^2\right ) \log (x) \log ^2\left (\frac {16}{e^{25}}+x\right )+\left (-x \log (x)+\left (-\frac {16}{e^{25}}-x\right ) \log (x) \log \left (\frac {16}{e^{25}}+x\right )\right ) \log (\log (x))}{x^2 \left (\frac {16}{e^{25}}+x\right ) \log (x) \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx\\ &=\int \frac {x-\frac {e^{25} x \log (\log (x))}{\left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )}+\frac {1-\log (x) \log (\log (x))}{\log (x) \log \left (\frac {16}{e^{25}}+x\right )}}{x^2} \, dx\\ &=\int \left (\frac {1+x \log (x) \log \left (\frac {16}{e^{25}}+x\right )}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )}-\frac {\left (e^{25} x+16 \log \left (\frac {16}{e^{25}}+x\right )+e^{25} x \log \left (\frac {16}{e^{25}}+x\right )\right ) \log (\log (x))}{x^2 \left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )}\right ) \, dx\\ &=\int \frac {1+x \log (x) \log \left (\frac {16}{e^{25}}+x\right )}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )} \, dx-\int \frac {\left (e^{25} x+16 \log \left (\frac {16}{e^{25}}+x\right )+e^{25} x \log \left (\frac {16}{e^{25}}+x\right )\right ) \log (\log (x))}{x^2 \left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx\\ &=\int \frac {x+\frac {1}{\log (x) \log \left (\frac {16}{e^{25}}+x\right )}}{x^2} \, dx-\int \frac {\left (\frac {e^{25} x}{16+e^{25} x}+\log \left (\frac {16}{e^{25}}+x\right )\right ) \log (\log (x))}{x^2 \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {1}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )}\right ) \, dx-\int \left (\frac {e^{25} \log (\log (x))}{x \left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )}+\frac {\log (\log (x))}{x^2 \log \left (\frac {16}{e^{25}}+x\right )}\right ) \, dx\\ &=\log (x)-e^{25} \int \frac {\log (\log (x))}{x \left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx+\int \frac {1}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )} \, dx-\int \frac {\log (\log (x))}{x^2 \log \left (\frac {16}{e^{25}}+x\right )} \, dx\\ &=\log (x)-e^{25} \int \left (\frac {\log (\log (x))}{16 x \log ^2\left (\frac {16}{e^{25}}+x\right )}-\frac {e^{25} \log (\log (x))}{16 \left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )}\right ) \, dx+\int \frac {1}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )} \, dx-\int \frac {\log (\log (x))}{x^2 \log \left (\frac {16}{e^{25}}+x\right )} \, dx\\ &=\log (x)-\frac {1}{16} e^{25} \int \frac {\log (\log (x))}{x \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx+\frac {1}{16} e^{50} \int \frac {\log (\log (x))}{\left (16+e^{25} x\right ) \log ^2\left (\frac {16}{e^{25}}+x\right )} \, dx+\int \frac {1}{x^2 \log (x) \log \left (\frac {16}{e^{25}}+x\right )} \, dx-\int \frac {\log (\log (x))}{x^2 \log \left (\frac {16}{e^{25}}+x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 20, normalized size = 1.00 \begin {gather*} \log (x)+\frac {\log (\log (x))}{x \log \left (\frac {16}{e^{25}}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((16/E^25 + x)*Log[16/E^25 + x] + ((16*x)/E^25 + x^2)*Log[x]*Log[16/E^25 + x]^2 + (-(x*Log[x]) + (-1
6/E^25 - x)*Log[x]*Log[16/E^25 + x])*Log[Log[x]])/(((16*x^2)/E^25 + x^3)*Log[x]*Log[16/E^25 + x]^2),x]

[Out]

Log[x] + Log[Log[x]]/(x*Log[16/E^25 + x])

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fricas [A]  time = 0.65, size = 34, normalized size = 1.70 \begin {gather*} \frac {x \log \left (x + e^{\left (4 \, \log \relax (2) - 25\right )}\right ) \log \relax (x) + \log \left (\log \relax (x)\right )}{x \log \left (x + e^{\left (4 \, \log \relax (2) - 25\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4*log(2)-25)-x)*log(x)*log(exp(4*log(2)-25)+x)-x*log(x))*log(log(x))+(x*exp(4*log(2)-25)+x^2
)*log(x)*log(exp(4*log(2)-25)+x)^2+(exp(4*log(2)-25)+x)*log(exp(4*log(2)-25)+x))/(x^2*exp(4*log(2)-25)+x^3)/lo
g(x)/log(exp(4*log(2)-25)+x)^2,x, algorithm="fricas")

[Out]

(x*log(x + e^(4*log(2) - 25))*log(x) + log(log(x)))/(x*log(x + e^(4*log(2) - 25)))

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giac [A]  time = 0.24, size = 36, normalized size = 1.80 \begin {gather*} \frac {x \log \left (x e^{25} + 16\right ) \log \relax (x) - 25 \, x \log \relax (x) + \log \left (\log \relax (x)\right )}{x \log \left (x e^{25} + 16\right ) - 25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4*log(2)-25)-x)*log(x)*log(exp(4*log(2)-25)+x)-x*log(x))*log(log(x))+(x*exp(4*log(2)-25)+x^2
)*log(x)*log(exp(4*log(2)-25)+x)^2+(exp(4*log(2)-25)+x)*log(exp(4*log(2)-25)+x))/(x^2*exp(4*log(2)-25)+x^3)/lo
g(x)/log(exp(4*log(2)-25)+x)^2,x, algorithm="giac")

[Out]

(x*log(x*e^25 + 16)*log(x) - 25*x*log(x) + log(log(x)))/(x*log(x*e^25 + 16) - 25*x)

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maple [A]  time = 0.28, size = 20, normalized size = 1.00

method	result	size
risch	\(\frac {\ln \left (\ln \relax (x )\right )}{x \ln \left (16 \,{\mathrm e}^{-25}+x \right )}+\ln \relax (x )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(4*ln(2)-25)-x)*ln(x)*ln(exp(4*ln(2)-25)+x)-x*ln(x))*ln(ln(x))+(x*exp(4*ln(2)-25)+x^2)*ln(x)*ln(exp
(4*ln(2)-25)+x)^2+(exp(4*ln(2)-25)+x)*ln(exp(4*ln(2)-25)+x))/(x^2*exp(4*ln(2)-25)+x^3)/ln(x)/ln(exp(4*ln(2)-25
)+x)^2,x,method=_RETURNVERBOSE)

[Out]

1/x/ln(16*exp(-25)+x)*ln(ln(x))+ln(x)

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maxima [A]  time = 0.41, size = 22, normalized size = 1.10 \begin {gather*} \frac {\log \left (\log \relax (x)\right )}{x \log \left (x e^{25} + 16\right ) - 25 \, x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4*log(2)-25)-x)*log(x)*log(exp(4*log(2)-25)+x)-x*log(x))*log(log(x))+(x*exp(4*log(2)-25)+x^2
)*log(x)*log(exp(4*log(2)-25)+x)^2+(exp(4*log(2)-25)+x)*log(exp(4*log(2)-25)+x))/(x^2*exp(4*log(2)-25)+x^3)/lo
g(x)/log(exp(4*log(2)-25)+x)^2,x, algorithm="maxima")

[Out]

log(log(x))/(x*log(x*e^25 + 16) - 25*x) + log(x)

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mupad [B]  time = 4.91, size = 19, normalized size = 0.95 \begin {gather*} \ln \relax (x)+\frac {\ln \left (\ln \relax (x)\right )}{x\,\ln \left (x+16\,{\mathrm {e}}^{-25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + exp(4*log(2) - 25))*(x + exp(4*log(2) - 25)) - log(log(x))*(x*log(x) + log(x + exp(4*log(2) - 25)
)*log(x)*(x + exp(4*log(2) - 25))) + log(x + exp(4*log(2) - 25))^2*log(x)*(x*exp(4*log(2) - 25) + x^2))/(log(x
+ exp(4*log(2) - 25))^2*log(x)*(x^2*exp(4*log(2) - 25) + x^3)),x)

[Out]

log(x) + log(log(x))/(x*log(x + 16*exp(-25)))

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sympy [A]  time = 0.66, size = 17, normalized size = 0.85 \begin {gather*} \log {\relax (x )} + \frac {\log {\left (\log {\relax (x )} \right )}}{x \log {\left (x + \frac {16}{e^{25}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(4*ln(2)-25)-x)*ln(x)*ln(exp(4*ln(2)-25)+x)-x*ln(x))*ln(ln(x))+(x*exp(4*ln(2)-25)+x**2)*ln(x)
*ln(exp(4*ln(2)-25)+x)**2+(exp(4*ln(2)-25)+x)*ln(exp(4*ln(2)-25)+x))/(x**2*exp(4*ln(2)-25)+x**3)/ln(x)/ln(exp(
4*ln(2)-25)+x)**2,x)

[Out]

log(x) + log(log(x))/(x*log(x + 16*exp(-25)))

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