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

Optimal. Leaf size=25 \[ \left (e^{8 e}+\frac {4}{4+x^2+\frac {\log (5)}{x}}\right ) \log (\log (x)) \]

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Rubi [F]  time = 1.12, 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+4 x^4+4 x \log (5)+e^{8 e} \left (16 x^2+8 x^4+x^6+\left (8 x+2 x^3\right ) \log (5)+\log ^2(5)\right )+\left (-8 x^4+4 x \log (5)\right ) \log (x) \log (\log (x))}{\left (16 x^3+8 x^5+x^7+\left (8 x^2+2 x^4\right ) \log (5)+x \log ^2(5)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(4*(1 + E^(8*E))*x + E^(8*E)*x^3 + E^(8*E)*Log[5])/(x*(4*x + x^3 + Log[5])*Log[x]), x] + 4*Log[125]
*Defer[Int][Log[Log[x]]/(4*x + x^3 + Log[5])^2, x] + 32*Defer[Int][(x*Log[Log[x]])/(4*x + x^3 + Log[5])^2, x]
- 8*Defer[Int][Log[Log[x]]/(4*x + x^3 + Log[5]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {\left (4 x+x^3+\log (5)\right ) \left (4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)\right )}{x \log (x)}+4 \left (-2 x^3+\log (5)\right ) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2} \, dx\\ &=\int \left (\frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)}-\frac {4 \left (2 x^3-\log (5)\right ) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {\left (2 x^3-\log (5)\right ) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2} \, dx\right )+\int \frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)} \, dx\\ &=-\left (4 \int \left (\frac {2 \log (\log (x))}{4 x+x^3+\log (5)}-\frac {(8 x+\log (125)) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2}\right ) \, dx\right )+\int \frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)} \, dx\\ &=4 \int \frac {(8 x+\log (125)) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2} \, dx-8 \int \frac {\log (\log (x))}{4 x+x^3+\log (5)} \, dx+\int \frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)} \, dx\\ &=4 \int \left (\frac {8 x \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2}+\frac {\log (125) \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2}\right ) \, dx-8 \int \frac {\log (\log (x))}{4 x+x^3+\log (5)} \, dx+\int \frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)} \, dx\\ &=-\left (8 \int \frac {\log (\log (x))}{4 x+x^3+\log (5)} \, dx\right )+32 \int \frac {x \log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2} \, dx+(4 \log (125)) \int \frac {\log (\log (x))}{\left (4 x+x^3+\log (5)\right )^2} \, dx+\int \frac {4 \left (1+e^{8 e}\right ) x+e^{8 e} x^3+e^{8 e} \log (5)}{x \left (4 x+x^3+\log (5)\right ) \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 24, normalized size = 0.96 \begin {gather*} \left (e^{8 e}+\frac {4 x}{4 x+x^3+\log (5)}\right ) \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(8*E) + (4*x)/(4*x + x^3 + Log[5]))*Log[Log[x]]

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fricas [A]  time = 0.57, size = 39, normalized size = 1.56 \begin {gather*} \frac {{\left ({\left (x^{3} + 4 \, x + \log \relax (5)\right )} e^{\left (2 \, e^{\left (2 \, \log \relax (2) + 1\right )}\right )} + 4 \, x\right )} \log \left (\log \relax (x)\right )}{x^{3} + 4 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5)-8*x^4)*log(x)*log(log(x))+(log(5)^2+(2*x^3+8*x)*log(5)+x^6+8*x^4+16*x^2)*exp(exp(1+2*lo
g(2)))^2+4*x*log(5)+4*x^4+16*x^2)/(x*log(5)^2+(2*x^4+8*x^2)*log(5)+x^7+8*x^5+16*x^3)/log(x),x, algorithm="fric
as")

[Out]

((x^3 + 4*x + log(5))*e^(2*e^(2*log(2) + 1)) + 4*x)*log(log(x))/(x^3 + 4*x + log(5))

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giac [A]  time = 0.17, size = 53, normalized size = 2.12 \begin {gather*} \frac {x^{3} e^{\left (8 \, e\right )} \log \left (\log \relax (x)\right ) + 4 \, x e^{\left (8 \, e\right )} \log \left (\log \relax (x)\right ) + e^{\left (8 \, e\right )} \log \relax (5) \log \left (\log \relax (x)\right ) + 4 \, x \log \left (\log \relax (x)\right )}{x^{3} + 4 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5)-8*x^4)*log(x)*log(log(x))+(log(5)^2+(2*x^3+8*x)*log(5)+x^6+8*x^4+16*x^2)*exp(exp(1+2*lo
g(2)))^2+4*x*log(5)+4*x^4+16*x^2)/(x*log(5)^2+(2*x^4+8*x^2)*log(5)+x^7+8*x^5+16*x^3)/log(x),x, algorithm="giac
")

[Out]

(x^3*e^(8*e)*log(log(x)) + 4*x*e^(8*e)*log(log(x)) + e^(8*e)*log(5)*log(log(x)) + 4*x*log(log(x)))/(x^3 + 4*x
+ log(5))

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maple [A]  time = 0.10, size = 28, normalized size = 1.12

method	result	size
risch	\(\frac {4 x \ln \left (\ln \relax (x )\right )}{x^{3}+\ln \relax (5)+4 x}+{\mathrm e}^{8 \,{\mathrm e}} \ln \left (\ln \relax (x )\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x*ln(5)-8*x^4)*ln(x)*ln(ln(x))+(ln(5)^2+(2*x^3+8*x)*ln(5)+x^6+8*x^4+16*x^2)*exp(exp(1+2*ln(2)))^2+4*x*
ln(5)+4*x^4+16*x^2)/(x*ln(5)^2+(2*x^4+8*x^2)*ln(5)+x^7+8*x^5+16*x^3)/ln(x),x,method=_RETURNVERBOSE)

[Out]

4*x/(x^3+ln(5)+4*x)*ln(ln(x))+exp(8*exp(1))*ln(ln(x))

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maxima [A]  time = 0.47, size = 43, normalized size = 1.72 \begin {gather*} \frac {{\left (x^{3} e^{\left (8 \, e\right )} + 4 \, x {\left (e^{\left (8 \, e\right )} + 1\right )} + e^{\left (8 \, e\right )} \log \relax (5)\right )} \log \left (\log \relax (x)\right )}{x^{3} + 4 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*log(5)-8*x^4)*log(x)*log(log(x))+(log(5)^2+(2*x^3+8*x)*log(5)+x^6+8*x^4+16*x^2)*exp(exp(1+2*lo
g(2)))^2+4*x*log(5)+4*x^4+16*x^2)/(x*log(5)^2+(2*x^4+8*x^2)*log(5)+x^7+8*x^5+16*x^3)/log(x),x, algorithm="maxi
ma")

[Out]

(x^3*e^(8*e) + 4*x*(e^(8*e) + 1) + e^(8*e)*log(5))*log(log(x))/(x^3 + 4*x + log(5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {4\,x\,\ln \relax (5)+{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,\ln \relax (2)+1}}\,\left (\ln \relax (5)\,\left (2\,x^3+8\,x\right )+{\ln \relax (5)}^2+16\,x^2+8\,x^4+x^6\right )+16\,x^2+4\,x^4+\ln \left (\ln \relax (x)\right )\,\ln \relax (x)\,\left (4\,x\,\ln \relax (5)-8\,x^4\right )}{\ln \relax (x)\,\left (\ln \relax (5)\,\left (2\,x^4+8\,x^2\right )+x\,{\ln \relax (5)}^2+16\,x^3+8\,x^5+x^7\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*log(5) + exp(2*exp(2*log(2) + 1))*(log(5)*(8*x + 2*x^3) + log(5)^2 + 16*x^2 + 8*x^4 + x^6) + 16*x^2 +
4*x^4 + log(log(x))*log(x)*(4*x*log(5) - 8*x^4))/(log(x)*(log(5)*(8*x^2 + 2*x^4) + x*log(5)^2 + 16*x^3 + 8*x^
5 + x^7)),x)

[Out]

int((4*x*log(5) + exp(2*exp(2*log(2) + 1))*(log(5)*(8*x + 2*x^3) + log(5)^2 + 16*x^2 + 8*x^4 + x^6) + 16*x^2 +
4*x^4 + log(log(x))*log(x)*(4*x*log(5) - 8*x^4))/(log(x)*(log(5)*(8*x^2 + 2*x^4) + x*log(5)^2 + 16*x^3 + 8*x^
5 + x^7)), x)

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sympy [A]  time = 0.49, size = 29, normalized size = 1.16 \begin {gather*} \frac {4 x \log {\left (\log {\relax (x )} \right )}}{x^{3} + 4 x + \log {\relax (5 )}} + e^{8 e} \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x*ln(5)-8*x**4)*ln(x)*ln(ln(x))+(ln(5)**2+(2*x**3+8*x)*ln(5)+x**6+8*x**4+16*x**2)*exp(exp(1+2*ln
(2)))**2+4*x*ln(5)+4*x**4+16*x**2)/(x*ln(5)**2+(2*x**4+8*x**2)*ln(5)+x**7+8*x**5+16*x**3)/ln(x),x)

[Out]

4*x*log(log(x))/(x**3 + 4*x + log(5)) + exp(8*E)*log(log(x))

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