3.89.10 \(\int \frac {-16 x^4 \log ^2(5)+(-16 x^3 \log ^2(5)+(-4 x^2+32 x^4) \log ^2(5) \log (x)) \log (\log (x))+((4 x^2+48 x^3) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)) \log ^2(\log (x))+((4 x+16 x^2) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{6} \log ^2(5) \left (\log (x)+4 x \left (1+\frac {x}{\log (\log (x))}\right )\right )^2 \]

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Rubi [F]  time = 1.27, 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^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{3 x \log (x) \log ^3(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5]^2*Log[x])*Log[Log[x]] + ((4*x^2 + 48*x^3)
*Log[5]^2*Log[x] + 8*x^2*Log[5]^2*Log[x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*Log[5]
^2*Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]

[Out]

(Log[5]^2*(4*x + Log[x])^2)/6 - (16*Log[5]^2*Defer[Int][x^3/(Log[x]*Log[Log[x]]^3), x])/3 - (4*Log[5]^2*Defer[
Int][x/Log[Log[x]]^2, x])/3 + (32*Log[5]^2*Defer[Int][x^3/Log[Log[x]]^2, x])/3 - (16*Log[5]^2*Defer[Int][x^2/(
Log[x]*Log[Log[x]]^2), x])/3 + (4*Log[5]^2*Defer[Int][x/Log[Log[x]], x])/3 + 16*Log[5]^2*Defer[Int][x^2/Log[Lo
g[x]], x] + (8*Log[5]^2*Defer[Int][(x*Log[x])/Log[Log[x]], x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-16 x^4 \log ^2(5)+\left (-16 x^3 \log ^2(5)+\left (-4 x^2+32 x^4\right ) \log ^2(5) \log (x)\right ) \log (\log (x))+\left (\left (4 x^2+48 x^3\right ) \log ^2(5) \log (x)+8 x^2 \log ^2(5) \log ^2(x)\right ) \log ^2(\log (x))+\left (\left (4 x+16 x^2\right ) \log ^2(5) \log (x)+(1+4 x) \log ^2(5) \log ^2(x)\right ) \log ^3(\log (x))}{x \log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{3} \int \frac {\log ^2(5) \left (-16 x^4+4 x^2 \left (-4 x+\left (-1+8 x^2\right ) \log (x)\right ) \log (\log (x))+4 x^2 \log (x) (1+12 x+2 \log (x)) \log ^2(\log (x))+(1+4 x) \log (x) (4 x+\log (x)) \log ^3(\log (x))\right )}{x \log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{3} \log ^2(5) \int \frac {-16 x^4+4 x^2 \left (-4 x+\left (-1+8 x^2\right ) \log (x)\right ) \log (\log (x))+4 x^2 \log (x) (1+12 x+2 \log (x)) \log ^2(\log (x))+(1+4 x) \log (x) (4 x+\log (x)) \log ^3(\log (x))}{x \log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{3} \log ^2(5) \int \left (\frac {(1+4 x) (4 x+\log (x))}{x}-\frac {16 x^3}{\log (x) \log ^3(\log (x))}+\frac {4 x \left (-4 x-\log (x)+8 x^2 \log (x)\right )}{\log (x) \log ^2(\log (x))}+\frac {4 x (1+12 x+2 \log (x))}{\log (\log (x))}\right ) \, dx\\ &=\frac {1}{3} \log ^2(5) \int \frac {(1+4 x) (4 x+\log (x))}{x} \, dx+\frac {1}{3} \left (4 \log ^2(5)\right ) \int \frac {x \left (-4 x-\log (x)+8 x^2 \log (x)\right )}{\log (x) \log ^2(\log (x))} \, dx+\frac {1}{3} \left (4 \log ^2(5)\right ) \int \frac {x (1+12 x+2 \log (x))}{\log (\log (x))} \, dx-\frac {1}{3} \left (16 \log ^2(5)\right ) \int \frac {x^3}{\log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{6} \log ^2(5) (4 x+\log (x))^2+\frac {1}{3} \left (4 \log ^2(5)\right ) \int \left (-\frac {x}{\log ^2(\log (x))}+\frac {8 x^3}{\log ^2(\log (x))}-\frac {4 x^2}{\log (x) \log ^2(\log (x))}\right ) \, dx+\frac {1}{3} \left (4 \log ^2(5)\right ) \int \left (\frac {x}{\log (\log (x))}+\frac {12 x^2}{\log (\log (x))}+\frac {2 x \log (x)}{\log (\log (x))}\right ) \, dx-\frac {1}{3} \left (16 \log ^2(5)\right ) \int \frac {x^3}{\log (x) \log ^3(\log (x))} \, dx\\ &=\frac {1}{6} \log ^2(5) (4 x+\log (x))^2-\frac {1}{3} \left (4 \log ^2(5)\right ) \int \frac {x}{\log ^2(\log (x))} \, dx+\frac {1}{3} \left (4 \log ^2(5)\right ) \int \frac {x}{\log (\log (x))} \, dx+\frac {1}{3} \left (8 \log ^2(5)\right ) \int \frac {x \log (x)}{\log (\log (x))} \, dx-\frac {1}{3} \left (16 \log ^2(5)\right ) \int \frac {x^3}{\log (x) \log ^3(\log (x))} \, dx-\frac {1}{3} \left (16 \log ^2(5)\right ) \int \frac {x^2}{\log (x) \log ^2(\log (x))} \, dx+\frac {1}{3} \left (32 \log ^2(5)\right ) \int \frac {x^3}{\log ^2(\log (x))} \, dx+\left (16 \log ^2(5)\right ) \int \frac {x^2}{\log (\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 31, normalized size = 1.24 \begin {gather*} \frac {\log ^2(5) \left (4 x^2+(4 x+\log (x)) \log (\log (x))\right )^2}{6 \log ^2(\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*x^4*Log[5]^2 + (-16*x^3*Log[5]^2 + (-4*x^2 + 32*x^4)*Log[5]^2*Log[x])*Log[Log[x]] + ((4*x^2 + 4
8*x^3)*Log[5]^2*Log[x] + 8*x^2*Log[5]^2*Log[x]^2)*Log[Log[x]]^2 + ((4*x + 16*x^2)*Log[5]^2*Log[x] + (1 + 4*x)*
Log[5]^2*Log[x]^2)*Log[Log[x]]^3)/(3*x*Log[x]*Log[Log[x]]^3),x]

[Out]

(Log[5]^2*(4*x^2 + (4*x + Log[x])*Log[Log[x]])^2)/(6*Log[Log[x]]^2)

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fricas [B]  time = 0.69, size = 76, normalized size = 3.04 \begin {gather*} \frac {16 \, x^{4} \log \relax (5)^{2} + {\left (16 \, x^{2} \log \relax (5)^{2} + 8 \, x \log \relax (5)^{2} \log \relax (x) + \log \relax (5)^{2} \log \relax (x)^{2}\right )} \log \left (\log \relax (x)\right )^{2} + 8 \, {\left (4 \, x^{3} \log \relax (5)^{2} + x^{2} \log \relax (5)^{2} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{6 \, \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="fricas")

[Out]

1/6*(16*x^4*log(5)^2 + (16*x^2*log(5)^2 + 8*x*log(5)^2*log(x) + log(5)^2*log(x)^2)*log(log(x))^2 + 8*(4*x^3*lo
g(5)^2 + x^2*log(5)^2*log(x))*log(log(x)))/log(log(x))^2

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giac [B]  time = 0.26, size = 71, normalized size = 2.84 \begin {gather*} \frac {8}{3} \, x^{2} \log \relax (5)^{2} + \frac {4}{3} \, x \log \relax (5)^{2} \log \relax (x) + \frac {1}{6} \, \log \relax (5)^{2} \log \relax (x)^{2} + \frac {4 \, {\left (2 \, x^{4} \log \relax (5)^{2} + 4 \, x^{3} \log \relax (5)^{2} \log \left (\log \relax (x)\right ) + x^{2} \log \relax (5)^{2} \log \relax (x) \log \left (\log \relax (x)\right )\right )}}{3 \, \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="giac")

[Out]

8/3*x^2*log(5)^2 + 4/3*x*log(5)^2*log(x) + 1/6*log(5)^2*log(x)^2 + 4/3*(2*x^4*log(5)^2 + 4*x^3*log(5)^2*log(lo
g(x)) + x^2*log(5)^2*log(x)*log(log(x)))/log(log(x))^2

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maple [B]  time = 0.06, size = 62, normalized size = 2.48

method	result	size
risch	\(\frac {\ln \relax (x )^{2} \ln \relax (5)^{2}}{6}+\frac {4 x \ln \relax (5)^{2} \ln \relax (x )}{3}+\frac {8 x^{2} \ln \relax (5)^{2}}{3}+\frac {4 x^{2} \ln \relax (5)^{2} \left (2 x^{2}+4 x \ln \left (\ln \relax (x )\right )+\ln \relax (x ) \ln \left (\ln \relax (x )\right )\right )}{3 \ln \left (\ln \relax (x )\right )^{2}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(((4*x+1)*ln(5)^2*ln(x)^2+(16*x^2+4*x)*ln(5)^2*ln(x))*ln(ln(x))^3+(8*x^2*ln(5)^2*ln(x)^2+(48*x^3+4*x^2
)*ln(5)^2*ln(x))*ln(ln(x))^2+((32*x^4-4*x^2)*ln(5)^2*ln(x)-16*x^3*ln(5)^2)*ln(ln(x))-16*x^4*ln(5)^2)/x/ln(x)/l
n(ln(x))^3,x,method=_RETURNVERBOSE)

[Out]

1/6*ln(x)^2*ln(5)^2+4/3*x*ln(5)^2*ln(x)+8/3*x^2*ln(5)^2+4/3*x^2*ln(5)^2*(2*x^2+4*x*ln(ln(x))+ln(x)*ln(ln(x)))/
ln(ln(x))^2

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maxima [B]  time = 0.50, size = 82, normalized size = 3.28 \begin {gather*} \frac {8}{3} \, x^{2} \log \relax (5)^{2} + \frac {1}{6} \, \log \relax (5)^{2} \log \relax (x)^{2} + \frac {4}{3} \, {\left (x \log \relax (x) - x\right )} \log \relax (5)^{2} + \frac {4}{3} \, x \log \relax (5)^{2} + \frac {4 \, {\left (2 \, x^{4} \log \relax (5)^{2} + {\left (4 \, x^{3} \log \relax (5)^{2} + x^{2} \log \relax (5)^{2} \log \relax (x)\right )} \log \left (\log \relax (x)\right )\right )}}{3 \, \log \left (\log \relax (x)\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*log(5)^2*log(x)^2+(16*x^2+4*x)*log(5)^2*log(x))*log(log(x))^3+(8*x^2*log(5)^2*log(x)^2
+(48*x^3+4*x^2)*log(5)^2*log(x))*log(log(x))^2+((32*x^4-4*x^2)*log(5)^2*log(x)-16*x^3*log(5)^2)*log(log(x))-16
*x^4*log(5)^2)/x/log(x)/log(log(x))^3,x, algorithm="maxima")

[Out]

8/3*x^2*log(5)^2 + 1/6*log(5)^2*log(x)^2 + 4/3*(x*log(x) - x)*log(5)^2 + 4/3*x*log(5)^2 + 4/3*(2*x^4*log(5)^2
+ (4*x^3*log(5)^2 + x^2*log(5)^2*log(x))*log(log(x)))/log(log(x))^2

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mupad [B]  time = 6.03, size = 73, normalized size = 2.92 \begin {gather*} \frac {8\,x^2\,{\ln \relax (5)}^2}{3}+\frac {{\ln \relax (5)}^2\,{\ln \relax (x)}^2}{6}+\frac {4\,x\,{\ln \relax (5)}^2\,\ln \relax (x)}{3}+\frac {16\,x^3\,{\ln \relax (5)}^2}{3\,\ln \left (\ln \relax (x)\right )}+\frac {8\,x^4\,{\ln \relax (5)}^2}{3\,{\ln \left (\ln \relax (x)\right )}^2}+\frac {4\,x^2\,{\ln \relax (5)}^2\,\ln \relax (x)}{3\,\ln \left (\ln \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((16*x^4*log(5)^2)/3 - (log(log(x))^3*(log(5)^2*log(x)*(4*x + 16*x^2) + log(5)^2*log(x)^2*(4*x + 1)))/3 -
(log(log(x))^2*(8*x^2*log(5)^2*log(x)^2 + log(5)^2*log(x)*(4*x^2 + 48*x^3)))/3 + (log(log(x))*(16*x^3*log(5)^
2 + log(5)^2*log(x)*(4*x^2 - 32*x^4)))/3)/(x*log(log(x))^3*log(x)),x)

[Out]

(8*x^2*log(5)^2)/3 + (log(5)^2*log(x)^2)/6 + (4*x*log(5)^2*log(x))/3 + (16*x^3*log(5)^2)/(3*log(log(x))) + (8*
x^4*log(5)^2)/(3*log(log(x))^2) + (4*x^2*log(5)^2*log(x))/(3*log(log(x)))

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sympy [B]  time = 0.34, size = 83, normalized size = 3.32 \begin {gather*} \frac {8 x^{2} \log {\relax (5 )}^{2}}{3} + \frac {4 x \log {\relax (5 )}^{2} \log {\relax (x )}}{3} + \frac {8 x^{4} \log {\relax (5 )}^{2} + \left (16 x^{3} \log {\relax (5 )}^{2} + 4 x^{2} \log {\relax (5 )}^{2} \log {\relax (x )}\right ) \log {\left (\log {\relax (x )} \right )}}{3 \log {\left (\log {\relax (x )} \right )}^{2}} + \frac {\log {\relax (5 )}^{2} \log {\relax (x )}^{2}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(((4*x+1)*ln(5)**2*ln(x)**2+(16*x**2+4*x)*ln(5)**2*ln(x))*ln(ln(x))**3+(8*x**2*ln(5)**2*ln(x)**2
+(48*x**3+4*x**2)*ln(5)**2*ln(x))*ln(ln(x))**2+((32*x**4-4*x**2)*ln(5)**2*ln(x)-16*x**3*ln(5)**2)*ln(ln(x))-16
*x**4*ln(5)**2)/x/ln(x)/ln(ln(x))**3,x)

[Out]

8*x**2*log(5)**2/3 + 4*x*log(5)**2*log(x)/3 + (8*x**4*log(5)**2 + (16*x**3*log(5)**2 + 4*x**2*log(5)**2*log(x)
)*log(log(x)))/(3*log(log(x))**2) + log(5)**2*log(x)**2/6

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